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Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 359-377
www.serialspublications.com
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
LEONID MYTNIK* AND EYAL NEUMAN*
Abstract. We consider the regularity of sample paths of Volterra processes.
These processes are defined as stochastic integrals
M (t) =
Z
t
F (t, r)dX(r), t ∈ R+ ,
0
where X is a semimartingale and F is a deterministic real-valued function.
We derive the information on the modulus of continuity for these processes
under regularity assumptions on the function F and show that M (t) has
“worst” regularity properties at times of jumps of X(t). We apply our results
to obtain the optimal H¨
older exponent for fractional L´
evy processes.
1. Introduction and Main Results
1.1. Volterra Processes. A Volterra process is a process given by
Z t
M (t) =
F (t, r)dX(r), t ∈ R+ ,
(1.1)
0
where {X(t)}t≥0 is a semimartingale and F (t, r) is a bounded deterministic realvalued function of two variables which sometimes is called a kernel. One of the
questions addressed in the research of Volterra and related processes is studying
their regularity properties. It is also the main goal of this paper. Before we describe our results let us give a short introduction to this area. First, let us note that
one-dimensional fractional processes, which are the close relative of Volterra processes, have been extensively studied in the literature. One-dimensional fractional
processes are usually defined by
Z ∞
X(t) =
F (t, r)dL(r),
(1.2)
−∞
where L(r) is some stochastic process and F (t, r) is some specific kernel. For
example in the case
of L(r) being a two-sided
standard Brownian motion and
H−1/2
H−1/2
1
F (t, r) = Γ(H+1/2) (t − s)+
− (−s)+
, X is called fractional Brownian
motion with Hurst index H (see e.g. Chapter 1.2 of [3] and Chapter 8.2 of [11]). It
is also known that the fractional Brownian motion with Hurst index H is H¨older
Received 2011-11-3; Communicated by F. Viens.
2000 Mathematics Subject Classification. Primary 60G17, 60G22 ; Secondary 60H05.
Key words and phrases. Sample path properties, fractional processes, L´
evy processes.
* This research is partly supported by a grant from the Israel Science Foundation.
359
360
LEONID MYTNIK AND EYAL NUEMAN
continuous with any exponent less than H (see e.g. [8]). Another prominent example is the case of fractional α-stable L´evy process which can be also defined
via (1.2) with L(r) being two-sided α-stable L´evy process and
F (t, r) = a{(t − r)d+ − (−r)d+ } + b{(t − r)d− − (−r)d− }.
Takashima in [15] studied path properties of this process. Takashima set the
following conditions on the parameters: 1 < α < 2, 0 < d < 1 − α−1 and −∞ <
a, b < ∞, |a| + |b| 6= 0. It is proved in [15] that X is a self-similar process. Denote
the jumps of L(t) by ∆L (t): ∆L (t) = L(t) − L(t−), −∞ < t < ∞. It is also proved
in [15] that:
lim(X(t + h) − X(t))h−d = a∆L (t), 0 < t < 1, P − a.s.,
h↓0
lim(X(t) − X(t − h))h−d = −b∆L (t), 0 < t < 1, P − a.s.
h↓0
Note that in his proof Takashima strongly used the self-similarity of the process
X.
Another well-studied process is the so-called fractional L´evy process, which again
is defined via (1.2) for a specific kernel F (t, r) and L(r) being a two-sided L´evy
process. For example, Marquardt in [10] defined it as follows.
Definition 1.1. (Definition 3.1 in [10]): Let L = {L(t)}t∈R be a two-sided L´evy
process on R with E[L(1)] = 0, E[L(1)2 ] < ∞ and without a Brownian component.
Let F (t, r) be the following kernel function:
1
[(t − r)d+ − (−r)d+ ].
F (t, r) =
Γ(d + 1)
For fractional integration parameter 0 < d < 0.5 the stochastic process
Z ∞
Md (t) =
F (t, r)dLr , t ∈ R,
−∞
is called a fractional L´evy process.
As for the regularity properties of fractional L´evy process Md defined above,
Marquardt in [10] used an isometry of Md and the Kolmogorov continuity criterion
in order to prove that the sample paths of Md are P -a.s. local H¨
older continuous
of any order β < d. Moreover she proved that for every modification of Md and
for every β > d:
P ({ω ∈ Ω : Md (·, ω) 6∈ C β [a, b]}) > 0,
where C β [a, b] is the space of H¨
older continuous functions of index β on [a, b]. Note
that in this paper we are going to improve the result of Marquardt and show that
for d ∈ (0, 0.5) the sample paths of Md are P -a.s. H¨
older continuous of any order
β ≤ d.
The regularity properties of the analogous multidimensional processes have been
also studied. For example, consider the process
Z
ˆ (t) =
F (t, r)L(dr), t ∈ RN ,
(1.3)
M
Rm
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
361
where L(dr) is some random measure and F is a real valued function of two
variables. A number of important results have been derived recently by Ayache,
ˆ (t) for some particular
Roueff and Xiao in [1], [2], on the regularity properties of M
choices of F and L. As for the earlier work on the subject we can refer to Kˆono
and Maejima in [5] and [6]. Recently, the regularity of related fractional processes
was studied by Maejima and Shieh in [7]. We should also mention the book of
Samorodnitsky and Taqqu [13] and the work of Marcus and Rosi´
nsky in [9] where
ˆ (t) in (1.3) were also studied.
the regularity properties of processes related to M
1.2. Functions of Smooth Variation. In this section we make our assumptions
on the kernel function F (s, r) in (1.1). First we introduce the following notation.
Denote
∂ n+m f (s, r)
, ∀n, m = 0, 1, . . . .
f (n,m) (s, r) ≡
∂sn ∂rm
We also define the following sets in R2 :
E = {(s, r) : −∞ < r ≤ s < ∞},
˜ = {(s, r) : −∞ < r < s < ∞}.
E
We denote by K a compact set in E, E˜ or R, depending on the context. We define
the following spaces of functions that are essential for the definition of functions
of smooth variation and regular variation.
(k)
Definition 1.2. Let C+ (E) denote the space of functions f from a domain E in
R2 to R1 satisfying
1. f is continuous on E;
˜
2. f has continuous partial derivatives of order k on E.
˜
3. f is strictly positive on E.
Note that functions of smooth variation of one variable have been studied extensively in the literature; [4] is the standard reference for these and related functions.
Here we generalize the definition of functions of smooth variation to functions on
R2 .
(2)
Definition 1.3. Let ρ > 0. Let f ∈ C+ (E) satisfy, for every compact set K ⊂ R,
a)
(0,1)
hf
(t, t − h)
+ ρ = 0,
lim sup h↓0 t∈K
f (t, t − h)
b)
c)
d)
(1,0)
hf
(t + h, t)
lim sup − ρ = 0,
h↓0 t∈K
f (t + h, t)
2 (1,1)
h f
(t, t − h)
+ ρ(ρ − 1) = 0,
lim sup h↓0 t∈K
f (t, t − h)
2 (0,2)
h f
(t, t − h)
lim sup − ρ(ρ − 1) = 0.
h↓0 t∈K
f (t, t − h)
362
LEONID MYTNIK AND EYAL NUEMAN
Then f is called a function of smooth variation of index ρ at the diagonal and is
denoted as f ∈ SRρ2 (0+).
It is easy to check that f ∈ SRρ2 (0+), for ρ > 0 satisfies f (t, t) = 0 for all t. The
trivial example for a function of smooth variation SRρ2 (0+) is f (t, r) = (t − r)ρ .
Another example would be f (t, r) = (t − r)ρ | log(t − r)|η where η ∈ R.
1.3. Main Results.
Convention: From now on we consider a semimartingale {X(t)}t≥0 such that
X(0) = 0 P -a.s. Without loss of generality we assume further that X(0−) = 0,
P -a.s.
In this section we present our main results. The first theorem gives us information
about the regularity of increments of the process M .
Theorem 1.4. Let F (t, r) be a function of smooth variation of index d ∈ (0, 1)
and let {X(t)}t≥0 be a semimartingale. Define
Z t
M (t) =
F (t, r)dX(r), t ≥ 0.
0
Then,
M (s + h) − M (s)
= ∆X (s), ∀s ∈ [0, 1], P − a.s.,
F (s + h, s)
where ∆X (s) = X(s) − X(s−).
lim
h↓0
Information about the regularity of the sample paths of M given in the above
theorem is very precise in the case when the process X is discontinuous. In fact,
it shows that at the point of jump s, the increment of the process behaves like
F (s + h, s)∆X (s).
In the next theorem we give a uniform in time bound on the increments of the
process M .
Theorem 1.5. Let F (t, r) and {M (t)}t≥0 be as in Theorem 1.4. Then
lim
h↓0
|M (t) − M (s)|
= sup |∆X (s)|, P − a.s.
F (t, s)
|t−s|≤h
s∈[0,1]
sup
0<s<t<1,
Our next result, which in fact is a corollary of the previous theorem, improves
the result of Marquardt from [10].
Theorem 1.6. Let d ∈ (0, 0.5). The sample paths of {Md(t)}t≥0 , a fractional
L´evy process, are P -a.s. H¨
older continuous of order d at any point t ∈ R.
In Sections 2,3 we prove Theorems 1.4 , 1.5. In Section 4 we prove Theorem
1.6.
2. Proof of Theorem 1.4
The proof of Theorems 1.4 and 1.5 uses ideas of Takashima in [15], but does
not use the self-similarity assumed there.
The goal of this section is to prove Theorem 1.4. First we prove the integration by
parts formula in Lemma 2.1. Later, in Lemma 2.2, we decompose the increment
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
363
M (t + h) − M (t) into two components and then we analyze the limiting behavior
of each of the components. This allows us to prove Theorem 1.4.
In the following lemma we refer to functions in C(1) (E), which is the space of
˜ It is easy
functions from Definition 1.2, without the condition that f > 0 on E.
to show that functions of smooth variation satisfy the assumptions of this lemma.
Lemma 2.1. Let X be a semimartingale such that X(0) = 0 a.s. Let F (t, r)
be a function in C(1) (E) satisfying F (t, t) = 0 for all t ∈ R. Denote f (t, r) ≡
F (0,1) (t, r). Then,
Z t
Z t
F (t, r)dX(r) = −
f (t, r)X(r)dr, P − a.s.
0
0
Proof. Denote Ft (r) = F (t, r). By Corollary 2 in Section 2.6 of [12] we have
Z t
Z t
X(r−)dFt (r) − [X, Ft ]t .
(2.1)
Ft (r−)dX(r) = X(t)Ft (t) −
0
0
By the hypothesis we get X(t)Ft (t) = 0. Since Ft (·) has continuous derivative
and therefore is of bounded variation, it is easy to check that [X, Ft ]t = 0, P -a.s.
Finally, since X is a semimartingale, it has c`
adl`
ag sample paths (see definition in
Chapter 2.1 of [12]) and we immediately have
Z t
Z t
f (t, r)X(r−)dr =
f (t, r)X(r)dr.
0
0
Convention and Notation
In this section we use the notation F (t, r) for a smoothly varying function of
index d (that is, F ∈ SRd2 (0+)), where d is some number in (0, 1). We denote by
2
(0+) denote
f (t, r) ≡ F (0,1) (t, r), a smooth derivative of index d − 1 and let SDd−1
the set of smooth derivative functions of index d − 1.
In the following lemma we present the decomposition of the increments of the
process Y (t) that will be the key for the proof of Theorem 1.4.
Lemma 2.2. Let
Y (t) =
Z
t
f (t, r)X(r)dr,
t ≥ 0.
0
Then we have
Y (t + δ) − Y (t) = J1 (t, δ) + J2 (t, δ),
where
J1 (t, δ) = δ
Z
∀t ≥ 0, δ > 0,
1
f (t + δ, t + δ − δv)X(t + δ − δv)dv
(2.2)
[f (t + δ, t − δv) − f (t, t − δv)]X(t − δv)dv.
(2.3)
0
and
J2 (t, δ) = δ
Z
0
t/δ
364
LEONID MYTNIK AND EYAL NUEMAN
Proof. For any t ∈ [0, 1], δ > 0 we have
Z t+δ
Z t
Y (t + δ) − Y (t) =
f (t + δ, r)X(r)dr −
f (t, r)X(r)dr
0
Z
=
(2.4)
0
t+δ
f (t + δ, r)X(r)dr +
t
Z
t
[f (t + δ, r) − f (t, r)]X(r)dr.
0
By making a change of variables we are done.
The next propositions are crucial for analyzing the behavior of J1 and J2 from
the above lemma.
2
Proposition 2.3. Let f (t, r) ∈ SDd−1
(0+) where d ∈ (0, 1). Let X(r) be a
semimartingale. Denote
gδ (t, v) =
f (t + δ, t − δv) − f (t, t − δv)
, t ∈ [0, 1], v ≥ 0, δ > 0.
f (t + δ, t)
Then
Z
lim δ↓0
0
t/δ
1
gδ (t, v)X(t − δv)dv + X(t−) = 0, ∀t ∈ [0, 1], P − a.s.
d
2
Proposition 2.4. Let f (t, r) ∈ SDd−1
(0+) where d ∈ (0, 1). Let X(r) be a
semimartingale. Denote
fδ (t, v) =
f (t + δ, t + δ − δv)
, t ∈ [0, 1], v ≥ 0, δ > 0.
f (t + δ, t)
Then
Z
lim δ↓0
1
0
1
fδ (t, v)X(t + δ(1 − v))dv − X(t) = 0, ∀t ∈ [0, 1], P − a.s.
d
We first give a proof of Theorem 1.4 based on the above propositions and then
get back to the proofs of the propositions.
Proof of Theorem 1.4: From Lemma 2.1 we have
M (t + δ) − M (t) = −(Y (t + δ) − Y (t)), P − a.s.
where
Y (t) =
Z
(2.5)
t
f (t, r)X(r)dr.
0
By Lemma 2.2, for every t ≥ 0, δ > 0, we have
Y (t + δ) − Y (t) = J1 (t, δ) + J2 (t, δ),
(2.6)
For the first integral we get
J1 (t, δ)
=
δf (t + δ, t)
Z
1
fδ (t, v)X(t + δ(1 − v))dv.
0
Now we apply Proposition 2.4 to get
lim
δ↓0
1
J1 (t, δ)
= X(t), ∀t ∈ [0, 1], P − a.s.
δf (t + δ, t)
d
(2.7)
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
365
For the second integral we get
J2 (t, δ)
=
δf (t + δ, t)
Z
t/δ
gδ (t, v)X(t − δv)dv.
(2.8)
0
By Proposition 2.3 we get
lim
δ↓0
J2 (t, δ)
1
= − X(t−), ∀t ∈ [0, 1], P − a.s.
δf (t + δ, t)
d
(2.9)
Combining (2.7) and (2.9) with (2.6) we get
lim
δ↓0
Y (t + δ) − Y (t)
1
= ∆X (t), ∀t ∈ [0, 1], P − a.s.
δf (t + δ, t)
d
(2.10)
Recall that F (0,1) (t, r) = f (t, r) where by our assumptions F ∈ SRd2 (0+). It is
trivial to verify that
F (t, t − h)
1 = 0.
(2.11)
+
lim sup h↓0
hf (t, t − h) d t∈[0,1]
Then by (2.5), (2.10) and (2.11) we get
lim
δ↓0
M (t + δ) − M (t)
= ∆X (t), ∀t ∈ [0, 1], P − a.s.
F (t + δ, t)
(2.12)
Now we are going to prove Propositions 2.3 and 2.4. First let us state a few
properties of SRρ2 (0+) functions. These properties are simple extensions of some
properties of smoothly varying functions (see Chapter 1 of [4]).
Lemma 2.5. Let f be a SRd2 (0+) function for some d ∈ (0, 1). Then f ∈ Rd2 (0+),
2
f (0,1) ∈ Rd−1
(0+), and
a)
f (t, t − hv)
d
lim sup − v = 0, uniformly on v ∈ (0, a],
h↓0 t∈[0,1] f (t, t − h)
b)
c)
d)
f (t + hv, t)
− v d = 0, uniformly on v ∈ (0, a],
lim sup h↓0 t∈[0,1] f (t + h, t)
(0,1)
f
(t, t − hv)
d−1 lim sup (0,1)
−v
= 0, uniformly on v ∈ [a, ∞),
h↓0 t∈[0,1] f
(t, t − h)
(0,1)
f
(t + hv, t)
− v d−1 = 0, uniformly on v ∈ [a, ∞),
lim sup (0,1)
h↓0 t∈[0,1] f
(t + h, t)
for any a, b such that a ∈ (0, ∞).
Next, we state two lemmas which are dealing with the properties of functions fδ
and gδ . We omit the proofs as they are pretty much straightforward consequences
of Lemma 2.5 and properties of smoothly varying functions.
366
LEONID MYTNIK AND EYAL NUEMAN
2
Lemma 2.6. Let f (t, r) ∈ SDd−1
(0+) where d ∈ (0, 1). Let gδ (t, v) be defined as
in Proposition 2.3. Then for every h0 ∈ (0, 1]
(a)
t/δ
Z
lim sup
δ↓0 h0 ≤t≤1
|gδ (t, v)|dv = 0;
h0 /δ
(b)
h0 /δ
1 gδ (t, v)dv + = 0;
d
t/δ
1 gδ (t, v)dv + = 0;
d
(c)
Z
lim sup δ↓0 0≤t≤1
(d)
Z
lim sup δ↓0 h ≤t≤1
0
0
0
h0 /δ
Z
lim sup δ↓0 h ≤t≤1
0
Lemma 2.7. Let f (t, r) ∈
in Proposition 2.4. Then,
0
2
SDd−1
(0+)
1
|gδ (t, v)|dv − = 0.
d
where d ∈ (0, 1). Let fδ (t, v) be defined as
(a)
1
Z
lim sup δ↓0 0≤t≤1
0
(b)
Z
lim sup δ↓0 0≤t≤1
1 |fδ (t, v)|dv − = 0;
d
1
0
1
fδ (t, v)dv − = 0.
d
Now we will use Lemmas 2.6, 2.7 to prove Propositions 2.3, 2.4. At this point
we also need to introduce the notation for the supremum norm on c`adl`
ag functions
on [0, 1]:
kf k∞ = sup |f (t)|, f ∈ DR [0, 1],
0≤t≤1
where DR [0, 1] is the class of real valued c`
adl`
ag functions on [0, 1]. Since X is a
c`
adl`
ag process we have
kXk∞ < ∞, P − a.s.
(2.13)
Note that for every I ⊂ R, DR (I) will denotes the class of real-valued c`
adl`
ag
functions on I.
Proof of Proposition 2.3: Let us consider the following decomposition
Z t/δ
Z t/δ
1
gδ (t, v)[X(t − δv) − X(t−)]dv
gδ (t, v)X(t − δv)dv + X(t−) =
d
0
0
Z t/δ
1
+ X(t−)
gδ (t, v)dv +
d
0
=: J1 (δ, t) + J2 (δ, t).
(2.14)
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
367
By (2.13) and Lemma 2.6(c) we immediately get that for any arbitrarily small
h0 > 0, we have
lim sup |J2 (δ, t)| = 0, P − a.s.
δ↓0 h0 ≤t≤1
Since h0 was arbitrary and X(0−) = X(0) = 0, we get
lim |J2 (δ, t)| = 0, ∀t ∈ [0, 1], P − a.s.
δ↓0
Now to finish the proof it is enough to show that, P − a.s., for every t ∈ [0, 1]
lim |J1 (δ, t)| = 0.
(2.15)
δ↓0
For any h0 ∈ [0, t] we can decompose J1 as follows
J1 (δ, t)
Z h0 /δ
Z
=
gδ (t, v)[X(t − δv) − X(t−)]dv +
0
t/δ
gδ (t, v)[X(t − δv) − X(t−)]dv
h0 /δ
=: J1,1 (δ, t) + J1,2 (δ, t).
(2.16)
Let ε > 0 be arbitrarily small. X is a c`
adl`
ag process therefore, P − a.s. ω, for
every t ∈ [0, 1] we can fix h0 ∈ [0, t] small enough such that
|X(t − δv, ω) − X(t−, ω)| < ε, for all v ∈ (0, h0 /δ].
(2.17)
Let us choose such h0 for the decomposition (2.16). Then by (2.17) and Lemma
2.6(d) we can pick δ ′ > 0 such that for every δ ∈ (0, δ ′ ) we have
2ε
.
(2.18)
|J1,1 (δ, t)| ≤
d
Now let us treat J1,2 . By (2.13) and Lemma 2.6(a) we get
Z t/δ
|J1,2 (δ, t)| ≤ 2kXk∞
|gδ (t, v)|dv
(2.19)
h0 /δ
→ 0, as δ ↓ 0, P − a.s.
Then by combining (2.18) and (2.19), we get (2.15) and this completes the proof.
Proof of Proposition 2.4: We consider the following decomposition
Z 1
1
fδ (t, v)X(t + δ(1 − v))dv − X(t)
d
0
Z 1
=
fδ (t, v)[X(t + δ(1 − v)) − X(t)]dv
0
Z 1
1
+ X(t)
fδ (t, v)dv −
d
0
=: J1 (δ, t) + J2 (δ, t).
Now the proof follows along the same lines as that of Proposition 2.3. By (2.13)
and Lemma 2.7(b) we have
lim sup |J2 (δ, t)| = 0, P − a.s.
δ↓0 0≤t≤1
368
LEONID MYTNIK AND EYAL NUEMAN
Hence to complete the proof it is enough to show that, P − a.s., for every t ∈ [0, 1]
lim |J1 (δ, t)| = 0.
δ↓0
Let ε > 0 be arbitrarily small. X is a c`
adl`
ag process; therefore, P − a.s. ω, for
every t ∈ [0, 1] we can fix h0 small enough such that
|X(t + δ(1 − v), ω) − X(t, ω)| < ε, for all v ∈ (0, h0 /δ].
(2.20)
Then by (2.20) and Lemma 2.7(a) we easily get
lim |J1 (δ, t)| = 0. ∀t ∈ [0, 1], P − a.s.
δ↓0
3. Proof of Theorem 1.5
Recall that by Lemma 2.1 we have
M (t) − M (s) = −(Y (t) − Y (s)), 0 ≤ s < t,
where
Y (s) =
Z
(3.1)
s
f (s, r)X(r)dr, s > 0.
0
Then by Lemma 2.2 we get:
Y (s + δ) − Y (s)
δf (s + δ, s)
J1 (s, δ)
J2 (s, δ)
+
, δ > 0.
δf (s + δ, s) δf (s + δ, s)
=
(3.2)
Recall that J1 and J2 are defined in (2.2) and (2.3).
Convention: Denote by Γ ⊂ Ω the set of paths of X(·, ω) which are right continuous and have left limit. By the assumptions of the theorem, P (Γ) = 1. In what
follows we are dealing with ω ∈ Γ. Therefore, for every ε > 0 and t > 0 there
exists η = η(ε, t, ω) > 0 such that:
|X(t−) − X(s)| ≤
ε,
for all s ∈ [t − η, t),
|X(t) − X(s)| ≤
ε,
for all s ∈ [t, t + η].
(3.3)
Let us fix an arbitrary ε > 0. The interval [0, 1] is compact; therefore there exist
points t1 , . . . , tm that define a cover of [0, 1] as follows:
m
[
ηk ηk
,
[0, 1] ⊂
tk − , tk +
2
2
k=1
where we denote ηk = η(ε, tk ). Note that if ∆X (s) > 2ε then s = tk for some k.
We can also construct this cover in a way that
ηk inf
tk −
≥ t1 .
(3.4)
2
k∈{2,...,m}
Also since X(t) is right continuous at 0, we can choose t1 sufficiently small such
that
|X(t)| ≤ ε.
(3.5)
sup
t∈(0,t1 +
Denote:
η1
2
)
ηk
ηk Bk = tk − ηk , tk + ηk , Bk∗ = tk − , tk +
.
2
2
(3.6)
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
369
Note that the coverings Bk and Bk∗ we built above are random—they depend on a
particular realization of X. For the rest of this section we will be working with the
particular realization of X(·, ω), with ω ∈ Γ and the corresponding coverings Bk ,
Bk∗ . All the constants that appear below may depend on ω and the inequalities
should be understood P -a.s.
Let s, t ∈ Bk∗ and denote δ = t − s. Recall the notation from Propositions 2.3 and
i (s,δ)
, i = 1, 2, as follows:
2.4. Let us decompose δfJ(s+δ,s)
J1 (s, δ) + J2 (s, δ)
δf (s + δ, s)
Z 1
Z
= X(tk −)
fδ (s, v)dv +
0
0
Z
1
s/δ
+
Z
1
+
Z
Z
s/δ
0
1
fδ (s, v)1{s+δ(1−v)≥tk } dv +
Z
0
s/δ
gδ (s, v)1{s−δv≥tk } dv
gδ (s, v)1{s−δv<tk } [X(s − δv) − X(tk −)]dv
0
+
fδ (s, v)1{s+δ(1−v)<tk } [X(s + δ(1 − v)) − X(tk −)]dv
0
0
gδ (s, v)dv
0
Z
+ ∆X (tk )
+
s/δ
fδ (s, v)1{s+δ(1−v)≥tk } [X(s + δ(1 − v)) − X(tk +)]dv
gδ (s, v)1{s−δv≥tk } [X(s − δv) − X(tk +)]dv
=: D1 (k, s, δ) + D2 (k, s, δ) + . . . + D6 (k, s, δ),
where 1 is the indicator function. The proof of Theorem 1.5 will follow as we
handle the terms Di , i = 1, 2, . . . , 6 via a series of lemmas.
Lemma 3.1. There exists a sufficiently small h3.1 > 0 such that
4
ε, ∀k ∈ {1, . . . , m}, s ∈ Bk∗ , δ ∈ (0, h3.1 ). (3.7)
|D1 (k, s, δ)| ≤ |X(tk −)| +
d
Proof. By Lemma 2.6(c), Lemma 2.7(b) and by our assumptions on the covering
we get (3.7) for k = 2, . . . , m. As for k = 1, we get by (3.5)
|X(t1 −)| ≤ ε.
(3.8)
By Lemma 2.6(b) we have
Z
sup 0≤s≤1
0
(t1 +η1 )/δ
1 gδ (s, v)dv + < ε/2.
d
(3.9)
370
LEONID MYTNIK AND EYAL NUEMAN
Hence by Lemma 2.6(c), (3.8) and (3.9), for a sufficiently small δ, we get
Z (t1 +η1 )/δ
Z 1
sup |D1 (1, s, δ)| ≤ ε sup fδ (s, v)dv +
|gδ (s, v)|dv
s∈B1∗
s∈B1∗
≤
0
0
4
ε ,
d
and (3.7) follows.
To handle the D2 term we need the following lemma.
Lemma 3.2. Let gδ (s, v) and fδ (s, v) be defined as in Propositions 2.3 and 2.4.
Then there exists h3.2 > 0 such that for all δ ∈ (0, h3.2 ),
Z 1
Z s/δ
≤ 1 + ε,
f
(s,
v)1
dv
+
g
(s,
v)1
dv
δ
δ
{s+δ(1−v)≥tk }
{s−δv≥tk }
d
0
0
∀k ≥ 1, s ∈ [0, 1].
Proof. We introduce the following notation
Z 1
I1 (s, δ) =
fδ (s, v)1{s+δ(1−v)≥tk } dv,
0
I2 (s, δ) =
Z
s/δ
0
gδ (s, v)1{s−δv≥tk } dv.
From Definition 1.3, it follows that there exists h1 > 0, such that for every δ ∈ h1 ,
v ∈ (0, h2δ1 ) and s ∈ [0, 1]
gδ (s, v) ≤
0,
(3.10)
≥ 0.
(3.11)
and
fδ (s, v)
By Lemma 2.6(a), we can fix a sufficiently small h2 ∈ (0, h1 /2) such that for every
δ ∈ (0, h2 ), we have
Z s/δ
|gδ (s, v)|dv ≤ ε/2, ∀s ∈ [h1 /2, 1],
(3.12)
h1 /(2δ)
where ε was fixed for building the covering {Bk∗ }m
k=1 . Then, by (3.12), we have
Z (s∧ h21 )δ
gδ (s, v)1{v≤ s−tk } dv + ε/2, (3.13)
|I1 (s, δ) + I2 (s, δ)| ≤ I1 +
δ
0
for all s ∈ [0, 1], δ ∈ (0, h2 ).
By (3.11) and the choice of h2 ∈ (0, h21 , we get
I1 (s, δ) ≥
By (3.10) we have
Z (s∧ h21 )δ
0
0, ∀s ∈ [0, 1], δ ∈ (0, h2 ).
gδ (s, v)1{v≤ s−tk } dv
δ
≤
0, ∀s ∈ [0, 1], δ ∈ (0, h2 ).
(3.14)
(3.15)
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
Then by (3.13), (3.14) and (3.15) we get
Z
Z 1
|I1 (s, δ) + I2 (s, δ)| ≤ max
fδ (s, v)dv, (s∧
h1
2
)/δ
0
0
371
gδ (s, v)dv + ε/2
(3.16)
for all s ∈ [0, 1], δ ∈ (0, h2 ).
By (3.16), Lemma 2.7(b) and Lemma 2.6(d) we can fix h3.2 sufficiently small such
that
1
|I1 (s, δ) + I2 (s, δ)| ≤ + ε
d
and we are done.
Note that
Z
|D2 (k, s, δ)| = ∆X (tk )
0
1
fδ (s, v)1{s+δ(1−v)≥tk } dv +
Z
s/δ
0
gδ (s, v)1{s−δv≥tk } dv
(3.17)
Then the immediate corollary of Lemma 3.2 and (3.17) is
Corollary 3.3.
1
, ∀k ∈ {1, . . . , m}, s ∈ [0, 1], δ ∈ (0, h3.2 ).
d
One can easily deduce the next corollary of Lemma 3.2.
|D2 (k, s, δ)| ≤ |∆X (tk )| ε +
Corollary 3.4. There exists h3.4 > 0 such that
sup |D2 (k, s, δ)| − |∆X (tk )| 1 ≤ ε|∆X (tk )|, ∀k ∈ {1, . . . , m}, δ ∈ (0, h3.4 ).
∗
d
s∈Bk
(3.18)
Proof. By Corollary 3.3 we have
1
sup |D2 (k, s, δ)| ≤ |∆X (tk )| + |∆X (tk )|ε, ∀k ∈ {1, . . . , m},
d
s∈Bk∗
s ∈ [0, 1], δ ∈ (0, h3.2 ).
Bk∗
To get (3.18) it is enough to find s ∈
and h3.4 ∈ (0, h3.2 ) such that for all
δ ∈ (0, h3.4 ).
1
|D2 (k, s, δ)| ≥ |∆X (tk )| − |∆X (tk )|ε, ∀k ∈ {1, . . . , m}.
(3.19)
d
By picking s = tk we get
Z 1
|D2 (k, tk , δ)| = ∆X (tk )
fδ (tk , v)dv .
0
Then by Lemma 2.7(b), (3.19) follows and we are done.
The term |D3 (k, s, δ)| + |D5 (k, s, δ)| is bounded by the following lemma.
Lemma 3.5. There exists a sufficiently small h3.5 such that
4
|D3 (k, tk , δ)| + |D5 (k, tk , δ)| ≤ ε · , ∀k ∈ {1, . . . , m}, s ∈ Bk∗ , ∀δ ∈ (0, h3.5 ).
d
372
LEONID MYTNIK AND EYAL NUEMAN
Proof. By the construction of Bk we get that
Z 1
|fδ (s, v)|1{s+δ(1−v)<tk } dv, ∀s ∈ Bk∗ ,
|D3 (k, s, δ)| ≤ ε
0
k = {1, . . . , m}, δ ∈ (0, ηk /2),
(3.20)
and
|D5 (k, s, δ)| ≤ ε
Z
1
0
|fδ (s, v)|1{s+δ(1−v)≥tk } dv, ∀s ∈ Bk∗ ,
k = {1, . . . , m}, δ ∈ (0, ηk /2).
(3.21)
From (3.20), (3.21) we get
|D3 (k, s, δ)| + |D5 (k, s, δ)| ≤ 2ε
Z
1
|fδ (s, v)|dv, ∀ k = {1, . . . , m},
0
s ∈ Bk∗ , δ ∈ (0, η/2).
By Lemma 2.7(a) the result follows.
Next we show that |D4 (k, s, δ)| is bounded in the following lemma.
Lemma 3.6. There exists h3.6 > 0 such that for all δ ∈ (0, h3.6 ):
2
|D4 (k, s, δ)| ≤ ε + 2kXk∞ , ∀s ∈ Bk∗ , k ∈ {1, . . . , m}.
d
(3.22)
Proof. Let ε > 0 be arbitrarily small and fix k ∈ {1, . . . , m}. First we consider the
case s − tk > 0, k ∈ {1, . . . , m}.
Z s/δ
gδ (s, v)1{s−δv<tk } [X(s − δv) − X(tk −)]dv (3.23)
0
≤
Z
s−tk
δ
η
k
+ 2δ
|gδ (s, v)||X(s − δv) − X(tk −)|dv
(s−tk )/δ
Z
+ s/δ
s−tk
δ
η
k
+ 2δ
1{tk >ηk /2} gδ (s, v)[X(s − δv) − X(tk −)]dv := |I1 (k, s, δ)| + |I2 (k, s, δ)|.
Note that the indicator in I2 (k, s, δ) makes sure that s/δ > (s − tk )/δ + ηk /(2δ).
By the definition of Bk in (3.6) and by Lemma 2.6(d)Tthere exists h1 > 0 such
that for every δ ∈ (0, h1 ) we have uniformly on s ∈ Bk∗ [tk , 1]
2
ε.
(3.24)
d
By Lemma 2.6(a), there
T exists h2 ∈ (0, h1 ) such that for every δ ∈ (0, h2 ) we have
uniformly on s ∈ Bk∗ [tk , 1] (note that if tk ≤ ηk /2 then I2 (k, s, δ) = 0)
|I1 (k, s, δ)| ≤
|I2 (k, s, δ)| ≤
By (3.23), (3.24) and (3.25) we get (3.22).
2kXk∞ε.
(3.25)
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
373
Consider the case s ≤ tk , s ∈ Bk∗ , k ∈ {1, . . . , m}. Then we have
Z s/δ
gδ (s, v)1{s−δv<tk } [X(s − δv) − X(tk −)]dv
0
=
Z
ηk /(2δ)
Z
s/δ
gδ (s, v)[X(s − δv) − X(tk −)]dv
0
=
gδ (s, v)[X(s − δv) − X(tk −)]dv
ηk /(2δ)
=: J1 (k, s, δ) + J2 (k, s, δ).
(3.26)
Note that if v ∈ (0, ηk /(2δ)), s ≤ tk and s ∈ Bk∗ , then s − δv ∈ (tk − ηk , tk ). Hence,
by the construction of Bk we have
sup
v∈(0,ηk /(2δ))
|X(s − δv) − X(tk −)| ≤ ε, ∀s ≤ tk , s ∈ Bk∗ .
(3.27)
By (3.27) and Lemma 2.6(d), there exists h4 ∈ (0, h3 ) such that
2
≤ ε , ∀δ ∈ (0, h4 ), s ≤ tk , s ∈ Bk∗ , k ∈ {1, . . . , m}.(3.28)
d
|J1 (k, s, δ)|
By Lemma 2.6(a), exists h3.6 ∈ (0, h4 ) such that
|J2 (k, s, δ)|
≤ 2kXk∞ε, ∀δ ∈ (0, h3.6 ), s ≤ tk , s ∈ Bk∗ , k ∈ {1, . . . , m}.
(3.29)
By combining (3.28) and (3.29) with (3.26), the result follows.
|D6 (k, s, δ)| is bounded in the following lemma.
Lemma 3.7. There exists h3.7 > 0 such that for all δ ∈ (0, h3.7 ):
|D6 (k, s, δ)| ≤
2ε
, ∀s ∈ Bk∗ , k ∈ {1, . . . , m}.
d
Proof. Recall that
Bk∗ = tk −
Note that
ηk
ηk , k ∈ {1, . . . , m}.
, tk +
2
2
|D6 (k, s, δ)| = 0, ∀s ∈ tk −
ηk , tk , k ∈ {1, . . . , m}.
2
(3.30)
Hence we handle only the case of s > tk , s ∈ Bk∗ . One can easily see that in this
case
Z (s−tk )/δ
D6 (k, s, δ) =
gδ (s, v)[X(s − δv) − X(tk +)]dv.
0
Then by the construction of Bk∗ , for every s ∈ Bk∗ , s > tk we have
|X(s − δv) − X(tk )| ≤ ε, for all v ∈ (0, (s − tk )/δ].
(3.31)
374
LEONID MYTNIK AND EYAL NUEMAN
We notice that if s ∈ Bk∗ and s > tk then 0 < s − tk < ηk /2, for all k = 1, . . . , m.
Denote by η = maxk=1,...,m ηk . Then by (3.31) and Lemma 2.6(d) we can pick
h3.7 > 0 such that for every δ ∈ (0, h3.7 ) we have
|D6 (k, s, δ)|
≤
ηk 2ε
, ∀s ∈ tk , tk +
, k ∈ {1, . . . , m}.
d
2
(3.32)
Then by (3.30) and (3.32) for all δ ∈ (0, h3.7 ), the result follows.
Now we are ready to complete the proof of Theorem 1.5. By Lemmas 3.1, 3.5,
3.6, 3.7 and by Corollary 3.4, there exists h∗ small enough and C3.33 = 6kXk∞ + 12
d
such that
sup |J1 (s, δ) + J2 (s, δ)| − 1 |∆X (tk )| ≤ ε · C3.33 , ∀k ∈ {1, . . . , m},
s∈B ∗
|δf (s + δ, s)|
d
k
δ ∈ (0, h∗ ), P − a.s.
(3.33)
Recall that F (0,1) (s, r) = f (s, r) where F (s, r) ∈ SRd2 (0+) is a positive function.
Then, by (2.11), we can choose h ∈ (0, h∗ ) to be small enough such that
sup |J1 (s, δ) + J2 (s, δ)| − |∆X (tk )| ≤ ε · C3.34 , ∀k ∈ {1, . . . , m}, δ ∈ (0, h),
s∈B ∗
F (s + δ, s)
k
(3.34)
where C3.34 = 2C3.33 + 1. By (3.34) and Lemma 2.2 we get,
|Y (t) − Y (s)|
sup
− |∆X (tk )| ≤ ε · C3.34 , ∀k ∈ {1, . . . , m}, δ ∈ (0, h).
∗
F (t, s)
|t−s|≤δ, s∈Bk
From Lemma 2.1 we have
|M (t) − M (s)|
sup
− |∆X (tk )| ≤ ε · C3.34 , ∀k ∈ {1, . . . , m},
∗
F (t, s)
|t−s|≤δ, s∈Bk
δ ∈ (0, h), P − a.s. (3.35)
By the construction of the covering Bk , for any point s 6∈ {t1 , . . . , tk }, |∆X (s)| ≤
2ε. Set C3.36 = C3.34 + 2. Then, by (3.35) we get
|M (t) − M (s)|
− sup |∆X (s)| ≤ ε · C3.36 ,
(3.36)
sup
F (t, s)
s∈[0,1]
0<s<t<1, |t−s|≤δ
∀k ∈ {1, . . . , m},
δ ∈ (0, h), P − a.s.
Since C3.36 is independent of m, and since ε was arbitrarily small the result follows.
4. Proof of Theorem 1.6
In this section we prove Theorem 1.6. In order to prove Theorem 1.6 we need
the following lemma.
SAMPLE PATH PROPERTIES OF VOLTERRA PROCESSES
375
Lemma 4.1. Let L(t) be a two-sided L´evy process with E[L(1)] = 0, E[L(1)2 ] < ∞
and without a Brownian component. Then for P -a.e. ω, for any t ∈ R, a ≤ 0
′
such that t > a there exists δ ∈ (0, t − a) such that
Z a
′
d−1
d−1
[(t + δ − r)
− (t − r)
]L(r)dr ≤ C · |δ|, ∀|δ| ≤ δ ,
(4.1)
−∞
′
where C is a constant that may depend on ω, t, δ .
′
Proof. Fix an arbitrary t ∈ R and pick δ ∈ (0, t − a). For all |δ| ≤ δ
Z a
[(t + δ − r)d−1 − (t − r)d−1 ]L(r)dr
′
−∞
=
N
Z
[(t + δ + u)d−1 − (t + u)d−1 ]L2 (u)du
−a
+
Z
∞
[(t + δ + u)d−1 − (t + u)d−1 ]L2 (u)du
N
=: I2,1 (N, δ) + I2,2 (N, δ).
(4.2)
Now we use the result on the long time behavior of L´evy processes. By Proposition
48.9 from [14], if E(L2 (1)) = 0 and E(L2 (1)2 ) < ∞, then
lim sup
s→∞
L2 (s)
= (E[L2 (1)2 ])1/2 , P − a.s.
(2s log log(s))1/2
(4.3)
Recall that d ≤ 0.5. Hence by (4.3) we can pick N = N (ω) > 0 large enough such
that
Z ∞
|I2,2 (N, δ)| ≤
|(t + δ + u)d−1 − (t + u)d−1 |u1/2+ε du
N
≤
′
′
C · |δ|, ∀δ ∈ (−δ , δ ), P − a.s.
(4.4)
On the other hand, for δ small enough
Z N
d−1
d−1
|I2,1 (N, δ)| = [(t + δ + u)
− (t + u) ]L2 (u)du
(4.5)
−a
≤
′
′
C||L2 (u)||[0,N ] |δ| · (t − a)d−1 , ∀δ ∈ (−δ , δ ),
where
||L2 (u)||[0,N ] = sup |L2 (u)|.
u∈[0,N ]
Then, by (4.4) and (4.5) we get for d < 1/2
|I2,1 (N, δ) + I2,2 (N, δ)|
′
′
< C|δ|, ∀δ ∈ (−δ , δ ),
and by combining (4.2) with (4.6) the result follows.
(4.6)
376
LEONID MYTNIK AND EYAL NUEMAN
Proof of Theorem 1.6. By Theorem 3.4 in [10] we have
Z ∞
1
Md (t) =
[(t − r)d−1
− (−r)d−1
+
+ ]L(r)dr, t ∈ R, P − a.s.
Γ(d) −∞
(4.7)
We prove the theorem for the case of t > 0. The proof for the case of t ≤ 0 can
be easily adjusted along the similar lines. We can decompose Md (t) as follows:
Z t
Z 0
1
1
(t − r)d−1 L(r)dr +
[(t − r)d−1 − (−r)d−1 ]L(r)dr
Md (t) =
Γ(d) 0
Γ(d) −∞
= Md1 (t) + Md2 (t), t ∈ (0, 1), P − a.s.
By Lemma 2.1 we have
Md1 (t) =
1
Γ(d + 1)
Z
t
(t − r)d dLr , t ∈ R+ , P − a.s.
0
By Theorem 1.5 we have
lim
h↓0
sup
0<s<t<1, |t−s|≤h
Γ(d + 1)
|Md1 (t) − Md1 (s)|
= sup |∆X (s)|, P − a.s.
hd
s∈[0,1]
Therefore, P -a.s. ω, for any t ∈ (0, 1), there exists δ1 > 0 and C1 > 0 such that
|Md1 (t + δ) − Md1 (t)| ≤ C1 |δ|d , ∀δ ∈ (−δ1 , δ1 ).
(4.8)
By Lemma 4.1, P -a.s. ω, for any t ∈ (0, 1), there exists δ2 > 0 and C2 = C2 (ω, t) >
0 such that
|Md2 (t + δ) − Md2 (t)| ≤ C2 |δ|, ∀δ ∈ (−δ2 , δ2 ).
(4.9)
Hence by (4.8) and (4.9), P -a.s. ω, for any t ∈ (0, 1), we can fix δ3 and C = C(ω, t)
such that,
|Md (t + δ) − Md (t)| ≤ C|δ|d , ∀δ ∈ (−δ3 , δ3 ),
and we are done.
Acknowledgment. Both authors thank an anonymous referee for the careful
reading of the manuscript, and for a number of useful comments and suggestions
that improved the exposition.
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Leonid Mytnik: Faculty of Industrial Engineering and Management, Technion Institute of Technology, Haifa, 3200, Israel
E-mail address: [email protected]
Eyal Neuman: Faculty of Industrial Engineering and Management, Technion - Institute of Technology, Haifa, 3200, Israel
E-mail address: [email protected]
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 379-402
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STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
AND APPLICATION TO HITTING TIMES
PEDRO LEI AND DAVID NUALART*
Abstract. In this paper we establish a change-of-variable formula for a class
of Gaussian processes with a covariance function satisfying minimal regularity
and integrability conditions. The existence of the local time and a version of
Tanaka’s formula are derived. These results are applied to a general class of
self-similar processes that includes the bifractional Brownian motion. On the
other hand, we establish a comparison result on the Laplace transform of the
hitting time for a fractional Brownian motion with Hurst parameter H < 12 .
1. Introduction
There has been a recent interest in establishing change-of-variable formulas for a
general class of Gaussian process which are not semimartingales, using techniques
of Malliavin calculus. The basic example of such process is the fractional Brownian
¨ unel [8], different
motion, and, since the pioneering work by Decreusefond and Ust¨
versions of the Itˆ
o formula have been established (see the recent monograph by
Biagini, Hu, Øksendal and Zhang [4] and the references therein).
In [1] the authors have considered the case of a Gaussian Volterra process of
Rt
the form Xt = 0 K(t, s)dWs , where W is a Wiener process and K(t, s) is a square
integrable kernel satisfying some regularity and integrability conditions, and they
have proved a change-of-variable formula for a class of processes which includes
the fractional Brownian motion with Hurst parameter H > 41 . A more intrinsic
approach based on the covariance function (instead of the kernel K) has been
developed by Cheridito and Nualart in [5] for the fractional Brownian motion. In
this paper an extended divergence operator is introduced in order to establish an
Itˆ
o formula in the case of an arbitrary Hurst parameter H ∈ (0, 1). In [13], Kruk,
Russo and Tudor have developed a stochastic calculus for a continuous Gaussian
process X = {Xt , t ∈ [0, T ]} with covariance function R(s, t) = E(Xt Xs ) which
has a bounded planar variation. This corresponds to the case of the fractional
Brownian motion with Hurst parameter H ≥ 21 . In [12] Kruk and Russo have
extended the stochastic calculus for the Skorohod integral to the case of Gaussian
processes with a singular covariance, which includes the case of the fractional
Brownian motion with Hurst parameter H < 21 . The approach of [12] based on
Received 2012-1-29; Communicated by Hui-Hsiung Kuo.
2000 Mathematics Subject Classification. Primary 60H07, 60G15; Secondary 60G18.
Key words and phrases. Skorohod integral, Itˆ
o’s formula, local time, Tanaka’s formula, selfsimilar processes, fractional Brownian motion, hitting time.
* D. Nualart is supported by the NSF grant DMS-1208625.
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PEDRO LEI AND DAVID NUALART
the duality relationship of Malliavin calculus and the introduction of an extended
domain for the divergence operator is related with the method used in the present
paper, although there are clear differences in the notation and basic assumptions.
In [15], Mocioalca and Viens have constructed the Skorohod integral and developed a stochastic calculus for Gaussian processes having a covariance structure
of the form E[|Bt − Bs |2 ] ∼ γ 2 (|t − s|), where γ satisfies some minimal regularity
conditions. In particular, the authors have been able to consider processes with a
logarithmic modulus of continuity, and even processes which are not continuous.
The purpose of this paper is to extend the methodology introduced Cheridito
and Nualart in [5] to the case of a general Gaussian process whose covariance
function R is absolutely continuous in one variable and the derivative satisfies an
appropriate integrability condition, without assuming that R has planar bounded
variation. The main result is a general Itˆo’s formula formulated in terms of the
extended divergence operator, proved in Section 3. As an application we establish
the existence of a local time in L2 and a version of Tanaka’s formula in Section
4. In Section 5 the results of the previous sections are applied to the case of
a general class of self-similar processes that includes the bifractional Brownian
motion with parameters H ∈ (0, 1) and K ∈ (0, 1] and the extended bifractional
Brownian motion with parameters H ∈ (0, 1) and K ∈ (1, 2) such that HK ∈
(0, 1). Finally, using the stochastic calculus developed in Section 3, we have been
able, in Section 6, to generalize the results by Decreusefond and Nualart (see [7])
on the distribution of the hitting time, to the case of a fractional Brownian motion
with Hurst parameter H < 12 . More precisely, we prove that if the Hurst parameter
is less than 21 , then the hitting time τa , for a > 0, satisfies E(exp(−ατa2H )) ≥
√
e−a 2α for any α > 0.
2. Preliminaries
Let X = {Xt , t ∈ [0, T ]} be a continuous Gaussian process with zero mean and
covariance function R(s, t) = E(Xt Xs ), defined on a complete probability space
(Ω, F , P ). For the sake of simplicity we will assume that X0 = 0. Consider the
following condition on the covariance function:
(H1) For all t ∈ [0, T ], the map s 7→ R(s, t) is absolutely continuous on [0, T ],
and for some α > 1,
α
Z T
∂R
sup
(s, t) ds < ∞.
∂s
0≤t≤T 0
Our aim is to develop a stochastic calculus for the Gaussian process X, assuming
condition (H1). In this section we introduce some preliminaries.
Denote by E the space of step functions on [0, T ]. We define in E the scalar
product
1[0,t] , 1[0,s] H = R(t, s).
Let H be the Hilbert space defined as the closure of E with respect to this scalar
product. The mapping 1[0,t] → Xt can be extended to a linear isometry from H
into the Gaussian subspace of L2 (Ω) spanned by the random variables {Xt , t ∈
[0, T ]}. This Gaussian subspace is usually called the first Wiener chaos of the
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
381
Gaussian process X. The image of an element ϕ ∈ H by this isometry will be a
Gaussian random variable denoted by X(ϕ). For example, if X = B is a standard
Brownian motion, then the Hilbert space H is isometric to L2 ([0, T ]), and B(ϕ) is
RT
the Wiener integral 0 ϕt dBt . A natural question is whether the elements of the
space H can be indentified with real valued functions on [0, T ], and in this case,
X(ϕ) will be interpreted as the stochastic integral of the function ϕ with respect
to the process X. For instance, in the case of the fractional Brownian motion with
Hurst parameter H ∈ (0, 1), this question has been discussed in detail by Pipiras
and Taqqu in the references [18, 19].
We are interested in extending the inner product hϕ, 1[0,t] iH to elements ϕ that
are not necessarily in the space H. Suppose first that ϕ ∈ E has the form
n
X
ϕ=
ai 1[0,ti ] ,
i=1
where 0 ≤ ti ≤ T . Then the inner product hϕ, 1[0,t] iH can be expressed as follows
Z T
Z ti
n
n
X
X
∂R
∂R
(s, t)ds =
ϕ(s)
(s, t)ds. (2.1)
hϕ, 1[0,t] iH =
ai R(ti , t) =
ai
∂s
∂s
0
0
i=1
i=1
If β is the conjugate of α, i.e. α1 + β1 = 1, applying H¨older’s inequality, we obtain
Z
! α1
Z T
T
∂R
∂R
hϕ, 1[0,t] iH = ϕ(s)
(s, t)ds ≤ kϕkβ sup
|
(s, t)|α ds
.
0
∂s
∂s
0≤t≤T
0
Therefore, if (H1) holds, we can extend the inner product hϕ, 1[0,t] iH to functions
ϕ ∈ Lβ ([0, T ]) by means of formula (2.1), and the mapping ϕ → hϕ, 1[0,t] iH is
continuous in Lβ ([0, T ]). This leads to the following definition.
P
Definition 2.1. Given ϕ ∈ Lβ ([0, T ]) and ψ = m
j=1 bj 1[0,tj ] ∈ E, we set
Z
m
T
X
∂R
hϕ, ψiH =
bj
ϕ(s)
(s, tj )ds.
∂s
0
j=1
In particular, this implies that for any ϕ and ψ as in the above definition,
Z t
hϕ1[0,t] , ψiH =
ϕ(s)dh1[0,s] , ψiH .
(2.2)
0
3. Stochastic Calculus for the Skorohod Integral
Following the argument of Al´os, Mazet and Nualart in [1], in this section we
establish a version of Itˆ
o’s formula. In order to do this we first discuss the extended
divergence operator for a continuous Gaussian stochastic process X = {Xt , t ∈
[0, T ]} with mean zero and covariance function R(s, t) = E(Xt Xs ), defined in
a complete probability space (Ω, F , P ), satisfying condition (H1), and such that
X0 = 0. The Gaussian family {X(ϕ), ϕ ∈ H} introduced in the Section 2 is
an isonormal Gaussian process associated with the Hilbert space H, and we can
construct the Malliavin calculus with respect to this process (see [17] and the
references therein for a more complete presentation of this theory).
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PEDRO LEI AND DAVID NUALART
We denote by S the space of smooth and cylindrical random variables of the
form
F = f (X(ϕ1 ), . . . , X(ϕn )),
(3.1)
where f ∈ Cb∞ (Rn ) (f is an infinitely differentiable function which is bounded
together with all its partial derivatives), and, for 1 ≤ i ≤ n, ϕi ∈ E. The derivative
operator, denoted by D, is defined by
DF =
n
X
∂f
(X(ϕ1 ), . . . , X(ϕn ))ϕi ,
∂x
i
i=1
if F ∈ S is given by (3.1). In this sense, DF is an H-valued random variable. For
any real number p ≥ 1 we introduce the seminorm
1
kF k1,p = (E(|F |p ) + E(kDF kpH )) p ,
and we denote by D1,p the closure of S with respect to this seminorm. More
generally, for any integer k ≥ 1, we denote by Dk the kth derivative operator, and
Dk,p the closure of S with respect to the seminorm

 p1
k
X
E(kDj F kpH⊗j ) .
kF kk,p = E(|F |p ) +
j=1
The divergence operator δ is introduced as the adjoint of the derivative operator.
More precisely, an element u ∈ L2 (Ω; H) belongs to the domain of δ if there exists
a constant cu depending on u such that
|E(hu, DF iH )| ≤ cu kF k2 ,
for any smooth random variable F ∈ S. For any u ∈ Domδ, δ(u) ∈ L2 (Ω) is then
defined by the duality relationship E(F δ(u)) = E(hu, DF iH ), for any F ∈ D1,2
and in the above inequality we can take cu = kδ(u)k2 . The space D1,2 (H) is
included in the domain of the divergence.
If the process X is a Brownian motion, then H is L2 ([0, T ]) and δ is an extension
of the Itˆ
o stochastic integral. Motivated by this example, we would like to interpret δ(u) as a stochastic integral for u in the domain of the divergence operator.
However, it may happen that the process X itself does not belong to L2 (Ω; H). For
example, this is true if X is a fractional Brownian motion with Hurst parameter
H ≤ 41 (see [5]). For this reason, we need to introduce an extended domain of the
divergence operator.
Definition 3.1. We say that a stochastic process u ∈ L1 (Ω; Lβ ([0, T ])) belongs
to the extended domain of the divergence DomE δ if
|E(hu, DF iH )| ≤ cu kF k2 ,
for any smooth random variable F ∈ S, where cu is some constant depending on
u. In this case, δ(u) ∈ L2 (Ω) is defined by the duality relationship
E(F δ(u)) = E(hu, DF iH ),
for any F ∈ S.
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
383
Note that the pairing hu, DF iH is well defined because of Definition 2.1.
In general, the domains Domδ and DomE δ are not comparable because u ∈
Domδ takes values in the abstract Hilbert space H and u ∈ DomE δ takes values
in Lβ ([0, T ]). In the particular case of the fractional Brownian motion with Hurst
parameter H < 21 we have (see [7])
1
−H
1
H = IT2 − (L2 ) ⊂ L H ([0, T ]),
1
and assumption (H1) holds for any α < 2H−1
. As a consequence, if β is the
1
β
conjugate of α, then β > 2H , so H ⊂ L ([0, T ]) and Domδ ⊂ DomE δ.
If u belongs to DomE δ, we will make use of the notation
Z T
δ(u) =
us δXs ,
0
Rt
and we will write 0 us δXs for δ(u1[0,t] ), provided u1[0,t] ∈ DomE δ.
We are going to prove a change-of-variable formula for F (t, Xt ) involving the
extended divergence operator. Let F (t, x) be a function in C 1,2 ([0, T ]×R) (the par∂2F
∂F
tial derivatives ∂F
∂x , ∂x2 and ∂t exist and are continuous). Consider the following
growth condition.
(H2) There exist positive constants c and λ < 14 (sup0≤t≤T R(t, t))−1 such that
∂2F
∂F
∂F
sup |F (t, x)| + |
(t, x)| + | 2 (t, x)| + |
(t, x)| ≤ c exp(λ|x|2 ). (3.2)
∂x
∂x
∂t
0≤t≤T
Using the integrability properties of the supremum of a Gaussian process, condition (3.2) implies
E sup |F (t, Xt )|2 ≤ c2 E exp(2λ sup |Xt |2 ) < ∞,
0≤t≤T
0≤t≤T
2
∂ F
and the same property holds for the partial derivatives ∂F
∂x , ∂x2 and
the following additional condition on the covariance function.
(H3) The function Rt := R(t, t) has bounded variation on [0, T ].
∂F
∂t
. We need
Theorem 3.2. Let F be a function in C 1,2 ([0, T ]×R) satisfying (H2). Suppose that
X = {Xt , t ∈ [0, T ]} is a zero mean continuous Gaussian process with covariance
function R(t, s), such that X(0) = 0, satisfying (H1) and (H3). Then for each
t ∈ [0, T ] the process { ∂F
∂x (s, Xs )1[0,t] (s), 0 ≤ t ≤ T } belongs to extended domain
E
of the divergence Dom δ and the following holds
Z t
Z t
∂F
∂F
F (t, Xt ) = F (0, 0) +
(s, Xs )ds +
(s, Xs )δXs
∂s
0
0 ∂x
Z t 2
1
∂ F
+
(s, Xs )dRs .
(3.3)
2 0 ∂x2
Proof. Suppose that G is a random variable of the form G = In (h⊗n ), where In
denotes the multiple stochastic integral of order n with respect to X and h is a
step function in [0, T ]. The set of all these random variables form a total subset
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PEDRO LEI AND DAVID NUALART
of L2 (Ω). Taking into account Definition 3.1 of the extended divergence operator,
it is enough to show that for any such G,
E(GF (t, Xt )) − E(GF (0, 0)) −
Z
t
0
∂F
1
E(G
(s, Xs )ds −
∂s
2
∂F
= E(hDG, 1[0,t] (·)
(·, X· )iH ).
∂x
Z
t
E(G
0
∂2F
(s, Xs ))dRs
∂x2
(3.4)
First we reduce the problem to the case where the function F is smooth in x. For
this purpose we replace F by
Fk (t, x) = k
Z
1
−1
F (t, x − y)ε(ky)dy,
where ε is a nonnegative smooth function supported by [−1, 1] such that
R1
ε(y)dy = 1. The functions Fk are infinitely differentiable in x and their deriva−1
tives satisfy the growth condition (3.2) with some constants ck and λk .
Suppose first that G is a constant, that is, n = 0. The right-hand side of
Equality (3.4) vanishes. On the other hand, we can write
E(GFk (t, Xt )) = G
Z
Fk (t, x)p(Rt , x)dx,
R
where p(σ, y) = (2πσ)−1/2 exp(−x2 /2σ). We know that
quence, integrating by parts, we obtain
∂p
∂σ
=
1 ∂2 p
2 ∂x2 .
As a conse-
Z tZ
∂Fk
E(GFk (t, Xt )) − GF (0, 0) − G
(s, x)p(Rs , x)dxds
0
R ∂s
Z t Z
∂2p
1
Fk (s, x) 2 (Rs , x)dx dRs
= G
2
∂x
0
R
Z t Z 2
1
∂ Fk
= G
(s,
x)p(R
,
x)dx
dRs
s
2
2
0
R ∂x
Z t 2
1
∂ Fk
= G
E
(s, Xs ) dRs ,
2
∂x2
0
which completes the proof of (3.4), when G is constant.
Suppose now that n ≥ 1. In this case E(G) = 0. On the other hand, using the
fact that the multiple stochastic integral In is the adjoint of the iterated derivative
operator Dn we obtain
∂ n Fk
⊗n
⊗n
E(GFk (t, Xt )) = E(In (h⊗n )Fk (t, Xt )) = E hh⊗n ,
(t,
X
)1
i
t [0,t] H
∂xn
n
∂ Fk
=E
(t, Xt ) hh, 1[0,t] inH .
(3.5)
∂xn
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
385
Note that E(GFk (t, Xt )) is the product of two factors. Therefore, its differential
will be expressed as the sum of two terms
∂ n Fk
n
d (E(GFk (t, Xt ))) = hh, 1[0,t] iH d E(
(t, Xt ))
∂xn
∂ n Fk
+ E(
(t, Xt ))d hh, 1[0,t] inH .
(3.6)
n
∂x
Using again the integration by parts formula and the fact that the Gaussian density
satisfies the heat equation we obtain
Z n
∂ n Fk
∂ Fk
d E(
(t,
X
))
=
d
(t,
x)p(R
,
x)dx
t
t
n
∂xn
R ∂x
Z n+1
Z n
∂
Fk
∂ Fk
1
∂2p
=
(t,
x)p(R
,
x)dx
dt
+
(t,
x)
(R
,
x)dx
dRt
t
t
n
n
2
∂x2
R ∂t∂x
R ∂x
Z n+1
Z n+2
∂
Fk
1
Fk
∂
=
(t,
x)p(R
,
x)dx
dt
+
(t,
x)p(R
,
x)dx
dRt
t
t
n
n+2
2
R ∂t∂x
R ∂x
1 ∂ n+2 Fk
∂ n+1 Fk
(t, Xt ))dt + E(
(t, Xt ))dRt .
(3.7)
= E(
n
∂t∂x
2
∂xn+2
Equation (3.5) applied to
E(G
∂ 2 Fk
∂x2
and to
∂Fk
∂t
yields
∂ 2 Fk
∂ n+2 Fk
(t, Xt )) = E(
(t, Xt ))hh, 1[0,t] inH ,
2
∂x
∂xn+2
(3.8)
and
∂ n+1 Fk
∂Fk
(t, Xt )) = E(
(t, Xt ))hh, 1[0,t] inH ,
(3.9)
∂t
∂t∂xn
respectively. Then, substituting (3.8), (3.9) and (3.7) into the first summand in
the right-hand side of (3.6) we obtain
E(G
∂Fk
1
∂ 2 Fk
(t, Xt ))dt + E(G
(t, Xt ))dRt
∂t
2
∂x2
n
∂ Fk
+ E(
(t, Xt )))d hh, 1[0,t] inH .
(3.10)
n
∂x
Therefore, to show (3.4), it only remains to check that
Z t
∂Fk
∂ n Fk
E(hDG, 1[0,t] (·)
(·, X· )iH ) = n
E(
(s, Xs ))hh, 1[0,s] in−1
H d hh, 1[0,s] iH .
n
∂x
∂x
0
d(E(GFk (t, Xt ))) = E(G
Using the fact that DG = nIn−1 (h⊗(n−1) )h, we get
∂Fk
∂Fk
(·, X· )iH ) = nhh, 1[0,t] (·)E(In−1 (h⊗(n−1) )
(·, X· ))iH .
∂x
∂x
Then, taking into account (2.2), we can write
E(hDG, 1[0,t] (·)
∂Fk
∂Fk
(·, X· )iH )) = nE(In−1 (h⊗(n−1) )
(t, Xt ))d hh, 1[0,t] iH .
∂x
∂x
Finally, using again that In−1 is the adjoint of the derivative operator yields
d(E(hDG, 1[0,t] (·)
E(In−1 (h⊗(n−1) )
∂Fk
∂ n Fk
(t, Xt )) = E(
(t, Xt ))hh, 1[0,t] in−1
H ,
∂x
∂xn
386
PEDRO LEI AND DAVID NUALART
which allows us to complete the proof for the function Fk . Finally, it suffices to
let k tend to infinity.
4. Local Time
In this section, we will apply the Itˆo formula obtained in Section 3 to derive a
version of Tanaka’s formula involving the local time of the process X. In order to
do this we first discuss the existence of the local time for a continuous Gaussian
stochastic process X = {Xt , t ∈ [0, T ]} with mean zero defined on a complete probability space (Ω, F , P ), with covariance function R(s, t). We impose the following
additional condition which is stronger than (H3):
(H3a) The function Rt = R(t, t) is increasing on [0, T ], and Rt > 0 for any t > 0.
The local time Lt (x) of the process X (with respect to the measure induced
by the variance function) is defined, if it exists, as the density of the occupation
measure
Z
t
mt (B) =
1B (Xs )dRs ,
0
B ∈ B(R)
with respect to the Lebesgue measure. That is, for any bounded and measurable
function g we have the occupation formula
Z
Z t
g(x)Lt (x)dx =
g(Xs )dRs .
0
R
Following the computations in [6] based on Wiener chaos expansions we can get
sufficient conditions for the local time Lt (x) to exists and to belong to L2 (Ω) for
any fixed t ∈ [0, T ] and x ∈ R. We denote by Hn the nth Hermite polynomial
defined for n ≥ 1 by
(−1)n − x2 dn − x2
e 2
(e 2 ),
n!
dxn
Hn (x) =
and H0 = 1. For s, t 6= 0 set ρ(s, t) =
Z tZ
αn (t) =
0
0
R(s,t)
√
.
Rs Rt
u
For all n ≥ 1 and t ∈ [0, T ] we define
|ρ(u, v)|n dRv dRu
√
√
,
n
Ru Rv
and we introduce the following condition on the covariance function R(s, t):
∞
X
(H4)
αn (T ) < ∞.
n=1
The following proposition is an extension of the result on the existence and
Wiener chaos expansion of the local time for the fractional Brownian motion proved
by Coutin, Nualart and Tudor in [6]. Recall that for all ε > 0 and x ∈ R,
p(ε, x) = (2πε)−1/2 exp(−x2 /2ε).
Proposition 4.1. Suppose that X = {Xt , t ∈ [0, T ]} is a zero mean continuous
Gaussian process with covariance function R(t, s), satisfying conditions (H3a) and
(H4) and with X(0) = 0. Then, for each a ∈ R, and t ∈ [0, T ], the random
variables
Z
t
0
p(ε, Xs − a)dRs
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
387
converge in L2 (Ω) to the local time Lt (a), as ε tends to zero. Furthermore the
local time Lt (a) has the following Wiener chaos expansion
Lt (a) =
∞ Z
X
n=0
0
t
a
−n
Rs 2 p(Rs , a)Hn ( √ )In (1⊗n
[0,s] )dRs .
Rs
(4.1)
Proof. Applying Stroock’s formula we can compute the Wiener chaos expansion
of the random variable p(ε, Xs − a) for any s > 0 as it has been done in [6], and
we obtain
∞
X
p(ε, Xs − a) =
βn,ε (s)In (1⊗n
(4.2)
[0,s] ),
n=0
where
n
a
βn,ε (s) = (Rs + ε)− 2 p(Rs + ε, a)Hn ( √
).
Rs + ε
(4.3)
From (4.2), integrating with respect to the measure dRs , we deduce the Wiener
chaos expansion
Z
0
t
p(ε, Xs − a)dRs =
∞ Z
X
n=0
t
0
βn,ε (s)In (1⊗n
[0,s] (·))dRs .
(4.4)
We need to show this expression converges in L2 (Ω) to the right-hand side of
Equation (4.1), denoted by Λt (a), as ε tends to zero. For every n and s we have
limε→0 βn,ε (s) = βn (s), where
a
−n
βn (s) = Rs 2 p(Rs , a)Hn ( √ ).
Rs
We claim that
2n/2
|βn,ε (s)| ≤ c
Γ
n!
n+1
2
− n+1
2
Rs
.
(4.5)
In fact, from the properties of Hermite polynomials it follows that
Z ∞
√
2
2
n
2
Hn (y)e−y /2 = (−1)[ 2 ] 2n/2 √
sn e−s g(ys 2)ds,
n! π 0
where g(r) = cos r for n even, and g(r) = sin r for n odd. Thus, |g| is dominated
by 1, and this implies
2
2n/2
n+1
|Hn (y)e−y /2 | ≤ c
Γ
.
n!
2
Substituting this estimate into (4.3) yields (4.5). The estimate (4.5) implies that,
Rt
for any n ≥ 1, the integral 0 βn (s)In (1⊗n
[0,s] )dRs is well defined as a random
Rt
variable in L2 (Ω), and it is the limit in L2 (Ω) of 0 βn,ε (s)In (1⊗n
[0,s] )dRs as ε tends
388
PEDRO LEI AND DAVID NUALART
to zero. In fact, (4.5) implies that
Z t
Z t
⊗n
⊗n ≤
β
(s)I
(1
)dR
|β
(s)|
(1
)
I
n
n
s
n
n
[0,s]
[0,s] dRs
2
0
0
2
Z t
√ n/2
n
n+1
− n+1
≤ c n!2 Γ
Rs 2 Rs2 dRs
2
0
√ n/2
n+1 p
≤= c n!2 Γ
Rt .
2
Rt
For n = 0, βn,ε (s) = p(Rs + ε, a), and clearly 0 p(Rs + ε, a)dRs converges to
Rt
0 p(Rs , a)dRs as ε tends to zero. In the same way, using dominated convergence,
we can prove that
Z t
⊗n
lim (β
(s)
−
β
(s))
I
(1
)dR
n,ε
n
n [0,s]
s = 0.
ε→0
0
2
Set
αn,ε = E
Z
0
t
βn,ε (s)In (1⊗n
[0,s] )dRs
2
.
2
To show the convergence in LP
(Ω) of the series (4.4) to the right-hand side of (4.1)
it suffices to prove that supε ∞
n=1 αn,ε < ∞. Using (4.5) and Stirling formula we
have
Z tZ t
⊗n
αn,ε =
E(In (1⊗n
[0,u] )In (1[0,v] ))βn,ε (u)βn,ε (v)dRv dRu
0
0
Z tZ u
= 2n!
R(u, v)n βn,ε (u)βn,ε (v)dRv dRu
0
0
2 Z t Z u
n+1
2n
n+1
|R(u, v)|n (Ru Rv )− 2 dRv dRu
≤c Γ
n!
2
0
0
Z tZ u
|ρ(u, v)|n dRv dRu
√
√
≤c
n
Rv Ru
0
0
= αn (t).
Therefore, taking into account hypothesis (H4), we conclude that
sup
ε
∞
X
αn,ε <
n=1
∞
X
n=1
αn (T ) < ∞,
and this proves the convergence in L2 (Ω) of the series (4.4) to a limit denoted by
Λt (a).
Finally, we have to show that Λt (a) is the local time Lt (a). The above estimates
are uniform in a ∈ R. Therefore, we can deduce that the convergence of
Rt
p(ε,
Xs − a)dRs to Λt (a) holds in L2 (Ω × R, P × µ), for any finite meausre µ.
0
As a consequence, for any continuous function g with compact support we have
that
Z Z
t
R
0
p(ε, Xs − a)dRs g(a)da
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
389
R
converges in L2 (Ω), as ε tends to zero, to R Λt (a)g(a)da. Clearly, this sequence
Rt
also converges to 0 g(Xs )dRs . Hence,
Z
Z t
Λt (a)g(a)da =
g(Xs )dRs ,
0
R
which imples that Λt (a) is a version of the local time Lt (a)
Corollary 4.2. Condition (H4) holds if
Z TZ T
1 − ln(1 − |ρ(u, v)|)
p
dRv dRu < ∞.
√
Rv Ru · 1 − |ρ(u, v)|
0
0
(4.6)
Proof. We can write
∞
X
g(x) =
≤
∞
X
xn √
√
1−x=
n
n=1
X
n(1−x)<1
1
2
T
T
dRv dRu
ϕ(|ρ(u, v)|) √
,
Rv Ru
0
0
n=1
√
P
xn
√
where ϕ(x) = ∞
n=1 n . If we define g(x) = ϕ(x) 1 − x for every x ∈ [0, 1), then
αn (T ) =
Z
X
Z
n(1−x)<1
∞
X
xn √
√
1−x+
n
X
n(1−x)≥1
xn √
√
1−x
n
xn
+
xn (1 − x) ≤ 1 − ln(1 − x),
n
n=0
and the result follows.
Notice that the Wiener chaos expansion (4.1) can also we written as
Z t
∞
X
a
−n
2
Lt (a) =
In
Rs p(Rs , a)Hn ( √ )dRs .
Rs
s1 ∨···∨sn
n=0
In the particular case a = 0, the Wiener chaos expansion of Lt (0) can be written
as
∞ Z t
k
X
−k− 1 (−1)
Lt (0) =
Rs 2 √
I2k (1⊗2k
[0,s] )dRs .
k k!
2π2
k=0 0
Using arguments of Fourier analysis, in [3] it is proved that if the covariance
function R(s, t) satisfies
Z TZ T
−1
(Ru + Rv − 2R(u, v)) 2 dRu dRv < ∞,
(4.7)
0
0
then
R for any t∈ [0, T ] the local time Lt of X exists and is square integrable, i.e.
E R L2t (x)dx < ∞. We can write
p
Ru + Rv − 2R(u, v) = Ru + Rv − 2ρ(u, v) Ru Rv
p
p
p
= ( Ru − Rv )2 + 2 Ru Rv (1 − ρ(u, v))
p
≥ 2 Ru Rv (1 − ρ(u, v)).
390
PEDRO LEI AND DAVID NUALART
Therefore, condition (4.7) is implied by
Z TZ T
1
p
dRu dRv < ∞,
√
4
Ru Rv · 1 − ρ(u, v)
0
0
(4.8)
which can be compared with the above assumption (4.6). Notice
conR that both
ditions have different consequences. In fact, (4.8) implies E R L2t (x)dx < ∞;
whereas (4.6) implies only that for each x, E(L2t (x)) < ∞.
We can now establish the following version of Tanaka formula.
Theorem 4.3. Suppose that X = {Xt , 0 ≤ t ≤ T } is a zero-mean continuous
Gaussian process,with X0 = 0, and such that the covariance function R(s, t) satisfies conditions (H1), (H3a) and (H4). Let y ∈ R. Then, for any 0 < t ≤ T ,
the process {1(y,∞) (Xs )1[0,t] (s), 0 ≤ s ≤ T } belongs to DomE δ and the following
holds
1
δ 1(y,∞) (X· )1[0,t] (·) = (Xt − y)+ − (−y)+ − Lt (y).
2
Proof. Let ε > 0 and for all x ∈ R set
Z x Z v
fε (x) =
p(ε, z − y)dzdv.
Theorem 3.2 implies that
fε (Xt ) = fε (0) +
−∞
Z
0
t
−∞
fε′ (Xs )δXs +
1
2
Z
0
t
fε′′ (Xs )dRs .
Then we have that fε′ (Xs )1[0,t] (s) converges to 1(y,∞) (Xs )1[0,t] (s) in L2 (Ω×R) and
Rt
fε (Xt ) converges to (Xt −y)+ in L2 (Ω). Finally, by Proposition 4.1, 0 fε′′ (Xs )dRs
converges to Lt (y) in L2 (Ω). This completes the proof.
5. Example: Self-Similar Processes
In this section, we are going to apply the results of the previous sections to the
case of a self-similar centered Gaussian process X. Suppose that X = {Xt , t ≥ 0}
is a stochastic process defined on a complete probability space (Ω, F , P ). We say
that X is self-similar with exponent H ∈ (0, 1) if for any a > 0, the processes
{X(at), t ≥ 0} and {aH X(t), t ≥ 0} have the same distribution. It is well-known
that fractional Brownian motion is the only H-self-similar centered Gaussian process with stationary increments. Suppose that X = {Xt , t ≥ 0} is a continuous
Gaussian centered self-similar process with exponent H. Let R(s, t) be the covariance function of X. To simplify the presentation we assume E(X12 ) = 1. The
process X satisfies the condition (H3a) because
Rt = R(t, t) = t2H R(1, 1) = t2H .
The function R is homogeneous of order 2H, that is, for a > 0 and s, t ≥ 0, we
have
R(as, at) = E(Xas Xat ) = E(aH Xs aH Xt ) = a2H R(s, t).
For any x ≥ 0, we define
ϕ (x) = R(1, x).
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
391
Notice that for any x > 0,
2H
ϕ(x) = R(1, x) = x
1
R( , 1) = x2H ϕ
x
1
.
x
On the other hand, applying Cauchy-Schwarz inequality we get that function ϕ
satisfies |ϕ(x)| ≤ xH for all x ∈ [0, 1]. The next proposition provides simple
sufficient conditions on the function ϕ for the process X to satisfy the assumptions
(H1) and (H4).
Proposition 5.1. Suppose that X = {Xt , t ≥ 0} is a zero mean continuous
self-similar Gaussian process with exponent of self-similarity H and covariance
function R(s, t). Let ϕ(x) = R(1, x). Then
(i) (H1) holds on any interval [0, T ] for α > 1 if α(2H − 1) + 1 > 0 and ϕ is
absolutely continuous and satisfies
Z 1
|ϕ′ (x)|α dx < ∞.
(5.1)
0
(ii) (H4) holds on any interval [0, T ] if for some ε > 0 and for all x ∈ [0, 1]
|ϕ(x)| ≤ xH+ε .
(5.2)
Proof. We first prove (i). We write
Z T
Z t
Z T
∂R
∂R
∂R
α
α
|
(s, t)| ds =
|
(s, t)| ds +
|
(s, t)|α ds.
∂s
∂s
∂s
0
0
t
2H−1 ′ s
ϕ ( t ). Applying (5.1) and the
For s ≤ t, R(s, t) = t2H ϕ( st ) and ∂R
∂s (s, t) = t
s
change of variables by x = t , we have
Z t
Z t
s
∂R
|
(s, t)|α ds =
tα(2H−1) |ϕ′
|α ds
∂s
t
0
0
Z 1
α(2H−1)+1
=t
|ϕ′ (x)|α dx.
(5.3)
0
For s > t, R(s, t) = s
2H
ϕ( st )
and
∂R
t
t
(s, t) = 2Hs2H−1 ϕ( ) − s2H−2 tϕ′ ( ).
∂s
s
s
Then,
Z T
t
∂R
|
(s, t)|α ds ≤ C
∂s
Z
T
t
s
2H−1
!
Z T
t α
t α
2H−2
′
|ϕ
| ds +
s
t|ϕ
| ds .
s
s
t
With the change of variables x = st we can write
Z T
Z 1
t α
2H−1
(2H−1)α+1
s
|ϕ
| ds ≤ kϕkα
t
xα(1−2H)−2 dx,
∞
t
s
t
T
=
kϕkα
∞
[t(2H−1)α+1 − T (2H−1)α+1 ]
α(1 − 2H) − 1
(5.4)
392
PEDRO LEI AND DAVID NUALART
and
Z
t
T
s2H−2 t|ϕ′
Z 1
t α
| ds ≤ t(2H−1)α+1
|ϕ′ (x) |α x(2−2H)α−2 dx
t
s
T
(2−2H)α−2 Z 1
t
≤ t(2H−1)α+1
|ϕ′ (x) |α dx
t
T
T
Z 1
tα−1
≤ (2−2H)α−2
|ϕ′ (x) |α dx.
(5.5)
t
T
T
Now, (H1) follows from (5.3), (5.4) and (5.5).
In order to show (ii) we need to show that
∞
∞ Z T Z u
X
X
1 |R(u, v)|n
√
αn (T ) =
dRv dRu < ∞.
n (Ru Rv ) n+1
2
0
n=1
n=1 0
For any 0 < v < u, we have R(u, v) = u2H ϕ( uv ), and the change of variable x = uv
yields
Z Z
(2H)2 T u
αn (T ) = √
|R(u, v)|n (uv)H(1−n)−1 dvdu
n 0 0
Z Z
(2H)2 T 1
= √
|R(1, x)|n u2H−1 xH(1−n)−1 dvdu
n 0 0
Z
2HT 2H 1
√
=
|ϕ(x)|n xH(1−n)−1 dx
n
0
Z
2HT 2H 1 nε+H−1
≤ √
x
dx
n
0
2HT 2H
1
= √
.
n nε + H
Therefore, we have
∞
∞
X
2HT 2H X − 3
αn (T ) ≤
n 2 < ∞.
(5.6)
ε
n=1
n=1
This completes the proof of (ii).
Example 5.2. The bifractional Brownian motion is a centered Gaussian process
X = {BtH,K , t ≥ 0}, with covariance
R(t, s) = RH,K (t, s) = 2−K ((t2H + s2H )K − |t − s|2HK ),
(5.7)
where H ∈ (0, 1) and K ∈ (0, 1]. We refer to Houdr´e and Villa [10] for the definition
and basic properties of this process. Russo and Tudor [20] have studied several
properties of the bifractional Brownian motion and analyzed the case HK = 12 .
Tudor and Xiao [21] have derived small ball estimates and have proved a version
of the Chung’s law of the iterated logarithm for the bifractional Brownian motion.
In [14], the authors have shown a decomposition of the bifractional Brownian
motion with parameters H and K into the sum of a fractional Brownian motion
with Hurst parameter HK plus a stochastic process with absolutely continuous
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
393
trajectories. The stochastic calculus with respect to the bifractional Brownian
motion has been recently developed in the references [13] and [12]. A Tanaka
formula for the bifractional Brownian motion in the case HK ≤ 12 by Es-Sebaiy
and Tudor in [9]. A multidimensional Itˆo’s formula for the bifractional Brownian
motion has been established in [2].
Note that, if K = 1 then B H,1 is a fractional Brownian motion with Hurst
parameter H ∈ (0, 1), and we denote this process by B H . Bifractional Brownian
motion is a self-similar Gaussian process with non-stationary increment if K is not
equal to 1.
Set
ϕ(x) = 2−K ((1 + x2H )K − (1 − x)2HK ).
Then
ϕ′ (x) = 21−K HK[x2HK−1 (1 + x2H )K + (1 − x)2HK−1 ],
which implies that (i) in Proposition 5.1 holds α such that α(2HK − 1) > −1.
Notice that
1
ϕ(x) ≤ K [1 + x2H − (1 − x)2H ]K .
(5.8)
2
Then, if 2H ≤ 1
1 + x2H − (1 − x)2H ≤ 2x2H ,
(5.9)
and when 2H > 1,
1 + x2H − (1 − x)2H ≤ x + x2H ≤ 2x.
(5.10)
From the inequalities (5.8), (5.9) and (5.10) we obtain
ϕ(x)
(1 + x2H )K − (1 − x)2HK
=
≤ xmin(H,1−H)K .
(5.11)
xHK
2K xHK
Then condition (ii) in Proposition 5.1 holds with ε = min(H, 1 − H)K. As a
consequence, the results in Sections 3, 4 and 5 hold for the bifractional Brownian
motion.
Bardina and Es-Sebaiy considered in [2] an extension of bifractional Brownian
motion with parameters H ∈ (0, 1), K ∈ (1, 2) and HK ∈ (0, 1) with covariance
function (5.7). By the same arguments as above, Proposition 5.1 holds in this case
with ε = min(H, 1 − H)K in condition (ii). Thus, the results in Sections 3, 4 and
5 hold for this extension of the bifractional Brownian motion.
6. Hitting Times
Suppose that X = {Xt , t ≥ 0} is a zero mean continuous Gaussian process with
covariance function R(t, s), satisfying (H1) and (H3) on any interval [0, T ]. We
also assume that X(0) = 0. Moreover, we assume the following conditions:
(H5) lim supt→∞ Xt = +∞ almost surely.
(H6) For any 0 ≤ s < t, we have E(|Xt − Xs |2 ) > 0.
(H7) For any continuous function f ,
Z t
∂R
r 7→
f (s)
(s, r)ds
∂s
0
is continuous on [0, ∞).
394
PEDRO LEI AND DAVID NUALART
For any a > 0, we denote by τa the hitting time defined by
τa = inf{t ≥ 0, Xt = a} = inf{t ≥ 0, Xt ≥ a}.
(6.1)
The map a 7→ τa is left continuous and increasing with right limits.
We are interested in the distribution of the random variable τa . The explicit
form of this distribution is known only in some special cases such as the standard
Brownian motion. In this case the Laplace transform of the hitting time τa is
given by
√
E(e−ατa ) = e−a 2α ,
for all α > 0. This can be proved, for instance, using the exponential martingale
1
2
Mt = eλXt − 2 λ t ,
and Doob’s optional stopping theorem. In the general case, the exponential process
1
(6.2)
Mt = exp(λXt − λ2 Rt ).
2
is no longer martingale. However, if we apply (3.2) for the divergence integral, we
have
Mt = 1 + λδ(M 1[0,t] ) = 1 + λδt (M ).
(6.3)
Substituting t by τa and taking the expectation in Equation (6.3), Decreusefond
√
and Nualart have established in [7] an inequality of the form E(e−αRτa ) ≤ e−a 2α ,
assuming that the partial derivative of the covariance ∂R
∂s (t, s) is nonnegative and
continuous. This includes the case of the fractional Brownian motion with Hurst
parameter H > 21 . The purpose of this section is derive the converse inequality in
the singular case where the partial derivative of the covariance is not continuous,
assuming ∂R
∂s (t, s) ≤ 0 for s < t (which includes the case of the fractional Brownian
motion with Hurst parameter H < 12 ), completing the analysis initiated in [7].
As in the case of the Brownian motion we would like to substitute t by τa in both
sides of Equation (6.3) and then take the mathematical expectation in both sides
of the equality. It is convenient to introduce also an integral in a, and, following
the approach developed in [7], we claim that the following result holds.
Proposition 6.1. Suppose X satisfies (H1), (H3), (H5), (H6) and (H7), then
Z ∞
E(Mτa )ψ(a)da = c−
0
lim λE
δ→0
Z
0
ST
dτy
Z
0
1
ψ(y)dη
Z
0
∞
pδ ((τy+ ∧ T )η + τy (1 − η) − s)Ms
∂R
(τy , s)ds,
∂s
(6.4)
where p is an infinitely differentiable function with support on [−1, 1] such that
R1
p(x)dx = 1, ψ(x) be a nonnegative smooth function with compact support
−1
R∞
contained in (0, ∞) such that 0 ψ(a)da = c and we use the notation pε (x) =
1
x
ε p( ε ).
Before proving this proposition we need several technical lemmas. The first
lemma is an integration by parts formula, and it is a consequence of the definition
of the extended divergence operator given in Definition 2.1.
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
395
Lemma 6.2. For any t > 0 and any random variable of the form
F = f (Xt1 , . . . , Xtn ), where f is an inifinitely differentiable function which is
bounded together with all its partial derivatives, we have
!
Z t
n
X
∂f
∂R
Ms
(ti , s)ds ,
(6.5)
E(F δt (M )) = E
(Xt1 , . . . , Xtn )
∂xi
∂s
0
i=1
where δt (M ) is given in Equation (6.3).
Proof. Using the Definition 2.1 of the extended divergence operator and Equation
(2.1) we can write
E(F δt (M )) = E(hDF, M 1[0,t] iH )
=E
=E
!
n
X
∂f
(Xt1 , . . . , Xtn )h1[0,ti ] , M 1[0,t] iH
∂xi
i=1
!
Z t
n
X
∂f
∂R
(Xt1 , . . . , Xtn )
Ms
(ti , s)ds ,
∂xi
∂s
0
i=1
which completes the proof of the lemma.
For any a > 0, we know that P (τa < ∞) = 1 by condition (H5). Set
St = sup Xs .
s∈[0,t]
We know that for all t > 0, St belongs to D1,2 and DSt = 1[0,τSt ] (see [7] and
[11]). Following the approach developed in [7], we introduce a regularization of
the hitting time τa , and we establish its differentiability in the sense of Malliavin
calculus.
Lemma 6.3. Suppose that ϕ is a nonnegative smooth function with compact support in (0, ∞) and define for any T > 0,
Y =
Z
∞
0
ϕ(a)(τa ∧ T )da.
The random variable Y belongs to the space D1,2 , and
Dr Y = −
Z
ST
ϕ(y)1[0,τy ] (r)dτy .
0
Proof. First, it is clear that Y is bounded because ϕ has compact support. On
the other hand, for any r > 0, we can write
{τa > r} = {Sr < a}.
396
PEDRO LEI AND DAVID NUALART
Apply Fubini’s theorem, we have
Y =
Z
∞
Z
ϕ(a)
0
=
Z
∞
Z
0
T
Z
Z
∞
0
∞
!
dθ da =
0
0
=
τa ∧T
Z
∞
0
∞
Z
ϕ(a)1{θ<τa ∧T } dθda
0
ϕ(a)1{θ<τa } 1{θ<T } dadθ =
T
Z
0
Z
∞
0
ϕ(a)1{Sθ <a} dadθ
ϕ(a)dadθ.
Sθ
R∞
The function ψ(x) = x ϕ(a)da is continuously differentiable with a bounded
derivative, so ψ(Sθ ) ∈ D1,2 for any θ ∈ [0, T ] because Sθ ∈ D1,2 (see, for instance,
RT
[17]). Finally, we can show that Y = 0 ψ(Sθ )dθ belongs to D1,2 approximatig
the integral by Riemann sums. Hence, taking the Malliavin derivative of Y , we
obtain
Z T
Dr Y = −
ϕ(Sθ )Dr Sθ dθ
0
=−
Z
T
ϕ(Sθ )1[0,τSθ ] (r)dθ = −
0
Z
ST
ϕ(y)1[0,τy ] (r)dτy ,
0
where the last equality holds by changing variable Sθ = y, which is equivalent to
θ = τy .
The following lemma provides an explicit formula for the expectation
E(p(Y )δt (M )), where p is a smooth function with compact support.
Lemma 6.4. Suppose X satisfies (H1), (H3), (H5), (H6) and (H7). Then, for
any infinitely differentiable function p with compact support,
!
Z t
Z ST
∂R
′
ϕ(y)
E(p(Y )δt (M )) = −E
Ms p (Y )
(τy , s)dτy ds ,
(6.6)
∂s
0
0
Proof. Consider the random variable
Y =
Z
∞
0
=
Z
ϕ(a)(τa ∧ T )da =
T
Z
0
T
Z
∞
ϕ(a)dadθ
Sθ
ξ(Sθ )dθ,
0
R∞
where ξ(x) = x ϕ(a)da. Let {DN , N ≥ 1} be an increasing sequence of finite
subsets of [0, T ] such ∪∞
N =1 DN is dense in [0, T ]. Set DN = {σi , 0 = σ0 < σ1 <
θ
· · · < σN = T } and DN
= DN ∩ [0, θ], and
θ
SθN = max{Xt , t ∈ DN
} = max{Xσ0 , . . . , Xσ(θ) },
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
397
θ
where σ(θ) = sup DN
. We also write SkN = SσNk . Define
Z T
N
X
(σk − σk−1 )ξ(max{Xσ0 , . . . , Xσk−1 })
YN =
ξ(SθN )dθ =
0
=
k=1
N
X
k=1
N
(σk − σk−1 )ξ(Sk−1
).
Then, taking into account that Xσ0 = X0 = 0, p(YN ) is a Lipschitz function F of
the N − 1 variables {Xσ1 , . . . , XσN −1 }, namely,
!
N
X
N
p(YN ) = F (Xσ1 , . . . , XσN −1 ) = p
(σk − σk−1 )ξ(Sk−1
) ,
k=2
and, for all 1 ≤ i ≤ N − 1 the derivative of F respect to xi is
N
X
∂F
N
= −p′ (YN )
(σk − σk−1 )ϕ(Sk−1
)1{Sk−1
N
=Xσi } .
∂xi
k=i+1
By (6.5), we have
E(p(YN )δt (M ))
=E
= −E
= −E
−p′ (YN )
p′ (YN )
N
−1
X
i=1 k=i+1
N
X
k=2
p′ (YN )
N
X
Z
N
(σk − σk−1 )ϕ(Sk−1
)1{Sk−1
N =X
σi }
N
(σk − σk−1 )ϕ(Sk−1
)
T
σ1
ϕ(SθN )
Z
t
Ms
0
Z
0
t
Z
k−1
X
Ms (
0
t
!
∂R
Ms
(σi , s)ds
∂s
!
∂R
(σi , s)1{Sk−1
N
=Xσi } )ds
∂s
i=1
!
∂R θ,N
(σ , s)dsdθ
∂s
+ RN ,
where
σ θ,N =
N k−1
X
X
k=1 i=0
σi 1(σk−1 ,σk ] (θ)1{max(Xσ0 ,...,Xσk−1 )=Xσi } ,
and the reminder term RN is given by
Z t
N
X
∂R
Ms
(0, s)ds .
RN = −ϕ(0)
(σk − σk−1 )E p′ (YN )1{max(Xσ0 ,...,Xσk−1 =0}
∂s
0
k=2
As N tends to infinity, RN converges to
Z T Z t
∂R
−ϕ(0)
E p′ (YN )1{Sθ =0}
Ms
(0, s)ds dθ = 0,
∂s
0
0
because Sθ has an absolutely continuous distribution for any θ > 0. On the other
hand, we claim that for all θ, σ θ,N converges to τSθ almost suterly as N goes to
infinite. This is a consequence of the fact that X is continuous and the maximum
is almost surely attained in a unique point by condition (H6). In addition, p′ (YN )
converges to p′ (Y ) and ϕ(SθN ) converges to ϕ(Sθ ) almost surely. Therefore, by
398
PEDRO LEI AND DAVID NUALART
Rt
Rt
∂R
condition (H7), 0 Ms ∂R
∂s (s, τSθN )ds converges pointwise to 0 Ms ∂s (s, τSθ )ds. On
the other hand, by condition (H1),
! β1
α ! α1
Z t
Z T
Z T
∂R
∂R
(τSnθ , s)ds ≤
Msβ ds
(s, t) ds
Ms
sup
,
∂s
∂s
0≤t≤T
0
0
0
so by the dominated convergence theorem, we obtain
!
Z T
Z t
∂R
′
E(p(Y )δt (M )) = −E p (Y )
ϕ(Sθ )
Ms
(τSθ , s)dsdθ .
∂s
0
0
Finally, the change of variable Sθ = y yields
Z t
Z
′
E(p(Y )δt (M )) = −E
Ms p (Y )
0
0
ST
!
∂R
(τy , s)dτy ds ,
ϕ(y)
∂s
which completes the proof of the lemma.
Proof of Proposition 6.1. Define
Z ∞
Z
Z 1
1 a
Yε,a =
ϕε (x − a)(τx ∧ T )dx =
(τx ∧ T )dx =
(τa−εξ ∧ T )dξ,
ε a−ε
0
0
where ϕε (x) = 1ε 1[−1,0] ( xε ), and by convention τx = 0 if x < 0. Lemma 6.4 can be
extended to the function x 7→ ϕε (x − a) and to the random variable Yε,a for any
fixed a. Therefore, from (6.3) and Lemma 6.4 we optain
Z ∞
E(pδ (Yε,a − t)Mt )dt
0
Z ∞
=1+λ
E(pδ (Yε,a − t)δ(M 1[0,t] ))dt
0
!
Z ∞
Z t
Z ST
∂R
Ms p′δ (Yε,a − t)
ϕε (y − a)
=1−λ
E
(τy , s)dτy ds dt
∂s
0
0
0
!
Z ∞
Z ST
∂R
=1−λ
E pδ (Yε,a − s)Ms
ϕε (y − a)
(τy , s)dτy ds,
(6.7)
∂s
0
0
where the last inequality holds by integration by parts. Multiplying by ψ(a) and
integrating with respect to the variable a yields
Z
Z ∞
ψ(a)
E(pδ (Yε,a − t)Mt )dtda
R
0
!
Z ∞
Z Z ST
∂R
dτy
pδ (Yε,a − s)Ms
= c − λE
(τy , s)ds ϕε (y − a)ψ(a)da
∂s
R 0
0
Z ∞
!
Z ST
Z
1 y+ε
∂R
= c − λE
dτy
ψ(a)da
pδ (Yε,a − s)Ms
(τy , s)ds
ε y
∂s
0
0
Z ∞ Z 1
!
Z ST
∂R
= c − λE
dτy
dηψ(y + εη)pδ (Yε,y−εη − s) Ms
(τy , s)ds
,
∂s
0
0
0
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
399
where the last equation holds by the change of variable a = y + εη. Next, consider
Z 1
Z η
Z 1
Yε,y+εη =
(τy+εη−εξ ∧ T )dξ =
(τy+εη−εξ ∧ T )dξ +
(τy+εη−εξ ∧ T )dξ.
0
0
η
Taking the limit as ε goes to zero, and using the fact that τ is left continuous and
with right limit, we obtain
Z η
Z η
(τy+ ∧ T )dξ = (τy+ ∧ T )η,
lim
(τy+εη−εξ ∧ T )dξ =
ε→0
lim
ε→0
0
Z
0
1
η
(τy−εη+εξ ∧ T )dξ =
Z
This implies that
Z 1
Z
lim
ψ(y + εη)pδ (Yε,y+εη − s)dη =
ε→0
0
1
η
(τy ∧ T )dξ = τy (1 − η).
1
0
ψ(y)pδ ((τy+ ∧ T )η + τy (1 − η) − s)dη.
This allows us to compute the limit of the right-hand side of Equation (6.7) as ε
tends to zero, using the dominated convergence theorem. In fact,
Z 1
ψ(y + εη)pδ (Yε,y+εη − s)dη ≤ K,
0
where K is a constant, and assuming supp(pδ ) ⊆ [0, T +δ], we have using condition
(H1),
!
Z T +δ Z ST
∂R
dτy
E
Ms ∂s (τy , s) ds
0
0

! β1 Z
! α1 
Z ST
Z T +δ
T +δ
∂R

≤E
dτy
|Ms |β ds
|
(τy , s)|α ds
∂s
0
0
0

! β1 
! α1
Z T +δ
Z T +δ
∂R
β
α
≤ TE 
|Ms |
ds sup
|
(z, s)| ds
< ∞.
∂s
z∈[0,T +δ]
0
0
On the other hand, we know that limε→0 Yε,a = τa ∧ T = τa since τa ≤ T .
Therefore,
Z
Z ∞
ψ(a)
E(pδ (τa − t)Mt )dtda
R
0
= c − λE
Z
ST
dτy
0
Z
1
ψ(y)dη
0
Z
0
∞
pδ ((τy+ ∧ T )η + τy (1 − η) − s)Ms
Finally, for the left hand side of (6.8) we have
Z
Z ∞
Z
lim
ψ(a)
E(pδ (τa − t)Mt )dtda =
δ→0
R
0
which implies the desired result.
∞
∂R
(τy , s)ds.
∂s
(6.8)
E(Mτa )ψ(a)da,
0
Proposition 6.1 implies the following inequalities which are the main result of
this section.
400
PEDRO LEI AND DAVID NUALART
Theorem 6.5. Assume that X satisfies (H1), (H3), (H5), (H6) and (H7).
(i) If ∂R
∂s (t, s) ≥ 0 for all s > t, then for all α, a > 0, we have
(ii) If
∂R
∂s (t, s)
E(exp(−αRτa ) ≤ e−a
√
2α
.
(6.9)
≤ 0 for all s > t, then for all α, a > 0, we have
E(exp(−αRτa )) ≥ e−a
Proof. If we assume
∂R
∂s (t, s)
√
2α
.
(6.10)
≥ 0, Proposition 6.1 implies
Z ∞
E(Mτa )ψ(a)da ≤ c.
0
Therefore, E(Mτa ) ≤ 1, namely,
1
E(exp(λa − λ2 Rτa )) ≤ 1,
2
for any λ > 0, which implies (6.9).
To show (ii), we choose pδ such that pδ (x − y) = 0 if x > y. Then, in the
integral with respect to ds appearing in the right-hand side of (6.8) we can assume
that s > (τy+ ∧ T )η + τy (1 − η) ≥ τy , which implies ∂R
∂s (τy , s) ≤ 0. Then,
Z ∞
E(Mτa )ψ(a)da ≥ c,
0
which allows us to conclude the proof as in the case (i).
Theorem 6.5 tells that the Laplace transform of the random variable Rτa can
be compared with the Laplace transform of the hitting time of the ordinary Brownian motion at the level a, under some monotonicity conditions on the covariance
function. This implies some consequences on the moments of Rτa . In the case (i),
the inequality (6.9) implies for any r > 0,
Z ∞
1
−r
E(Rτa ) =
E(e−αRτa )αr−1 dα
Γ(r) 0
Z ∞
√
2r Γ(r + 21 ) −2r
1
√
≤
e−a 2α αr−1 dα =
a .
(6.11)
Γ(r) 0
π
On the other hand, for 0 < r < 1,
Z ∞
r
r
E(Rτa ) =
(1 − E(e−αRτa ))α−r−1 dα
Γ(1 − r) 0
Z ∞
√
r
≥
(1 − e−a 2α )α−r−1 dα.
Γ(1 − r) 0
As a consequence, E(Rτra ) = +∞ for r ∈ ( 12 , 1).
In the case (ii), the inequality (6.10) implies for any r > 0
Z ∞
1
E(Rτ−r
)
=
E(e−αRτa )αr−1 dα
a
Γ(r) 0
Z ∞
√
2r Γ(r + 21 ) −2r
1
√
≥
e−a 2α αr−1 dα =
a .
Γ(r) 0
π
(6.12)
(6.13)
STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES
401
On the other hand, for 0 < r < 1,
E(Rτra )
Z ∞
r
(1 − E(e−αRτa ))α−r−1 dα
=
Γ(1 − r) 0
Z ∞
√
r
≤
(1 − e−a 2α )α−r−1 dα,
Γ(1 − r) 0
(6.14)
and, hence, E(Rτra ) < ∞ for r ∈ (0, 12 ).
Example 6.6. Consider the case of a fractional Brownian motion Hurst parameter
H > 12 . Recall that
1 2H
(t + s2H − |t − s|2H ).
2
Conditions (H5), (H6) and (H7) are satisfied. We can write
RH (t, s) =
∂RH
(t, s) = H(s2H−1 + sign(t − s)|t − s|2H−1 )
∂s
for all s, t ∈ [0, T ].
H
(t, s) ≥ 0 for all s, t, and by (6.9) in Theorem 6.5,
If H > 12 , then ∂R
∂s
√
2H
−a 2α
E(exp(−ατa )) ≤ e
. This implies that E(τap ) = +∞ for any H < p and τa
has finite negative moments of all order.
H
(t, s) ≤ 0 for s > t, and by (6.10) in Theorem 6.5,
If H < 21 , then ∂R
∂s
√
2H
−a 2α
E(exp(−ατa )) ≥ e
. This implies that E(τap ) < +∞ for any p < H.
In ([16]), Molchan proved that for the fractional Brownian motion with Hurst
parameter H ∈ (0, 1),
P (τa > t) = tH−1+o(1) ,
as t tends to infinity. As a consequence, E(τap ) > ∞ if p < 1 − H and E(τap ) = ∞
if p > 1 − H, which is stronger than the integrability results mentioned above.
Acknowledgment. We would like to thank an anonymous referee for this helpful
comments.
References
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os, E., Mazet, O. and Nualart, D.: Stochastic calculus with respect to Gaussian processes,
Ann. Probab. 29 (2001) 766–801.
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on Stoch. Anal. 5 (2011) 333–340.
3. Berman, B. M.: Local Times and Sample Function Properties of stationary Gaussian Processes, Trans. Amer. Math. Soc. 137 (1969) 277–299.
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motions and applications, Springer, 2008.
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fractional Brownian motion with Hurst parameter H < 12 , Ann. Institut Henri Poincar´
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(2005) 1049–1081.
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motion, Stoch. Process. Appl. 94 (2001) 301–315.
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(2008) 319–330.
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PEDRO LEI AND DAVID NUALART
¨ unel, A. S.: Stochastic analysis of the fractional Brownian motion,
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Potential Anal. 10 (1999) 177–214.
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o and
Tanaka formulas, Stoch. Dyn. 7 (2007) 365–388.
10. Houdr´
e, C. and Villa, J.: An example of infinite dimensional quasi-helix, Contemporary
Mathematics 366 (2003) 195–201.
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12. Kruk, I. and Russo, F.: Malliavin-Skorohod calculus and Paley-Wiener integral for covariance
singular processes. Preprint.
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structure measure, Journal of Functional Analysis 249 (2007) 92–142.
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applications, Statist. Probab. Lett. 79 (2009) 619–624.
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fractional Brownian scale, Journal of Functional Analysis 222 (2005) 385–434.
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44 (2000) 97–102.
17. Nualart, D.: The Malliavin Calculus and Related Topics (Probability and Its Applications),
2nd ed. Springer, 2006.
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Probab. Theory Related Fields 118 (2000) 251–291.
19. Pipiras, V. and Taqqu, M. S.: Are classes of deterministic integrands for fractional Brownian
motion on an interval complete, Bernoulli 6 (2001) 873–897.
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(2006) 830–856.
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Pedro Lei: Department of Mathematics, University of Kansas, Lawrnece, KS 66045
E-mail address: [email protected]
David Nualart: Department of Mathematics, University of Kansas, Lawrnece, KS
66045
E-mail address: [email protected]
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 403-407
www.serialspublications.com
AN ESTIMATE FOR BOUNDED SOLUTIONS OF THE
HERMITE HEAT EQUATION
BISHNU PRASAD DHUNGANA
Abstract. An estimate result on the partial derivatives of the Mehler kernel
E(x, ξ, t) for t > 0 is first established. Particularly for 0 < t < 1, it extends
the estimate result given by S. Thangavelu in his monograph A lecture notes
on Hermite and Laguerre expansions on the order of the partial derivative
of the Mehler kernel with respect to the space variable. Furthermore, for
∂ m U (x,t)
each m ∈ N0 , a growth estimate on the partial derivative
of all
∂xm
bounded solutions U (x, t) of the Cauchy Dirichlet problem for the Hermite
heat equation is established.
1. Introduction
As introduced in [1], we denote by E(x, ξ, t) the Mehler kernel defined by
P∞ −(2k+1)t
hk (x)hk (ξ), t > 0,
k=0 e
E(x, ξ, t) =
0,
t ≤ 0,
where hk ’s are L2 – normalized Hermite functions defined by
2
(−1)k ex /2 dk −x2
hk (x) = p
e
,
√
2k k! π dxk
x ∈ R.
Moreover the explicit form of E(x, ξ, t) for t > 0 is
− 21
E(x, ξ, t) =
e−t e
1+e−4t
1−e−4t
√
−2t
(x−ξ)2 − 1−e−2t xξ
1+e
1
π(1 − e−4t ) 2
.
We note that for each ξ ∈ R, E(x, ξ, t) satisfies the Hermite heat equation. In
(Theorem 3.1, [2]), we proved that
Z ∞
U (x, t) =
{E(x, ξ, t) − E(x, −ξ, t)} φ(ξ)dξ
(1.1)
0
is a unique bounded solution of the following Cauchy Dirichlet problem for the
Hermite heat equation

∂
∂2
2
 ( ∂t
− ∂x
x > 0, t > 0,
2 + x )U (x, t) = 0,
(1.2)
U (x, 0) = φ(x),
x > 0,

U (0, t) = 0,
t > 0,
Received 2012-1-10; Communicated by K. Saitˆ
o.
2000 Mathematics Subject Classification. Primary 33C45; Secondary 35K15.
Key words and phrases. Hermite functions, Mehler kernel, Hermite heat equation.
403
404
BISHNU PRASAD DHUNGANA
where φ is a continuous and bounded function on [0, ∞) with φ(0) = 0.
It is not necessary that every bounded solution of the Hermite heat equation
should satisfy a fixed growth behavior on its mth partial derivative with respect to
the space variable. However, since the solution U (x, t) in (1.1) is a unique solution
of (1.2), it is natural to make an effort for obtaining a fixed growth estimate
m
U (x,t)
on ∂ ∂x
. But it is not as easy as we anticipate. To find a growth estimate
m
m
m
∂ U (x,t)
on ∂xm , we require first to obtain an estimate on ∂ E(x,ξ,t)
. Note that an
∂xm
estimate on the partial derivatives of the heat kernel
(
1
x2
(4πt)− 2 e− 4t , t > 0,
E(x, t) =
0,
t ≤ 0,
with respect to the space variable has been given in [3]:
m
∂ E(x, t) (1+m)
1
ax2
m − 2
m! 2 e− 4t ,
∂xm ≤ C t
t > 0,
where C is some constant and a can be taken as close as desired to 1 such that
0 < a < 1.
Though the estimates of the following types on the Mehler kernel for 0 < t < 1
and B independent of x, ξ and t
∂E(x, ξ, t) ≤ Ct−1 e− Bt |x−ξ|2 ,
(1.3)
∂x
2
∂ E(x, ξ, t) 2
− 32 − B
e t |x−ξ| ,
∂x ∂ξ ≤ Ct
are provided in [4], the estimate on the partial derivatives of the Mehler kernel of
all order with respect to the space variable is yet to be established.
m
Lemma 2.1 that gives an estimate on ∂ E(x,ξ,t)
for each nonnegative integer m,
∂xm
is therefore a novelty of this paper which as an application yields
m
∂ U (x, t) m
≤ M in [0, ∞) × [0, ∞)
t 2 e−mt ∂xm for some constant M , the main objective and the final part of this paper.
2. Main Results
Lemma 2.1. Let E(x, ξ, t) be the Mehler kernel and m ∈ N0 . Then for some
constants a with 0 < a < 1 and A := A(a) > 0
√
m
(x−ξ)2
∂ E(x, ξ, t) m! e(A+1)m emt − ae−2t
≤ √
1−e−4t
.
m
m+1 e
1+
m
∂x
π2 2 t 2
AN ESTIMATE FOR BOUNDED SOLUTIONS OF THE HERMITE HEAT EQUATION 405
Proof. By the Cauchy integral formula, we have
∂ m E(x, ξ, t)
m
∂xZ
m!
E(ζ, ξ, t)
=
dζ
2πi ΓR (ζ − x)m+1
−4t
−2t
Z
− 1 1+e
(ζ−ξ)2 − 1−e−2t ζξ
1+e
m!
e−t e 2 1−e−4t
=
dζ,
3
1
2π 2 i ΓR (ζ − x)m+1 (1 − e−4t ) 2
where ΓR is a cirle of radius R in the complex plane C with center at x. With
ζ = x + Reiθ , we have
m
Z 2π
−4t
−2t
∂ E(x, ξ, t) m!e−t
(x−ξ+Reiθ )2 − 1−e−2t (x+Reiθ )ξ
− 1 1+e
≤
1+e
√
e 2 1−e−4t
dθ.
3
m
∂x
2π 2 Rm 1 − e−4t 0
Then, writing S for x + R cos θ, we have
m
Z 2π −
∂ E(x, ξ, t) m! e−t
e
≤
2π 32 Rm √1 − e−4t
∂xm
0
−4t
1+e
Let P = 12 1−e
−4t and Q =
using the inequality
1−e−2t
1+e−2t .
1 1+e−4t
2 1−e−4t
−2t
{ξ−S}2 + 1−e−2t ξS
− 12
e
i
1+e
1+e−4t
1−e−4t
dθ.
R2
Then P > 0 and Q > 0 since t is positive. Now
2
P {ξ − (x + R cos θ)} + Qξ(x + R cos θ) ≥
we have
m
∂ E(x, ξ, t) ≤
∂xm
h
Q
P−
2
2
{ξ − (x + R cos θ)} ,
Z 2π
−4t
−2t
m! e−t
− e
(x−ξ+R cos θ)2 + 12 1+e−4t R2
1−e
√
e 1−e−4t
dθ
3
2π 2 Rm 1 − e−4t 0
−4t
−2t
m! e−t
x
˜2 + 12 1+e−4t R2
− e
1−e
,
e 1−e−4t
√ m√
πR
1 − e−4t
≤
exp
1 1+e−4t
2 1−e−4t
Rm
R2
where x
˜ = x − ξ − R or 0 or x − ξ + R. Since the ratio
attains its
pm
1 1+e−4t
minimum at R = 2b where b = 2 1−e−4t , we have
m
m
m
∂ E(x, ξ, t) e−2t
m! e−t e 2
1 + e−4t 2 − 1−e
˜2
−4t x
≤ √ √
e
. (2.1)
m
m
−4t
−4t
∂x
1−e
π 1−e m2
But with 0 < a < 1 and |β| ≤ 1
−
e
e−2t
1−e−4t
(x−ξ+βR)2
−
ae−2t
1−e−4t
(x−ξ)2 −
e−2t
1−e−4t
−
ae−2t
1−e−4t
(x−ξ)2 −
(1−a)e−2t
1−e−4t
−
ae−2t
1−e−4t
(x−ξ)2
= e
= e
≤ e
where A =
a
1−a .
e
e
Ae−2t
e 1−e−4t
[(1−a)(x−ξ)2 +2(x−ξ)βR+β 2 R2 ]
R2
h
i
βR 2
aβ 2 R2
(x−ξ+ 1−a
) − (1−a)
2
,
Then clearly
−
e
e−2t
1−e−4t
x
˜2
−
≤e
ae−2t
1−e−4t
(x−ξ)2
Ae−2t
e 1−e−4t
R2
.
406
BISHNU PRASAD DHUNGANA
−4t
−2t
−4t
2t
)
e
1 1+e
e
Using R2 = m(1−e
and the inequalities 1+e
−4t ≤ 2 , 1−e−4t ≤ 2t for every
1+e−4t
t > 0, (2.1) reduces to
√
m
ae−2t (x−ξ)2
∂ E(x, ξ, t) m! e(A+1)m
e−t
emt
−
≤
1−e−4t
√
√
e
. (2.2)
m
m
∂xm
π
1 − e−4t 2 2 t 2
Furthermore, since
−t
√ e
1−e−4t
≤
1
√
2 t
m
∂ E(x, ξ, t) ≤
∂xm
for every t > 0 we obtain
√
(x−ξ)2
m! e(A+1)m emt − ae−2t
1−e−4t
.
√ 1+ m
m+1 e
π2 2 t 2
(2.3)
This completes the proof.
−2t
e
1
Remark 2.2. For 0 < t < 1, in view of (2.3) and − 1−e
−4t ≤ − 8t it is easy to see
that
√
m
(A+3)m
∂ E(x, ξ, t) a(x−ξ)2
≤ √ m! e
e− 8t
m m+1
m
1+
∂x
π2 2t 2
which extends the estimate result (1.3) on the order m > 1 of the partial derivative
of E(x, ξ, t) with respect to the variable x.
Theorem 2.3. Every bounded solution of the Cauchy Dirichlet problem for the
Hermite heat equation

∂2
∂
2
 ( ∂t
− ∂x
x > 0, t > 0,
2 + x )U (x, t) = 0
(2.4)
U (x, 0) = φ(x)
x > 0,

U (0, t) = 0
t > 0,
satisfies the following growth estimate
m
∂ U (x, t) m
≤ M,
t 2 e−mt ∂xm in [0, ∞) × [0, ∞),
where m ∈ N0 and M is some constant.
Proof. From (Theorem 3.1, [2]), every bounded solution of the Cauchy Dirichlet
problem (2.4) for the Hermite heat equation is of the form
Z ∞
U (x, t) =
{E(x, ξ, t) − E(x, −ξ, t)} φ(ξ)dξ,
0
where φ is a continuous and bounded function on [0, ∞) with φ(0) = 0 and
E(x, ξ, t), the Mehler kernel. We write
Z ∞
{E(x, ξ, t) − E(x, −ξ, t)} φ(ξ)dξ
U (x, t) =
0
Z
=
E(x, ξ, t)h(ξ)dξ,
R
where
h(ξ) =
φ(ξ),
ξ ≥ 0,
−φ(−ξ), ξ < 0.
AN ESTIMATE FOR BOUNDED SOLUTIONS OF THE HERMITE HEAT EQUATION 407
From (2.2), we have
m
∂ U (x, t) ∂xm ≤
Z m
∂ E(x, ξ, t) |h(ξ)|dξ
∂xm
R
√
Z
ae−2t (x−ξ)2
khk∞ m!e(A+1)m e(m−1)t
−
1−e−4t
e
dξ.
≤
√ √
m
π 1 − e−4t (2t) 2
R
√
−t
a e
Under the change of variable √1−e
(ξ − x) = s and integrating, we have
−4t
√
m
∂ U (x, t) khk∞ m!e(A+1)m emt
≤
m√
m .
∂xm 22 a
t2
Clearly
m
∂ U (x, t) m
≤ M in [0, ∞) × [0, ∞)
t 2 e−mt ∂xm if we take M =
khk∞
√
m!e(A+1)m
.
m√
22 a
References
1. Dhungana, B. P.: An example of nonuniqueness of the Cauchy problem for the Hermite heat
equation, Proc. Japan. Acad. 81, Ser. A, no. 3 (2005), 37–39.
2. Dhungana, B. P. and Matsuzawa, T.: An existence result of the Cauchy Dirichlet problem
for the Hermite heat equation, Proc. Japan. Acad. 86, Ser. A, no. 2 (2010), 45–47.
3. Matsuzawa, T.: A calculus approach to hyperfunctions. II, Trans. Amer. Math. Soc. 313,
No. 2 (1989), 619–654.
4. Thangavelu, S.: Lectures on Hermite and Laguerre Expansions, Princeton University Press,
Princeton, 1993.
Bishnu Prasad Dhungana: Department of Mathematics, Mahendra Ratna Campus,
Tribhuvan University, Kathmandu Nepal
E-mail address: [email protected]
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 409-419
www.serialspublications.com
FEYNMAN-KAC FORMULA FOR THE SOLUTION OF
CAUCHY’S PROBLEM WITH TIME DEPENDENT
´
LEVY
GENERATOR
´
AROLDO PEREZ
Abstract. We exploit an equivalence between the well-posedness of the homogeneous Cauchy problem for a time dependent L´
evy generator L, and the
well-posedness of the martingale problem for L, to obtain the Feynman-Kac
representation of the solution of
∂u (t, x)
∂t
=
u (0, x)
=
L (t) u (t, x) + c(t, x)u (t, x) , t > 0, x ∈ Rd ,
ϕ (x) , ϕ ∈ C02 Rd ,
where c is a bounded continuous function.
1. Introduction
Solutions of many partial differential equations can be represented as expectation functionals of stochastic processes known as Feynman-Kac formulas; see [8],
[12] and [13] for pioneering work of these representations.
Feynman-Kac formulas are useful to investigate properties of partial differential equations in terms of appropriate stochastic models, as well as to study probabilistic properties of Markov processes by means of related partial differential
equations; see e.g. [9] for the case of diffusion processes. Feynman-Kac formulas
naturally arise in the potential theory for Sch¨
odinger equations [4], in systems of
relativistic interacting particles with an electromagnetic field [11], and in mathematical finance [6], where they provide a bridge between the probabilistic and
the PDE representations of pricing formulae. In recent years there has been a
growing interest in the use of L´evy processes to model market behaviours (see
e.g. [1] and references therein). This leads to consider Feynman-Kac formulas
for Cauchy’s problems with L´evy generators. Also, Feynman-Kac representations
have been used recently to determine conditions under which positive solutions
of semi-linear equations exhibit finite-time blow up, see [2] for the autonomous
case and [14] and [17] for the nonautonomous one. A well-known reference on
the interplay between the Cauchy problem for second-order differential operators,
and the martingale problem for diffusion processes is [20]. In particular, in [20]
it is proved that existence and uniqueness for the Cauchy problem associated to
Received 2012-1-31; Communicated by V. Perez-Abreu.
2000 Mathematics Subject Classification. 60H30, 60G51, 60G46, 35K55.
Key words and phrases. Feynman-Kac formula, L´
evy generator, martingale problem, Cauchy
problem.
409
´
AROLDO PEREZ
410
a diffusion operator is equivalent to existence and uniqueness of the martingale
problem for the same operator.
The purpose of this paper is to prove that such an equivalence also holds for
equations with L´evy generators and their corresponding martingale problems, and
to provide a Feynman-Kac
representation of the solution of the Cauchy problem.
Let ϕ ∈ C02 Rd be the space of C 2 -functions ϕ : Rd → R, vanishing at infinite.
The L´evy generators we consider here are given by the expresion
L (t) ϕ (x)
=
d
d
1 X
∂ 2 ϕ (x) X
∂ϕ (x)
+
(1.1)
aij (t, x)
bi (t, x)
2 i,j=1
∂xi ∂xj
∂xi
i=1
!
Z
hy, ∇ϕ (x)i
+
ϕ (x + y) − ϕ (x) −
µ (t, x, dy) ,
2
1 + |y|
Rd
d
P
t ≥ 0, ϕ ∈ C02 Rd , where hx, yi =
xi yi , a : [0, ∞) × Rd → Sd+ . Here Sd+
i=1
is the space of symmetric, non-negative definite, square real matrices of order d,
b : [0, ∞) × Rd → Rd and µ (t, x, ·) is a L´evy measure,
that is, µ (t, x, ·) is a σ-finite
R
measure on Rd that satisfies µ (t, x, {0}) = 0 and Rd |y|2 (1+|y|2 )−1 µ(t, x, dy) < ∞
for all t ≥ 0 and x ∈ Rd . These operators represent the infinitesimal generators
of the most general stochastically continuous, Rd -valued Markov processes with
independent increments. We establish the equivalence (see Theorem 4.3 below)
between the existence and uniqueness of solutions of the homogeneous Cauchy
problem
∂u (t, x)
∂t
u (s, x)
= L (t) u (t, x) , t > s ≥ 0, x ∈ Rd ,
= ϕs (x) , ϕs ∈ C02 Rd ,
(1.2)
and that of solutions of the martingale problem for {L (t)}t≥0 on C02 Rd . In
order to achieve this, we use some ideas introduced in [20], several properties
of the Howland semigroup (see e.g. [3]), and some results about the martingale
problem given in [7]. By means of this equivalence, using the Howland evolution
semigroup (whose definition is based on the classical idea of considering “time”to
be a new variable in order to transform a nonautonomous Cauchy problem into an
autonomous one) and Theorem 9.7, p. 298 from [5] (in the autonomous case), we
are able to prove (see Theorem 5.1 below) that the solution of the Cauchy problem
(5.1), given below, admits the representation
Z t
u(t, x) = Ex ϕ(X(t)) exp
c(t − s, X(s))ds ,
0
where X ≡ {X(t)}t≥0 is a strong Markov process on Rd with respect to the
filtration GtX = FtX+ , which is right continuous and quasi-left
continuous, and
2
d
solves the martingale problem for {L (t)}t≥0 on C0 R . Here Ex denotes the
expectation with respect to the process X starting at x.
FEYMAN-KAC FORMULA
411
2. Non-negativity of Solutions
Let us consider the L´
evy generator defined in (1.1). It is known (see e.g. [18])
that the space Cc∞ Rd of continuous functions g : Rd → R, having compact support and possessing continuous derivatives of all orders, is a core for the
common
domain D of the family of linear operators {L(t)}t≥0 , and that C02 Rd ⊂ D.
Notice that {L (t)}t≥0 satisfies the positive maximun principle, namely,
If ϕ (x) = sup ϕ (y) ≥ 0, ϕ ∈ D, then L (t) ϕ (x) ≤ 0 for all t ≥ 0.
y∈Rd
In fact, if ϕ (x) = sup ϕ (y) ≥ 0, then ∇ϕ (x) = 0 and
y∈Rd
∂ 2 ϕ(x)
∂xi ∂xj
(2.1)
is a
1≤i,j≤d
symmetric non-positive definite matrix. Since
1
L (t) ϕ (x) =
trace (a (t, x) Hϕ (x)) + hb (t, x) , ∇ϕ (x)i
2
!
Z
hy, ∇ϕ (x)i
+
ϕ (x + y) − ϕ (x) −
µ (t, x, dy) ,
2
1 + |y|
Rd
where b (t, x) = (bi (t, x))1≤i≤d , a (t, x) = (aij (t, x))1≤i,j≤d and Hϕ (x) is the
Hessian matrix, we have
L (t) ϕ (x) ≤ 0,
because trace (AB) ≤ 0 if A ∈ Sd+ , B ∈ Sd− , and by assumption ϕ (x + y) −
ϕ (x) ≤ 0 for all y ∈ Rd .
We note also that (2.1) implies that L (t) ϕ ≤ 0 for any nonnegative constant
function ϕ ∈ D. In fact, here L (t) ϕ = 0 for all functions which do not depend on
space.
Let us now turn to the differential equation (1.2).
We are going to assume that the homogeneous Cauchy problem (1.2) is wellposed, and that the evolution family {U (t, s)}t≥s≥0 that solves (1.2) is an evolution
family of contractions, i.e., a family of operators on C0 Rd such that
(i) U (t, s) = U (t, r)U (r, s) and U (t, t) = I for all t ≥ r ≥ s ≥ 0 (here I is the
identity operator).
(ii) For each ϕ ∈ C0 Rd , the function (t, s) 7→ U (t, s)ϕ is continuous for t ≥ s ≥ 0.
(iii) kU (t, s)k ≤ 1 for all t ≥ s ≥ 0.
Proposition
2.1. Assume that u is a classical solution of (1.2)
such that u (t, ·) ∈
C02 Rd for all t ≥ s ≥ 0, and that 0 ≤ ϕs ≡ ϕ ∈ C02 Rd . Then u (t, x) ≥ 0,
t ≥ s ≥ 0, x ∈ Rd .
Proof. Suppose that for some r > s,
inf u (r, x) = −c < 0.
x∈Rd
Fix T > r and let δ > 0 be such that δ <
c
T −s .
Define on [s, T ] × Rd the function
vδ (t, x) = u (t, x) + δ (t − s) ,
which coincides with u when t = s. Clearly, this function has a negative infimun
on [s, T ] × Rd and tends to the positive constant δ (t − s) as |x| → ∞ and t ∈
´
AROLDO PEREZ
412
(s, T ] is fixed. Consequently, vδ has a global (negative) minimun at some point
(t0 , x0 ) ∈ (s, T ] × Rd . This implies that
∂vδ
(t0 , x0 ) ≤ 0,
∂t
and by the positive maximun principle (2.1),
L (t) vδ (t0 , x0 ) ≥ 0.
Therefore
∂vδ
− L (t) vδ (t0 , x0 ) ≤ 0.
∂t
On the other hand, since u solves (1.2),
∂u
∂vδ
− L (t) vδ (t0 , x0 ) =
− L (t) u (t0 , x0 ) + δ − (t − s) (L (t) δ) (x0 ) ≥ δ.
∂t
∂t
This contradiction proves the proposition.
Corollary 2.2. The differential
equation (1.2) can have at most one solution u
such that u (t, ·) ∈ C02 Rd , t ≥ s ≥ 0.
3. Markov Process Associated to the Evolution Family
From
Proposition 2.1 we deduce that U (t, s) is a nonnegative contraction on
C0 Rd for t ≥ s ≥ 0, i.e., {U (t, s)}t≥s≥0 is an evolution family of contractions
such that U (t, s)ϕ
≥ 0 for each ϕ ≥ 0. This follows from the fact that C02 Rd is
dense in C0 Rd , and that, by definition of {U (t, s)}t≥s≥0 , for any ϕs ∈ C02 Rd
the function
u (t, x) = U (t, s) ϕs (x)
is a solution of (1.2) such that u (t, ·) ∈ C02 Rd . Thus, by Riesz representation
theorem for nonnegative functionals on C0 Rd (see e.g. [21], p. 5), we have that
for each t ≥ s ≥ 0 and x ∈ Rd , there exists a measure P (s, x, t, ·) on the Borel
σ-field B Rd on Rd , such that P s, x, t, Rd ≤ 1 and
Z
U (t, s) ϕ (x) =
ϕ (y) P (s, x, t, dy) , ϕ ∈ C0 Rd .
Rd
Since {U (t, s)}t≥s≥0 is an evolution family, in order to prove that P (s, x, t, Γ) ,
t ≥ s ≥ 0, x ∈ Rd , Γ ∈ B Rd , is a transition probability function it suffices to
d
show that P (s, x, t, ·) is a probability measure for all t ≥
s ≥ 0, x ∈ R . To this
d
end, from the evolution family {U (t, s)}t≥s≥0 on C0 R , we define the family of
operators {T (t)}t≥0 on C0 [0, ∞) × Rd by
U (r, r − t) f (r − t, x) , r > t ≥ 0, x ∈ Rd ,
(T (t) f ) (r, x) =
(3.1)
U (r, 0) f (0, x) , 0 ≤ r ≤ t.
Notice that {T (t)}t≥0 is a positivity-preserving, strongly continuous semigroup
of contractions on C0 [0, ∞) × Rd , which is called the Howland semigroup of
FEYMAN-KAC FORMULA
413
{U (t, s)}t≥s≥0 . Let us denote by A the infinitesimal generator of {T (t)}t≥0 , and
define the operator Aˆ by
ˆ (r, x) = − ∂f (r, x) + L (r) f (r, x) , r ≥ 0, x ∈ Rd ,
Af
∂r
(3.2)
whose domain is the space of functions f which
are differentiable at t, such that
ˆ ∈ C0 [0, ∞) × Rd . Let us denote by D the linear span
f (t, ·) ∈ C02 Rd and Af
of all functions f ∈ C0 [0, ∞) × Rd of the form
f (t, x) ≡ fα,ϕ (t, x) = α (t) ϕ (x) , α ∈ Cc1 ([0, ∞)) and ϕ ∈ C02 Rd ,
(3.3)
where Cc1 ([0, ∞)) is the space of continuous functions on [0, ∞) having compact
support and continuous first derivative. Then
(T (t) fα,ϕ ) (r, x) =
α (r − t) U (r, r − t) ϕ (x) , r > t ≥ 0, x ∈ Rd ,
α (0) U (r, 0) ϕ (x) , 0 ≤ r ≤ t, x ∈ Rd .
Thus,
d
(T (t) fα,ϕ ) (r, x) |t=0
dt
= −α0 (r) ϕ (x) + α (r) L (r) ϕ (x)
∂fα,ϕ (r, x)
= −
+ L (r) fα,ϕ (r, x)
∂r
ˆ α,ϕ (r, x) .
=
Af
(Afα,ϕ ) (r, x)
=
(3.4)
Since D is dense in C0 [0, ∞) × Rd , this proves that the operator A is the closure
of Aˆ |D .
Notice that the infinitesimal generator A of the Howland semigroup {T (t)}t≥0 is
conservative. Hence {T (t)}t≥0 is a Feller semigroup, i.e. {T (t)}t≥0 is a positivity
preserving, strongly continuous semigroup of contractions on C0 [0, ∞) × Rd ,
whose infinitesimal generator is conservative. Therefore (see [7], Theorem 2.7, p.
169), there exists a time-homogeneous Markov process {Z (t)}t≥0 with state space
[0, ∞) × Rd and sample paths in the Skorohod space D[0,∞)×Rd [0, ∞), such that
Z
f (υ, y) Pe (t, (r, x) , d (υ, y)) , f ∈ C0 [0, ∞) × Rd ,
(T (t) f ) (r, x) =
[0,∞)×Rd
e , t ≥ 0, (r, x) ∈ [0, ∞) × Rd , Γ
e ∈ B [0, ∞) × Rd , is a
where Pe t, (r, x) , Γ
transition probability function for {Z (t)}t≥0 . Recalling that
Z
U (t, s) ϕ (x) =
Rd
ϕ (y) P (s, x, t, dy) , ϕ ∈ C0 Rd ,
´
AROLDO PEREZ
414
we obtain, by definition of {T (t)}t≥0 , that for any r > t ≥ 0,
Z
f (υ, y) Pe (t, (r, x) , dυdy)
[0,∞)×Rd
Z
=
f (r − t, y) P (r − t, x, r, dy)
d
ZR
=
f (υ, y) P (r − t, x, r, dy) δr−t (dυ) , f ∈ C0 [0, ∞) × Rd ,
[0,∞)×Rd
where δl denotes the measure with unit mass at l, and for any 0 ≤ r ≤ t,
Z
f (υ, y) Pe (t, (r, x) , dυdy)
[0,∞)×Rd
Z
=
f (0, y) P (0, x, r, dy)
Rd
Z
=
f (υ, y) P (0, x, r, dy) δ0 (dυ) , f ∈ C0 [0, ∞) × Rd .
[0,∞)×Rd
Therefore
P (r − t, x, r, Γ) δr−t (C) , if r > t ≥ 0,
(3.5)
P (0, x, r, Γ) δ0 (C) , if 0 ≤ r ≤ t,
where C ∈ B ([0, ∞)) and Γ ∈ B Rd .
Since Pe (t, (r, x) , ·) is a probability measure on [0, ∞) × Rd , B [0, ∞) × Rd ,
it follows from (3.5) that P (s, x, t, ·) is a probability measure on Rd , B Rd .
Thus, if there exists an evolution family of contractions {U (t, s)}t≥s≥0 on C0 Rd
that solves the homogeneous Cauchy problem (1.2), then there also exists a transition probability function
P (s, x, t, Γ) , t ≥ s ≥ 0, x ∈ Rd , Γ ∈ B Rd ,
(3.6)
Pe (t, (r, x) , C × Γ) =
such that
Z
U (t, s) ϕ (x) =
Rd
ϕ (y) P (s, x, t, dy) , ϕ ∈ C0 Rd .
4. The Cauchy Problem and the Martingale Problem
Let {X (t)}t≥0 be an Rd -valued Markov process with sample paths in DRd [0, ∞),
and with the transition probability function given by (3.6) (see [10], Theorem 3,
p. 79).
Lemma 4.1. For any ϕ ∈ C02 Rd and s ≥ 0,
Z t
M (t) = ϕ (X (t)) −
L (υ) ϕ (X (υ)) dυ, t ≥ s,
s
is a martingale after time s with respect to the filtration FtX = σ (X (s) , s ≤ t).
FEYMAN-KAC FORMULA
415
Proof. Let ϕ ∈ C02 Rd and t > r ≥ s. Then, almost surely,
E M (t) | FrX
Z
Z tZ
=
ϕ (y) P (r, X (r) , t, dy) −
L (υ) ϕ (y) P (r, X (r) , υ, dy) dυ
Rd
r
Rd
Z r
−
L (υ) ϕ (X (υ)) dυ
s
Z t
Z r
U (υ, r) L (υ) ϕ (X (r)) dυ −
L (υ) ϕ (X (υ)) dυ
= U (t, r) ϕ (X (r)) −
r
s
Z t
Z r
∂U (υ, r) ϕ (X (r))
= U (t, r) ϕ (X (r)) −
dυ −
L (υ) ϕ (X (υ)) dυ
∂υ
r
s
Z r
= U (t, r) ϕ (X (r)) − [U (t, r) ϕ (X (r)) − ϕ (X (r))] −
L (υ) ϕ (X (υ)) dυ
s
Z r
= ϕ (X (r)) −
L (υ) ϕ (X (υ)) dυ = M (r) .
s
Let {Y (t)}t≥0 be the time-homogeneous Markov process on [0, ∞) with transition probabilities
P (t, r, Γ) = δr−t (Γ) , Γ ∈ B ([0, ∞)) , t, r ≥ 0.
Then the Markov semigroup {V (t)}t≥0 associated to {Y (t)}t≥0 is given by
V (t) f (r) = f (r − t) , r ≥ t ≥ 0,
and its generator Q satisfies
Qf = −f 0
on the space D (Q) = Cc1 ([0, ∞)).
Lemma 4.2. Let {X (t)}t≥0 and {Y (t)}t≥0 be as above. Then
{(Y (t) , X (t))}t≥0
is a Markov process with values in [0, ∞) × Rd , wich has the same distribution as
the Markov process {Z (t)}t≥0 whose semigroup is given by (3.1).
Proof. We consider the space of functions D defined in (3.3). Since the Markov
process {Y (t)}t≥0 is a solution of the martingale problem for Q and, by Lemma
4.1, the Markov process {X (t)}t≥0 is a solution of the martingale problem for
{L (t)}t≥0 on C02 Rd , we have (see [7], Theorem 10.1, p. 253) that the Markov
process {(Y (t) , X (t))}t≥0 is a solution of the martingale problem for Aˆ |D , where
Aˆ is the operator defined in (3.2). This implies (see [7], Theorem 4.1, p. 182) that
the processes {Z (t)}t≥0 and {(Y (t) , X (t))}t≥0 have the same finite-dimensional
distributions. But, since these processes have sample paths in D[0,∞)×Rd [0, ∞),
it follows (see [7], Corollary 4.3, p. 186) that they have the same distribution on
D[0,∞)×Rd [0, ∞).
´
AROLDO PEREZ
416
Theorem 4.3. There exists an evolution family of contractions {U (t, s)}t≥s≥0
on C0 Rd , which is unique and solves the homogeneous Cauchy problem (1.2), if
and only if, there exists a Markov process {X (t)}t≥0 on Rd , unique in distribution,
that solves the martingale problem for {L (t)}t≥0 on C02 Rd .
Proof. The necessity follows from Lemma 4.1. Assume that {X (t)}t≥0 is a Markov
process on Rd , unique
in distribution, that solves the martingale problem for
{L (t)}t≥0 on C02 Rd . Let P (s, x, t, Γ) be a transition function for {X (t)}t≥0 ,
and let
Z
U (t, s) f (x) =
P (s, x, t, dy) f (y) , t ≥ s ≥ 0, f ∈ C0 Rd .
Rd
Then {U (t, s)}t≥s≥0 is a positivity-preserving evolution family of contractions on
C0 Rd . Let {T (t)}t≥0 be the semigroup on C0 [0, ∞) × Rd defined from the
evolution family {U (t, s)}t≥s≥0 on C0 Rd by (3.1). Then, for ϕ ∈ C02 Rd and
α ∈ Cc1 ([s, ∞)) satisfying α (s) = 1,
u (t, x) ≡ U (t, s) ϕ (x) = (T (t − s) α (·) ϕ (·)) (t, x) , t ≥ s.
Hence, due to (3.2), (3.4) and the strong continuity of the semigoup {T (t)}t≥0 ,
∂u (t, x)
∂t
(T (t − s + h) α (·) ϕ (·)) (t + h, x) − (T (t − s) α (·) ϕ (·)) (t, x)
h→0
h
(T (t − s + h) α (·) ϕ (·)) (t + h, x) − (T (t − s + h) α (·) ϕ (·)) (t, x)
= lim
h→0
h
(T (t − s + h) α (·) ϕ (·)) (t, x) − (T (t − s) α (·) ϕ (·)) (t, x)
+ lim
h→0
h
(T (t − s) α (·) ϕ (·)) (t + h, x) − (T (t − s) α (·) ϕ (·)) (t, x)
= lim T (h)
h→0
h
ˆ
+A (T (t − s) α (·) ϕ (·)) (t, x)
=
lim
=
L (t) (T (t − s) α (·) ϕ (·)) (t, x) = L (t) U (t, s) ϕ (x) .
This shows clearly that the positivity-preserving evolution family of contractions
{U (t, s)}t≥s≥0 solves the homogeneous Cauchy problem (1.2). Uniqueness follows
from Corollary 2.2.
Conditions on the functions a : [0, ∞) × Rd → Sd+ , b : [0, ∞) × Rd → Rd
and the L´evy measure µ (t, x, Γ) that ensure existence of a unique Markov process
{X (t)}t≥0 on Rd that solves the martingale problem for {L (t)}t≥0 on Cc∞ Rd ,
are given in [15] and [19]. In particular, in [19] it is proved that if a : [0, ∞)×Rd →
Sd+ and b : [0, ∞) × Rd → Rd are bounded and continuous, and µ (t, x, ·) is a L´evy
−1
R
2
2
measure such that Γ |y| 1 + |y|
µ (t, x, dy) is bounded and continuous in
d
(t, x) for every Γ ∈ B R \ {0} , then there exists a unique (in law) strong Markov
process {X (t)}t≥0 on Rd that solves the martingale problem for {L (t)}t≥0 on
Cc∞ Rd .
FEYMAN-KAC FORMULA
417
5. The Feynman-Kac Formula
Assume now that there exists a Markov process X = {X (t)}t≥0 on Rd with
respect to the filtration GtX = FtX+ , right continuous and quasi-left continuous, that
solves the martingale problem for {L (t)}t≥0 on C02 Rd . Then, by Theorem 4.3,
there exists a unique evolution family of contractions {U (t, s)}t≥s≥0 on C0 Rd ,
that solves the homogeneous Cauchy problem (1.2).
We consider the Cauchy problem
∂u (t, x)
= L (t) u (t, x) + c(t, x)u (t, x) , t > 0, x ∈ Rd ,
∂t
u (0, x) = ϕ (x) , ϕ ∈ C02 Rd ,
(5.1)
where c is a given bounded continuous function. Any classical solution u (t, x) of
(5.1) satisfies the integral equation
Z t
u (t, x) = U (t, 0) ϕ(x) +
U (t, r)(cu)(r, x)dr.
(5.2)
0
Any solution of the integral equation (5.2) is called a mild solution of the Cauchy
problem (5.1).
Let us consider now the Cauchy problem
∂u (t, (r, x))
= Au (t, (r, x)) + c(r, x)u (t, (r, x)) ,
∂t
u (0, (r, x)) = ϕ(r,
e x),
(5.3)
where ϕ
e ∈ C0 ((0, ∞) × Rd ), is such that ϕ(0,
e x) = ϕ(x), x ∈ Rd . As before, the
classical solution of (5.3) satisfies the integral equation
Z t
u (t, (s, x)) = T (t)ϕ
e (s, x) +
T (t − r)(cu(r, ·)(s, x)dr.
(5.4)
0
Let u(t, (s, x)) be a solution of the integral equation (5.4). Using the definition of
T (t) given in (3.1) and the assumption that ϕ
e (0, x) = ϕ(x), x ∈ Rd , we obtain
Z t
u (t, (t, x)) = T (t)ϕ
e (t, x) +
T (t − r)(cu(r, ·))(t, x)dr
0
Z t
= U (t, 0)ϕ(x) +
U (t, r)(cu(r, ·))(r, x)dr.
0
Hence, u(t, x) ≡ u(t, (t, x)), t ≥ 0, satisfies the integral equation (5.2), i.e. u(t, x)
is the mild solution of the Cauchy problem (5.1).
Theorem 5.1. Let X = {X (t)}t≥0 be a strong Markov process on Rd with respect
to the filtration GtX = FtX+ , right continuous and quasi-left
continuous, that solves
2
d
the martingale problem for {L (t)}t≥0 on C0 R . Then the classical solution
u(t, x) of (5.1) on [0, T ) × Rd admits the representation
Z t
u(t, x) = Ex ϕ(X(t)) exp
c(t − s, X(s))ds .
(5.5)
0
´
AROLDO PEREZ
418
Proof. Let β >
sup
c(r, x), and consider the function
(r,x)∈[0,∞)×Rd
V (r, x) = β − c(r, x),
(r, x) ∈ [0, ∞) × Rd .
It follows from [5], Theorem 9.7, p. 298, that the classical solution v(t, (r, x)) of
the Cauchy problem
∂v (t, (r, x))
= Av (t, (r, x)) − V (r, x)v(t, (r, x)),
(5.6)
∂t
v (0, (r, x)) = ϕ(r,
e x),
with t ∈ [0, T ],and (r, x) ∈ [0, ∞) × Rd , admits the representation
Z t
e(r,x) ϕ(Z(t))
v(t, (r, x)) = E
e
exp −
V (Z(s))ds
0
Z t
−βt e
= e E(r,x) ϕ(Z(t))
e
exp
c(Z(s))ds ,
(5.7)
0
e(r,x) denotes the expectation with respect to the process {Z (t)}
where E
t≥0 starting
at (r, x).
On the other hand, if u(t, (r, x)) is a classical solution of (5.3) for t ∈ [0, T ],
(r, x) ∈ [0, ∞) × Rd , then clearly
e−βt u(t, (r, x)), t ∈ [0, T ], (r, x) ∈ [0, ∞) × Rd ,
is a classical solution of (5.6). Thus, using (5.7) and uniqueness of solutions of
problem (5.3), we obtain that u(t, (r, x)) admits the representation
Z t
e
u(t, (r, x)) = E(r,x) ϕ(Z(t))
e
exp
c(Z(s))ds ,
0
see [16], Theorem 1.2, p. 184. Thus, due to Lemma 4.2 and the definition of
ϕ
e (·, ·),
u(t, x)
≡ u (t, (t, x))
Z t
e
= E(t,x) ϕ
e (Z(t)) exp
c(Z(s))ds
0
Z t
= E ϕ(Z(t))
e
exp
c(Z(s))ds |Z(0) = (t, x)
0
Z t
= E ϕ(Y
e (t), X(t)) exp
c(Y (s), X(s))ds |(Y (0), X(0)) = (t, x))
0
Z t
c(t − s, X(s))ds |X(0) = x)
= E ϕ(X(t)) exp
0
Z t
= Ex ϕ(X(t)) exp
c(t − s, X(s))ds ,
0
and, since u(t, x) ≡ u(t, (t, x)) satisfies the integral equation (5.2), by uniqueness
of the mild solution to (5.1) (see e.g. [16]), it follows that the solution u(t, x) of
(5.1) on [0, T ] × Rd admits the representation (5.5).
FEYMAN-KAC FORMULA
419
References
1. Applebaum, D.: L´
evy Processes and Stochastic Calculus, Cambridge University Press, 2004.
2. Birkner, M., L´
opez-Mimbela, J. A. and Walkonbinger, A.: Blow-up of semilinear PDE’s
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2431–2442.
3. Chicone, C. and Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential
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odinger’s Equation, SpringerVerlag, 1995.
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Wiley & Sons, 1986.
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9. Freidlin, M.: Functional Integration and Partial Differential Equations, Princeton University
Press, 1985.
10. Gihman, I. I. and Skorohod, A. V.: The Theory of Stochastic Processes II, Springer-Verlag,
1975.
11. Ichinose, T. and Tamura, H.: Imaginary-time path integral for a relativistic spinless particle
in an electromagnetic field, Commun. Math. Phys. 105 (1986) 239–257.
12. Kac, M.: On distributions of certain Wiener functionals, Trans. Am. Math. Soc. 65 (1949)
1–13.
13. Kac, M.: On some connections between probability theory and differential and integral
equations, in: Proc. 2nd Berkeley Symp. Math. Stat. and Prob. 65 (1951) 189–215.
14. Kolkovska, E. T., L´
opez-Mimbela, J. A. and P´
erez, A.: Blow up and life span bounds for
a reaction-diffusion equation with a time-dependent generator, Electron. J. Diff. Eqns. 10
(2008) 1–18.
15. Komatsu, T.: Markov processes associated with certain integro-differential operators, Osaka
J. Math. 10 (1973) 271–303.
16. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
17. P´
erez, A. and Villa, J.: Blow-up for a system with time-dependent generators, ALEA, Lat.
Am. J. Probab. Math. Stat. 7 (2010) 207–215.
18. Sato Ken-Iti: L´
evy Processes and Infinitely Divisible Distributions, Cambridge University
Press, 1999.
19. Stroock, D. W.: Diffusion processes associated with L´
evy generators, Z. Wahrcheinlichkeitstheorie verw. 32 (1975) 209–244.
20. Stroock, D. W. and Varadhan, R. S.: Multidimensional Diffusion Processes, Springer-Verlag,
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21. Zimmer, R. J.: Essential Results of Functional Analysis, The University of Chicago Press,
1990.
´rez: Universidad Jua
´rez Auto
´ noma de Tabasco, Divisio
´ n Acade
´mica de
Aroldo Pe
´sicas, Km. 1 Carretera Cunduaca
´n-Jalpa de Me
´ndez, C.P. 86690 A.P. 24,
Ciencias Ba
´n, Tabasco, Me
´xico.
Cunduaca
E-mail address: [email protected]
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 421-435
www.serialspublications.com
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING
STOCHASTIC FLOWS
ANDREY A. DOROGOVTSEV*
Abstract. In this article we consider multiplicative operator-valued white
noise functionals related to a stochastic flow. A generalization of the Krylov–
Veretennikov expansion is presented. An analog of this expansion for the
Arratia flow is derived
Introduction
In this article we present the form of the kernels in the Itˆo–Wiener expansion
for functionals from a dynamical system driven by an additive Gaussian white
noise. The most known example of such expansion is the Krylov–Veretennikov
representation [11]:
∞ Z
X
f (y(t)) = Tt f (u) +
Tt−τk b∂Tτ2−τ1 . . .
k=1
∆k (0;t)
. . . b∂Tτ1 f (u)dw(τ1 ) . . . dw(τk ).
where f is a bounded measurable function, y is a solution of the SDE
dy(t) = a(y(t))dt + b(y(t))dw(t)
with smooth and nondegenerate coefficients, and {Tt ; t ≥ 0} is the semigroup of
operators related to the SDE and ∂ is the symbol of differentiation.
A family of substitution operators of the SDE’s solution into a function can be
treated as a multiplicative Gaussian white noise functional. In the first section
of this article we consider a family {Gs,t ; 0 ≤ s ≤ t < +∞} of strong random
operators (see Definition 1.1) in the Hilbert space which is an operator-valued
multiplicative functional from the Gaussian white noise. It turns out that the
precise form of the kernels in the Itˆo–Wiener expansion can be found for a wide
class of operator-valued multiplicative functionals using some simple algebraic relations. The obtained formula covers the Krylov–Veretennikov case and gives a
representation for different objects such as Brownian motion in Lie group etc.
The representation obtained in the first section may be useful in studing the
properties of a dynamical system with an additive Gaussian white noise. On the
Received 2011-7-22; Communicated by the editors.
2000 Mathematics Subject Classification. Primary 60J57; Secondary 60 H25, 60 K53.
Key words and phrases. Multiplicative functionals, white noise, stochastic semi-group, Arratia flow.
* This article was partially supported by the State fund for fundamental researches of Ukraine
and The Russian foundation for basic research, grant F40.1/023.
421
422
ANDREY A. DOROGOVTSEV
other hand, there exist cases when a dynamical system is obtained as a limit in
a certain sense of systems driven by the Gaussian white noise. A limiting system
could be highly irregular [2, 5, 10]. One example of such a system is the Arratia flow [2] of coalescing Brownian particles on the real line. The trajectories of
individual particles in this flow are Brownian motions, but the whole flow cannot be built from the Gaussian noise in a regular way [13]. Nevertheless, it is
possible to construct the n−point motion of the Arratia flow from the pieces of
the trajectories of n independent Wiener processes. Correspondingly a function
from the n-point motion of the Arratia flow has an Itˆo–Wiener expansion based on
the initial Wiener processes. This expansion depends on the way of construction
(coalescing description). We present such expansion in terms of an infinite family
of expectation operators related to all manner of coalescence of the trajectories in
the Arratia flow. To do this we first obtain an analog of the Krylov–Veretennikov
expansion for the Wiener process stopped at zero.
This paper is divided onto three parts. The first section is devoted to multiplicative operator-valued functionals from Gaussian white noise. The second part
contains the definition and necessary facts about the Arratia flow. In the last section we present a family of Krylov–Veretennikov expansions for the n-point motion
of the Arratia flow.
1. Multiplicative White Noise Functionals
In this part we present the Itˆo–Wiener expansion for the semigroup of strong
random linear operators in Hilbert space. Such operators in the space of functions
can be generated by the flow of solutions to a stochastic differential equation. In
this case our expansion turns into the well-known Krylov–Veretennikov representation [11]. In the case when these operators have a different origin, we obtain a
new representation for the semigroup.
Let us start with the definition and examples of strong random operators in the
Hilbert space. Let H denote a separable real Hilbert space with norm k · k and
inner product (·, ·). As usual (Ω, F , P ) denotes a complete probability space.
Definition 1.1. A strong linear random operator in H is a continuous linear map
from H to L2 (Ω, P, H).
Remark 1.2. The notion of strong random operator was introduced by A.V.Skorokhod [14]. In his definition Skorokhod used the convergence in probability, rather
than convergence in the square mean.
Consider some typical examples of strong random operators.
Example 1.3. Let H be l2 with the usual inner product and {ξn ; n ≥ 1} be an
i.i.d. sequence with finite second moment. Then the map
l2 ∋ x = (xn )n≥1 7→ Ax = (ξn xn )n≥1
is a strong random operator. In fact
2
EkAxk =
∞
X
n=1
x2n Eξ12
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 423
and the linearity is obvious. Note that pathwise the operator A can be not welldefined. For example, if {ξn : n ≥ 1} have the standard normal distribution, then
with probability one
sup |ξn | = +∞.
n≥1
An interesting set of examples of strong random operators can be found in the
theory of stochastic flows. Let us recall the definition of a stochastic flow on R
[12].
Definition 1.4. A family {φs,t ; 0 ≤ s ≤ t} of random maps of R to itself is
referred to as a stochastic flow if the following conditions hold:
(1) For any 0 ≤ s1 ≤ s2 ≤ . . . sn < ∞ : φs1 ,s2 , . . . ,φsn−1 ,sn are independent.
(2) For any s, t, r ≥ 0 : φs,t and φs+r,t+r are equidistributed.
(3) For any r ≤ s ≤ t and u ∈ R : φr,s φs,t (u) = φr,t (u), φr,r is an identity
map.
(4) For any u ∈ R : φ0,t (u) 7→ u in probability when t 7→ 0.
Stochastic flows arise as solutions to stochastic differential equations with
smooth coefficients. Namely, if φs,t (u) is a solution to the stochastic differential
equation
dy(t) = a(y(t))dt + b(y(t))dw(t)
(1.1)
starting at the point u in time s and considered in time t, then under smoothness
conditions on the coefficients a and b the family {φs,t } will satisfy the conditions
of Definition 1.4 [12]. Another example of a stochastic flow is the Harris flow
consisting of Brownian particles [5]. In this flow φ0,t (u) for every u ∈ R is a
Brownian martingale with respect to a common filtration and
dhφ0,t (u1 ), φ0,t (u2 )i = Γ(φ0,t (u1 ) − φ0,t (u2 ))dt
for some positive definite function Γ with Γ(0) = 1.
For a given stochastic flow one can try to construct a corresponding family of
strong random operators as follows.
Example 1.5. Let H = L2 (R). Define
Gs,t f (u) = f (φs,t (u)).
Let us check that in the both cases mentioned above Gs,t satisfies Definition 1.1.
For the Harris flow we have
Z
Z Z
Z
2
2
E
f (φs,t (u)) du =
f (v) pt−s (u − v)dudv =
f (v)2 dv.
R
R
R
R
Here pr denotes the Gaussian density with zero mean and variance r.
To get an estimation for the flow generated by a stochastic differential equation
let us suppose that the coefficients a and b are bounded Lipschitz functions and b
is separated from zero. Under such conditions φs,t (u) has a density, which can be
estimated from
density [1]. Consequently we will have the
R above by a Gaussian
R
inequality E R f (φs,t (u))2 du ≤ c R f (v)2 dv.
As it was shown in Example 1.3, a strong random operator in general is not
a family of bounded linear operators in H indexed by the points of probability
424
ANDREY A. DOROGOVTSEV
space. Despite of this the composition of such operators can be properly defined
(see [3] for detailed construction in case of dependent nonlinear operators via Wick
product). Here we will consider only the case when strong random operators A
and B are independent. In this case both A and B have measurable modifications
and one can define for u ∈ H, ω ∈ Ω
AB(u, ω) := A(B(u, ω), ω)
and prove that the value AB(u) does not depend on the choice of modifications.
Note that the operators from the previous example satisfy the semigroup property,
and that for the flow generated by a stochastic differential equation these operators
are measurable with respect to increments of the Wiener process. In this section
we will consider a general situation of this kind and study the structure of the
semigroup of strong random operators measurable with respect to a Gaussian
white noise. The white noise framework is presented in [3, 6, 8], here we just recall
necessary facts and definitions.
Let’s start with a description of the noise. Let H0 be a separable real Hilbert
˜ = H0 ⊗ L2 ([0; +∞]), where an inner
space. Consider a new Hilbert space H
product is defined by the formula
Z ∞
˜ ∋ f, g 7→< f, gi =
H
(f (t), g(t))0 dt.
0
˜ is a family of jointly Gaussian
Definition 1.6. Gaussian white noise ξ in H
˜ which is linear with respect to h ∈ H
˜ and for
random variables {hξ, hi; h ∈ H}
every h, hξ, hi has mean zero and variance khk2 .
˜ s,t be the product H0 ⊗ L2 ([s; t]), which can be naturally considered as a
Let H
˜ Define the σ-fields Fs,t = σ{hξ, hi; h ∈ H
˜ s,t }, 0 ≤ s ≤ t < +∞.
subspace of H.
Definition 1.7. A family {Gs,t ; 0 ≤ s ≤ t < +∞} of strong random operators in
H is a multiplicative functional from ξ if the following conditions hold:
1) Gs,t is measurable with respect to Fs,t ,
2) Gs,s is an identity operator for every s,
3) Gs1 ,s3 = Gs2 ,s3 Gs1 ,s2 for s1 ≤ s2 ≤ s3 .
Remark 1.8. Taking an orthonormal basis {en } in H0 one can replace ξ by a
sequence of independent Wiener processes {wn (t) = hen ⊗ 1[0; t] ; ξi; t ≥ 0}. We
use ξ in order to simplify notations and consider simultaneously both cases of finite
and infinite number of the processes {wn }.
Example 1.9. Let us define x(u, s, t) as a solution to the Cauchy problem for
(1.1) which starts from the point u at the moment s. Using the flow property one
can easily verify that the family of operators {Gs,t f (u) = f (x(u, s, t))} in L2 (R)
is a multiplicative functional from the Gaussian white noise w˙ in L2 ([0; +∞]).
Now we are going to introduce the notion of a homogeneous multiplicative functional. Let us recall, that every square integrable random variable α measurable
with respect to ξ can be uniquely expressed as a series of multiple Wiener integrals
[8]
∞ Z
X
α = Eα +
ak (τ1 , . . . , τk )ξ(dτ1 ) . . . ξ(dτk ),
k=1
∆k (0;+∞)
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 425
where
∆k (s; t) = {(τ1 , . . . , τk ) : s ≤ τ1 ≤ . . . ≤ τk ≤ t},
ak ∈ L2 (∆k (0; +∞), H0⊗k ), k ≥ 1.
Here in the multiple integrals we consider the white noise ξ as Gaussian H0 -valued
random measure on [0; +∞). In the terms of the mentioned above orthonormal
basis {en } in H0 and the sequence of the independent Wiener processes {wn } one
can rewrite the above multiple integrals as
Z
ak (τ1 , . . . , τk )ξ(dτ1 ) . . . ξ(dτk )
∆k (0;+∞)
=
X Z
n1 ,...,nk
ak (τ1 , . . . , τk )(en1 , ..., enk )dwn1 (τ1 )...dwnk (τk ).
∆k (0;+∞)
Define the shift of α for r ≥ 0 as follows
∞ Z
X
θr α = Eα +
ak (τ1 − r, . . . , τk − r)ξ(dτ1 ) . . . ξ(dτk ).
k=1
∆k (r;+∞)
Definition 1.10. A multiplicative functional {Gs,t } is homogeneous if for every
s ≤ t and r ≥ 0
θr Gs,t = Gs+r,t+r .
Note that the family {Gs,t } from Example 1.9 is a homogeneous functional.
From now on, we will consider only homogeneous multiplicative functionals from
ξ. For a homogeneous functional {Gs,t } one can define the expectation operators
Tt u = EG0,t u, u ∈ H, t ≥ 0.
Since the family {Gs,t } is homogeneous, then {Tt } is the semigroup of bounded
operators in H. Under the well-known conditions the semigroup {Tt } can be
described by its generator. However the family {Gs,t } cannot be recovered from
this semigroup. The following simple example shows this.
Example 1.11. Define {G1s,t } and {G2s,t } in the space L2 (R) as follows
G1s,t f (u) = Tt−s f (u),
where {Tt } is the heat semigroup, and
G2s,t f (u) = f (u + w(t) − w(s)),
where w is a standard Wiener processes. It is evident, that
EG2s,t f (u) = Tt−s f (u) = EG1s,t f (u).
To recover multiplicative functional uniquely we have to add some information to
{Tt }. It can be done in the following way. For f ∈ H define an operator which
acts from H0 to H by the rule
1
.
A(f )(h) = lim EG0,t f (ξ, h ⊗ 1[0; t] ).
t→0+ t
(1.2)
426
ANDREY A. DOROGOVTSEV
Example 1.12. Let the family {Gs,t } be defined as in Example 1.9. Now H =
L2 (R) and the noise ξ is defined on L2 ([0; +∞) as w.
˙ Then for f ∈ L2 (R) (now
H0 = R and it makes sense only to take h = 1)
1
A(f )(u) = lim Ef (x(u, t))w(t).
t→0+ t
Suppose that f has two bounded continuous derivatives. Then using Itˆo’s formula
one can get
Z t
Ef (x(u, t))w(t) =
Ef ′ (x(u, s))ϕ(x(u, s))ds,
0
and
1
L2 (R)
Ef (x(•, t))w(t) → f ′ (•)b(•), t → 0 + .
t
Consequently, for “good” functions
Af = bf ′ .
Definition 1.13. An element u of H belongs to the domain of definition D(A) of
A if the limit (1.2) exists for every h ∈ H0 and defines a Hilbert–Schmidt operator
A(u) : H0 → H. The operator A is refereed to as the random generator of {Gs,t }.
Now we can formulate the main statement of this section, which describes the
structure of homogeneous multiplicative functionals from ξ.
Theorem 1.14. Suppose, that for every t > 0, Tt (H) ⊂ D(A) and the kernels of
the Itˆ
o-Wiener expansion for G0,t are continuous with respect to time variables.
Then G0,t has the following representation
∞ Z
X
G0,t (u) = Tt u +
Tt−τk ATτk −τk−1 . . . ATτ1 udξ(τ1 ) . . . dξ(τk ).
(1.3)
k=1
∆k (0;t)
Proof. Let us denote the kernels of the Itˆo–Wiener expansion for G0,t (u) as
{atk (u, τ1 , . . . , τk ); k ≥ 0}. Since
at0 (u) = EG0,t (u),
then
at0 (u) = Tt u.
Since
G0,t+s (u) = Gt,t+s (G0,t (u)),
and Gt,t+s = θt G0,s , then
t
s
at+s
1 (u, τ1 ) = Ts a1 (u, τ1 )1τ1 <t + a1 (Tt u, τ1 − t)1t≤τ1 ≤t+s .
(1.4)
Using this relation one can get
at1 (u, 0) = Tt−τ1 aτ11 (u, 0),
1
at1 (u, τ1 ) = at−τ
(Tτ1 u, 0).
1
(1.5)
The condition of the theorem implies that for v = Tτ1 u and every h ∈ H0 there
exists the limit
1
A(v)h = lim EG0,t (v)(ξ, h ⊗ 1[0; t] )
t→0+ t
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 427
1
t→0+ t
= lim
Now, by continuity of a1 ,
Z
t
0
at1 (v, τ1 )hdτ1 .
a01 (Tτ1 u, 0) = A(Tτ1 u).
Finally,
at1 (u, τ1 ) = Tt−τ1 ATτ1 u.
The case k ≥ 2 can be proved by induction. Suppose, that we have the representation (1.3) for atj , j ≤ k. Consider at+s
k+1 . Using the multiplicative and homogeneity
properties one can get
at+s
k+1 (u, τ1 , . . . , τk+1 )1{0≤τ1 ≤...≤τk ≤t≤τk+1 ≤t+s}
= as1 (atk (u, τ1 , . . . , τk ), τk+1 − t)
= Ts+t−τk+1 ATτk+1 −t atk (u, τ1 , . . . , τk )
= Ts+t−τk+1 ATτk+1 −t Tt−τk A . . . ATτ1 u
= Ts+t−τk+1 ATτk+1 −τk A . . . ATτ1 u.
The theorem is proved.
Consider some examples of application of the representation (1.3).
Example 1.15. Consider the multiplicative functional from Example 1.9. Suppose that the coefficients a, b have infinitely many bounded derivatives. Now it
can be proved, that x(u, t) has infinitely many stochastic derivatives [15]. Consequently for a smooth function f the first kernel in the Itˆo–Wiener expansion of
f (x(u, t)) can be expressed as follows
at1 (u, τ ) = EDf (x(u, t))(τ ).
(1.6)
Indeed, for an arbitrary h ∈ L2 ([0; +∞))
Z t
Z t
at1 (u, τ )h(τ )dτ = Ef (x(u, t))
h(τ )dw(τ )
0
0
Z t
=E
Df (x(u, t))(τ )h(τ )dτ,
0
which gives us the expression (1.6). The required continuity of a1 follows from a
well-known expression for the stochastic derivative of x [8]. As it was mentioned
d
on smooth functions. Finally,
in Example 1.12, the operator A coincides with b du
the expression (1.3) turns into the well-known Krylov–Veretennikov expansion [11]
for f (x(u, t))
∞ Z
X
f (x(u, t)) = Tt f (u) +
Tt−τk b∂Tτ2 −τ1 . . .
k=1
∆k (0;t)
. . . b∂Tτ1 f (u)dw(τ1 ) . . . dw(τk ).
Remark 1.16. The expression (1.3) can be applied to multiplicative functionals,
which are not generated by a stochastic flow.
428
ANDREY A. DOROGOVTSEV
Example 1.17. Let L be a matrix Lie group with the corresponding Lie algebra
A with dim A = n. Consider an L-valued homogeneous multiplicative functional
{Gs,t } from ξ. Suppose that {G0,t } is a semimartingale with respect to the filtration generated by ξ. Let {Gs,t } be continuous with respect to s, t with probability
one. It means, in particular, that {G0,t } is a multiplicative Brownian motion in L
[7]. Then G0,t is a solution to the following SDE
dG0,t = G0,t dMt ,
G0,0 = I.
Here {Mt ; t ≥ 1} is an A-valued Brownian motion obtained from G by the rule [7]
Mt = P - lim
∆→0+
[ ∆t ]
X
(Gk∆,(k+1)∆ − I).
(1.7)
k=0
Since G0,t is a semimartingale with respect to the filtration of ξ, then Mt also has
the same property. The representation (1.7) shows that Mt − Ms is measurable
with respect to the σ-field Fs,t and for arbitrary r ≥ 0
θr (Mt − Ms ) = Mt+r − Ms+r .
Considering the Itˆ
o–Wiener expansion of Mt − Ms one can easily check, that
Z t
Mt =
Zdξ(τ )
(1.8)
0
with a deterministic matrix Z. We will prove (1.8) for the one-dimensional case.
Suppose that Mt has the following Itˆo-Wiener expansion with respect to ξ
∞ Z
X
Mt =
ak (t, τ1 , . . . , τk )dξ(τ1 ) . . . dξ(τk ).
k=1
∆k (t)
Then for k ≥ 2 the corresponding kernel ak satisfies relation
ak (t + s, τ1 , . . . , τk ) = ak (t, τ1 , . . . , τk )1{τ1 ,...,τk ≤t}
Pn
+ ak (s, τ1 − t, . . . , τk − t)1{τ1 ,...,τk ≥t} .
Iterating this relation for t = j=1
same arguments give ak ≡ const.
t
n
one can verify that ak ≡ 0. For k = 1 the
Consequently, the equation for G can be rewritten using ξ as
dG0,t = G0,t Zdξ(t).
(1.9)
Now the elements of the Itˆ
o–Wiener expansion from Theorem 1.14 can be determined as follows
Tt = EG0,t , A = Z.
Consequently,
G0,t = Tt +
∞ Z
X
k=1
∆k (0;t)
Tt−τk ZTτk −τk−1 . . . ZTτ1 dξ(τ1 ) . . . dξ(τk ).
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 429
2. The Arratia Flow
When trying to obtain an analog of the representation (1.3) for a stochastic
flow which is not generated by a stochastic differential equation with smooth coefficients, we are faced with the difficulty that there is no such a Gaussian random
vector field, which would generate the flow. This circumstance arise from the possibility of coalescence of particles in the flow. We will consider one of the best
known examples of such stochastic flows, the Arratia flow. Let us start with the
precise definition.
Definition 2.1. The Arratia flow is a random field {x(u, t); u ∈ R, t ≥ 0}, which
has the properties
1) all x(u, ·), u ∈ R are Wiener martingales with respect to the join filtration,
2) x(u, 0) = u, u ∈ R,
3) for all u1 ≤ u2 , t ≥ 0
x(u1 , t) ≤ x(u2 , t),
4) the joint characteristics of x(u1 , t) and x(u2 , t) equals
Z t
hx(u1 , ·), x(u2 , ·)it =
1{τ (u1 ,u2 )≤s} ds,
0
where
τ (u1 , u2 ) = inf{t : x(u1 , t) = x(u2 , t)}.
It follows from the properties 1)–3), that individual particles in the Arratia flow
move as Brownian particles and coalesce after meeting. Property 4) reflects the
independence of the particles before meeting. It was proved in [4], that the Arratia
flow has a modification, which is a cdlg process on R with the values in C([0; +∞)]).
From now on, we assume that we are dealing with such a modification. We will
construct the Arratia flow using a sequence of independent Wiener processes {wk :
k ≥ 1}. Suppose that {rk ; k ≥ 1} are rational numbers on R. To construct the
Arratia flow put wk (0) = rk , k ≥ 1 and define
x(r1 , t) = w1 (t), t ≥ 0.
If x(r1 , ·), . . . , x(rn , ·) have already been constructed, then define
σn+1 = inf{t :
n
Y
(x(rk , t) − wn+1 (t)) = 0},
k=1
x(rn+1 , t) =
(
wn+1 (t), t ≤ σn+1
x(rk∗ , t), t ≥ σn+1 ,
where
wn+1 (σn+1 ) = x(rk∗ , σn+1 ),
k = min{l : wn+1 (σn+1 ) = x(rl , σn+1 )}.
In this way we construct a family of the processes x(r, ·), r ∈ Q which satisfies
conditions 1)–4) from Definition 2.1.
430
ANDREY A. DOROGOVTSEV
Lemma 2.2. For every u ∈ R the random functions x(r, ·) uniformly converge
on compacts with probability one as r → u. For rational u the limit coincides with
x(u, ·) defined above. The resulting random field {x(u, t); u ∈ R, t ≥ 0} satisfies
the conditions of Definition 2.1.
Proof. Consider a sequence of rational numbers {rnk ; k ≥ 1} which converges to
some u ∈ R \ Q. Without loss of generality one can suppose that this sequence
decreases. For every t ≥ 0, {x(rnk , t); k ≥ 1} converges with probability one as a
bounded monotone sequence. Denote
x(u, t) = lim x(rnk , t).
k→∞
′
′′
Note that for arbitrary r , r ∈ Q and t ≥ 0
E sup (x(r′ , s) − x(r′′ , s))2 ≤ C · (|r′ − r′′ | + (r′ − r′′ )2 ).
(2.1)
[0; t]
Here the constant C does not depend on r′ and r′′ . Inequality (2.1) follows from
the fact, that the difference x(r′ , ·) − x(r′′ , ·) is a Wiener process with variance 2,
started at r′ − r′′ and stopped at 0. Monotonicity and (2.1) imply that the first
assertion of the lemma holds. Note that for every t ≥ 0
Ft = σ(x(r, s); r ∈ Q, s ∈ [0; t])
= σ(x(r, s); r ∈ R, s ∈ [0; t]).
Using standard arguments one can easily verify, that for every u ∈ R, x(u, ·) is a
Wiener martingale with respect to the flow (Ft )t≥0 , and that the inequality
x(u1 , t) ≤ x(u2 , t)
remains to be true for all u1 ≤ u2 . Consequently, for all u1 , u2 ∈ R, x(u1 , ·) and
x(u2 , ·) coincide after meeting. It follows from (2.1) and property 4) for x(r, ·)
with rational r, that
hx(u1 , ·), x(u2 , ·)it = 0
for
t < inf{s : x(u1 , s) = x(u2 , s)}.
Hence, the family {x(u, t); u ∈ R, t ≥ 0} satisfies Definition 2.1.
This lemma shows that the Arratia flow is generated by the initial countable
system of independent Wiener processes {wk ; k ≥ 1}.
From this lemma one can easily obtain the following statement.
Corollary 2.3. The σ-field
x
F0+
:=
\
σ(x(u, s); u ∈ R, 0 ≤ s ≤ t)
t>0
is trivial modulo P.
The proof of this statement follows directly from the fact that the Wiener
process has the same property [9].
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 431
3. The Krylov–Veretennikov Expansion for the n-point
Motion of the Arratia Flow
We begin this section with an analog of the Krylov–Veretennikov expansion for
the Wiener process stopped at zero. For the Wiener process w define the moment
of the first hitting zero
τ = inf{t : w(t) = 0}
and put w(t)
e = w(τ ∧ t). For a measurable bounded f : R → R define
Tet (f )(u) = Eu f (w(t)).
e
The following statement holds.
Lemma 3.1. For a measurable bounded function f : R → R and u ≥ 0
f (w(t))
e
= Tet f (u)
Z
∞
X
∂ e
∂ e
Trk −rk−1 . . .
Tr1 f (v1 )
+
Tet−rk
∂v
∂v
k
1
∆k (t)
k=1
dw(r1 ) . . . dw(rk ).
(3.1)
T
Proof. Let us use the Fourier–Wiener transform. Define for ϕ ∈ C([0; +∞), R)
L2 ([0; +∞), R) the stochastic exponential
(Z
)
Z
+∞
1 +∞
2
E(ϕ) = exp
ϕ(s)dw(s) −
ϕ(s) ds .
2 0
0
Suppose that a random variable α has the Itˆo–Wiener expansion
∞ Z
X
α = a0 +
aj (r1 , . . . , rk )dw(r1 ) . . . dw(rk ).
k=1
∆k (t)
Then
EαE(ϕ) = a0
∞ Z
X
+
ak (r1 , . . . , rk )ϕ(r1 ) . . . ϕ(rk )dr1 . . . drk .
k=1
(3.2)
∆k (t)
Consequently, to find the Itˆ
o–Wiener expansion of α it is enough to find EαE(ϕ)
as an analytic functional from ϕ. Note that
Eu f (w(t))E(ϕ)
e
= Eu f (e
y(t)),
where the process ye is obtained from the process
Z t
y(t) = w(t) +
ϕ(r)dr
0
in the same way as w
e from w. To find Eu f (e
y(t)) consider the case when f is
continuous bounded function with f (0) = 0. Let F be the solution to the following
boundary problem on [0; +∞) × [0; T ]
∂
1 ∂2
∂
F (u, t) = −
F (u, t) − ϕ(t) F (u, t),
∂t
2 ∂u2
∂u
F (u, T ) = f (u), F (0, s) = 0, s ∈ [0; T ],
(3.3)
432
ANDREY A. DOROGOVTSEV
F ∈ C 2 ((0; +∞) × (0; T )) ∩ C([0; +∞) × [0; T ]).
Then F (u, 0) = Eu f (e
y (T )). To check this relation note, that F satisfies the relation
∂
∂2
F (0, s) =
F (0, s) = 0, s ∈ [0; T ].
∂u
∂u2
Consider the process F (e
y (s), s) on the interval [0; T ]. Using Itˆo’s formula one can
get
!
Z T ∧τ
1 ∂2
F (e
y (T ), T ) = F (u, 0) +
F (e
y (s), s
2 ∂u2
0
!
∂
1 ∂2
∂
+ ϕ(s) F (e
y (s), s)) −
F (e
y (s), s + ϕ(s) F (e
y (s), s))ds
2
∂u
2 ∂u
∂u
+
Z
0
Consequently
T ∧τ
∂
F (e
y (s), s)dw(s).
∂u
F (u, 0) = Eu f (e
y(T )).
The problem (3.3) can be solved using the semigroup {Tet ; t ≥ 0}. It can be
obtained from (3.3) that
Z T
∂
e
F (u, s) = TT −s f (u) +
ϕ(r)Ter−s F (u, r)dr.
(3.4)
∂u
s
Solving (3.4) by the iteration method one can get the series
F (u, s) = TeT −s f (u)
+
∞ Z
X
k=1
∂ e
Tr2 −r1 . . .
Ter1 −s
∂v
1
∆k (s; T )
∂ e
TT −rk f (vk )ϕ(r1 ) . . . ϕ(rk )dr1 . . . drk .
∂vk
The last formula means that the Itˆo–Wiener expansion of f (w(t))
e
has the form
Z
∞
X
∂ e
Tr2 −r1 . . .
f (w(t))
e
= Tet f (u) +
Ter1
∂v
1
∆k (t)
k=1
∂ e
Tt−rk f (vk )dw(r1 ) . . . dw(rk ).
∂vk
To consider the general case note that for t > 0 and c ∈ R
∂ e
Tt c ≡ 0.
∂v
(3.5)
Consequently (3.5) remains to be true for an arbitrary bounded continuous f. Now
the statement of the lemma can be obtained using the approximation arguments.
The lemma is proved.
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 433
The same idea can be used to obtain the Itˆo–Wiener expansion for a function
from the Arratia flow. The n-point motion of the Arratia flow was constructed in
Section 2 from independent Wiener processes. Consequently, a function from this
n−point motion must have the Itˆo–Wiener expansion in terms of these processes.
We will treat such expansion as the Krylov–Veretennikov expansion for the Arratia
flow.
Here there is a new circumstance compared to the case when the flow is generated by SDE with smooth coefficients. Namely, there are many different ways to
construct the trajectories of the Arratia flow from the initial Wiener processes, and
the form of the Itˆ
o–Wiener expansion will depend on the way of constructing the
trajectories. In [2] Arratia described different ways of constructing the colliding
Brownian motions from independent Wiener processes. We present here a more
general approach by considering a broad class of constructions, and find the Itˆo–
Wiener expansion for it. To describe our method we will need some preliminary
notations and definitions.
Definition 3.2. An arbitrary set of the kind {i, i + 1, . . . , j}, where i, j ∈ N,i ≤ j
is called a block.
Definition 3.3. A representation of the block {1, 2, . . . , n} as a union of disjoint
blocks is called a partition of the block {1, 2, . . . , n}.
Definition 3.4. We say that a partition π2 follows from a partition π1 if it coincides with π1 or if it is obtained by the union of two subsequent blocks from
π1 .
We will consider a sequences of partitions {π0 , . . . , πl } where π0 is a trivial
partition, π0 = {{1}, {2}, . . . , {n}} and every πi+1 follows from πi . The set of all
such sequences will be denoted by R. Denote by Rk the set of all sequences from R
that have exactly k matching pairs: πi = πi+1 . The set R0 of strongly decreasing
˘ For every sequence {π0 , . . . , πk } from R
˘ each πi+1 is
sequences we denote by R.
obtained from πi by the union of two subsequent blocks. It is evident, that the
˘ is less or equal to n. Let us associate with every
length of every sequence from R
partition π a vector λ~π ∈ Rn with the next property. For each block {s, . . . , t}
from π the following relation holds
t
X
λ2πq = 1.
q=s
We will use the mapping ~λ as a rule of constructing the n−point motion of the
Arratia flow. Suppose now, that {wk ; k = 1, . . . , n} are independent Wiener
processes starting at the points u1 < . . . < un . We are going to construct the
trajectories {x1 , . . . , xn } of the Arratia flow starting at u1 < . . . < un from the
pieces of the trajectories of {wk ; k = 1, . . . , n}. Assume that we have already built
the trajectories of {x1 , . . . , xn } up to a certain moment of coalescence τ . At this
moment a partition π of {1, 2, . . . , n} naturally arise. Two numbers i and j belong
to the same block in π if and only if xi (τ ) = xj (τ ). Consider one block {s, . . . , t} in
π. Define the processes xs , . . . , xt after the moment τ and up to the next moment
434
ANDREY A. DOROGOVTSEV
of coalescence in the whole system {x1 , . . . , xn } by the rule
xi (t) = xi (τ ) +
t
X
λπq (wq (t) − wq (τ )).
q=s
Proceeding in the same way, we obtain the family {xk , k = 1, . . . , n} of continuous
square integrable martingales with respect to the initial filtration, generated by
{wk ; k = 1, . . . , n} with the following properties:
1) for every k = 1, . . . , n, xk (0) = uk ,
2) for every k = 1, . . . , n − 1, xk (t) ≤ xk+1 (t),
3) the joint characteristic of xi and xj satisfies relation
dhxi , xj i(t) = 1t≥τij ,
where τij = inf{s : xi (s) = xj (s)}.
It can be proved [10] that the processes {xk , k = 1, . . . , n} are the n−point
motion of the Arratia flow starting from the points u1 < . . . < un . We constructed
it from the independent Wiener processes {wk ; k = 1, . . . , n} and the way of
construction depends on the mapping ~λ. To describe the Itˆo–Wiener expansion for
functions from {xk (t), k = 1, . . . , n} it is necessary to introduce operators related
˘ Denote by τ0 = 0 < τ1 < . . . < τn−1 the
to a sequence of partitions π
˜ ∈ R.
moments of coalescence for {xk (t), k = 1, . . . , n} and by ν˜ = {π0 , ν1 , . . . , νn−1 }
related random sequence of partitions. Namely, the numbers i and j belong to the
same block in the partition νk if and only if xi (t) = xj (t) for τk ≤ t. Define for a
bounded measurable function f : Rn → R
Ttπ˜ f (u1 , . . . , un ) = Ef (x1 (t), . . . , xn (t))1{ν1 =π1 ,...,νk =πk , τk ≤t<τk+1 } .
Now let κ be an arbitrary partition and let u1 ≤ u2 ≤ . . . ≤ un be such, that
ui = uj if and only if i and j belong to the same block in κ. One can define
formally the n−point motion of the Arratia flow starting at u1 ≤ u2 ≤ . . . ≤ un ,
assuming that the trajectories that start at coinciding points, also coincide. Then
for the strongly decreasing sequence of partitions π
˜ = {κ, π1 , . . . , πk } the operator
Ttπ˜ is defined by the same formula as above.
The next theorem is the Krylov–Veretennikov expansion for the n−point motion
of the Arratia flow.
Theorem 3.5. For a bounded measurable function f : Rn → R the following
representation takes place
X
f (x1 (t), . . . , xn (t)) =
Ttπ˜ f (u1 , . . . , un )
˘
π
˜ ∈R
+
n X
X
λπ1 i
i=1 π
˜ ∈R1
+
n
X
X
i1 ,i2 =1 π
˜ ∈R2
Z
0
t
π
˜2
f (u1 , . . . , un )dwi (s1 )
Tsπ˜11 ∂i Tt−s
1
λπ1 i1 λπ2 i2
Z
△2 (t)
π
˜3
Tsπ˜11 ∂i1 Tsπ˜22−s1 ∂i2 Tt−s
2
f (u1 , . . . , un )dwi1 (s1 )dwi2 (s2 )
KRYLOV–VERETENNIKOV EXPANSION FOR COALESCING STOCHASTIC FLOWS 435
+
n
X
k
X Y
i1 ,...,ik =1 π
˜ ∈Rk j=1
λπj ij
Z
△k (t)
π
˜
k+1
Tsπ˜11 ∂i1 Tsπ˜22−s1 ...∂ik Tt−s
k
f (u1 , . . . , un )dwi1 (s1 )...dwik (sk ) + · · · · · ·
In this formula we use the following notations. For a sequence π
˜ ∈ Rk partitions
π1 , ..., πk are the left elements of equalities from π
˜ = {...π1 = ...π2 = ...πk = ...}
and π
˜1 , ..., π
˜k+1 are strictly decreasing pieces of π
˜ . The symbol ∂i denotes differentiation with respect to a variable corresponding toPthe block of partition, which
q=t
contains i. For example, if i ∈ {s, ..., t} then ∂i f = q=s fq′ .
The proof of the theorem can be obtained by induction, adopting ideas of
Lemma 3.1. One has to consider subsequent boundary value problems and then
use the probabilistic interpretation of the Green’s functions for these problems.
The corresponding routine calculations are omitted.
Acknowledgment. Author wish to thank an anonymous referee for careful reading of the article and helpful suggestions.
References
1. Aronson, D. G.: Bounds for the fundamental solution of a parabolic equation, Bull. Amer.
Math. Soc. 73 (1967) 890–896.
2. Arratia, R.: Coalescing Brownian motion on the line, in: University of Wisconsin–Madison.
PhD thesis (1979).
3. Dorogovtsev, A. A.: Stochastic Analysis and Random Maps in Hilbert Space VSP, Utrecht,
1994.
4. Dorogovtsev, A. A.: Some remarks on a Wiener flow with coalescence, Ukrainian mathematical journal 57 (2005) 1550–1558.
5. Harris, T. E.: Coalescing and noncoalescing stochastic flows in R1 , Stochastic Processes and
their Applications 17 (1984) 187–210.
6. Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White Noise – An Infinite Dimensional Calculus, Kluwer, Dordrecht, 1993.
7. Holevo, A. S.: An analog of the Itˆ
o decomposition for multiplicative processes with values
in a Lie group, The Indian Journal of Statistics 53, Ser. A, Pt.2 (1991) 158–161.
8. Janson, S.: Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, 1997.
9. Kallenberg, O.: Foundations of Modern Probability, Springer-Verlag, New-York, 1997.
10. Konarovskii, V. V.: On Infinite System of Diffusing Particles with Coalescing, Theory Probab.
Appl. 55 (2010) 159–169.
11. Krylov, N. V., Veretennikov, A. Yu.: Explicit formulae for the solutions of the stochastic
differential equations, Math. USSR Sb. 29, No. 2 (1976) 239–256.
12. Kunita, H.: Stochastic Flows and Stochastic Differential Equations, Cambridge University
Press, Cambridge, 1990.
13. Le Jan, Y., Raimond, O.: Flows, coalescence and noise, Ann.Probab. 32 (2004) 1247–1315.
14. Skorokhod, A. V.: Random Linear Operators, D. Reidel Publishing Company, Dordrecht,
Holland, 1983.
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Institute of Fundamental Research, Bombay, 1984.
Andrey A. Dorogovtsev: Institute of Mathematics, National Academy of Sciences
of Ukraine, Kiev-4, 01601, 3, Tereschenkivska st, Ukraine
E-mail address: [email protected]
URL: http://www.imath.kiev.ua/deppage/stochastic/people/doro/adoronew.html
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 437-450
www.serialspublications.com
QWN-CONSERVATION OPERATOR AND ASSOCIATED WICK
DIFFERENTIAL EQUATION
HABIB OUERDIANE AND HAFEDH RGUIGUI
Abstract. In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators
L(Fθ (SC′ (R)), Fθ∗ (SC′ (R))) endowed with the Wick product. Similarly to the
classical case, we give a useful integral representation in terms of the QWNderivatives {Dt− , Dt+ ; t ∈ R} for the QWN-conservation operator from which it
follows that the QWN-conservation operator is a Wick derivation. Via this property, a relation with the Cauchy problem associated to the QWN-conservation
operator and the Wick differential equation is worked out.
1. Introduction
Piech [21] initiated the study of an infinite dimensional analogue of a finite dimensional Laplacian on infinite dimensional abstract Wiener space. This infinite
dimensional Laplacian (called the number operator) has been extensively studied
in [16, 17] and the references cited therein. In particular, Kuo [15] formulated the
number operator as continuous linear operator acting on the space of test white
noise functionals (E). By virtue of the general theory based on an infinite dimensional analogue of Schwarz’s distribution theory, where the Lebesgue measure on
R and the Gel’fand triple
S(R) ⊂ L2 (R) ⊂ S ′ (R)
(1.1)
are replaced respectively by the Gaussian measure µ on S ′ (R) and the following
Gel’fand triple of test function space Fθ (SC′ (R)) and generalized function space
Fθ∗ (SC′ (R))
Fθ (SC′ (R)) ⊂ L2 (S ′ (R), µ) ⊂ Fθ∗ (SC′ (R)),
(1.2)
see for more details [5] and if we employ a discrete coordinate, the number operator
N has the following expressions:
N=
∞
X
∂e∗k ∂ek ,
(1.3)
k=1
where {en ; n ≥ 0} is an arbitrary orthonormal basis for L2 (R), ∂ek denotes
the derivative in the direction ek acting on Fθ (SC′ (Rd )) and ∂e∗k is the adjoint
Received 2012-1-22; Communicated by the editors.
2000 Mathematics Subject Classification. Primary 60H40; Secondary 46A32, 46F25, 46G20.
Key words and phrases. Wick differential equation, Wick derivation, QWN-conservation operator, QWN-derivatives.
437
438
HABIB OUERDIANE AND HAFEDH RGUIGUI
of ∂ek . For details see [16], [17]. In [2], the conservation operator N (K), for
K ∈ L(SC′ (R), SC′ (R)), is given by
N (K)ϕ(x) =
∞
X
b I ⊗(n−1) )ϕn i,
nhx⊗n , (K ⊗
(1.4)
n=0
from which it is obvious that N (I) = N . Using the S-transform it is well known
that N (K) is a wick derivation of distributions, see [4], moreover we have
Z
N (K) =
τK (s, t)x(s) ⋄ at dsdt.
(1.5)
R2
In the present paper, by using the new idea of QWN-derivatives pointed out by
Ji-Obata in [9, 8], we extend some results contained in [2] to the QWN domains. For
B1 , B2 ∈ L(SC′ (R), SC′ (R)), the QWN-analogous NBQ1 ,B2 stands for appropriate QWN
counterpart of the conservation operator in (1.3). In the first main result we show
that NBQ1 ,B2 has functional integral representations in terms of the QWN-derivatives
{Dt− , Dt+ ; t ∈ R} and a suitable Wick product ⋄ on the class of white noise
operators as a quantum white noise analogue of (1.5). The second remarkable
feature is that NBQ1 ,B2 behaviors as a Wick derivation of operators. This enable
us to give a relation between the Cauchy problem associated to NBQ1 ,B2
∂
Ut = NBQ1 ,B2 Ut , U0 ∈ L(Fθ (SC′ (R)), Fθ∗ (SC′ (R))),
∂t
and the Wick differential equation introduced in [10] as follows
DY = G ⋄ Y,
G ∈ L(Fθ (SC′ (R)), Fθ∗ (SC′ (R)))
(1.6)
(1.7)
where D is a Wick derivation. It is well known that (see [10]) if there exists
an operator Y in the algebra L(Fθ (SC′ (R)), Fθ∗ (SC′ (R))) such that DY = G and
wexpY := e⋄Y is defined in L(Fθ (SC′ (R)), Fθ∗ (SC′ (R))), then every solution to (1.7),
is of the form:
Ξ = (wexpY ) ⋄ F,
where F ∈ L(Fθ (SC′ (R)), Fθ∗ (SC′ (R))) satisfying DF = 0. More precisely, an
important example of the Wick differential equation associated with the QWNconservation operator is studied where the solution of a system of equations of
type (1.7) is given explicitly in terms of the solution of the associated Cauchy
problem associated to QWN-conservation operator.
The paper is organized as follows. In Section 2, we briefly recall well-known
results on nuclear algebra of entire holomorphic functions. In Section 3, we reformulate in our setting the creation derivative and annihilation derivative as
well as their adjoints. Then, we introduce the QWN-conservation operator acting on L(Fθ (SC′ (R)), Fθ∗ (SC′ (R))). As a main result, we give a useful integral
representation for the QWN-conservation operator from which it follows that the
QWN-conservation operator is a Wick derivation. In Section 4, we find a connection between the solution of a continuous system of QWN-differential equations and
the solution of the Cauchy problem associated to the QWN-conservation operator
NBQ1 ,B2 .
QWN-CONSERVATION OPERATOR AND ASSOCIATED DIFFERENTIAL EQUATION
439
2. Preliminaries
Let H be the real Hilbert space of square integrable functions on R with norm
| · |0 , E ≡ S(R) and E ′ ≡ S ′ (R) be the Schwarz space consisting of rapidly
decreasing C ∞ -functions and the space of the tempered distributions, respectively.
Then, the Gel’fand triple (1.1) can be reconstructed in a standard way (see Ref.
[17]) by the harmonic oscillator A = 1 + t2 − d2 /dt2 and H. The eigenvalues of A
are 2n + 2, n = 0, 1, 2, · · · , the corresponding eigenfunctions {en ; n ≥ 0} form an
orthonormal basis for L2 (R) and each en is an element of E. In fact E is a nuclear
space equipped with the Hilbertian norms
|ξ|p = |Ap ξ|0 ,
ξ ∈ E,
p∈R
and we have
E = proj lim Ep ,
p→∞
E ′ = ind lim E−p ,
p→∞
where, for p ≥ 0, Ep is the completion of E with respect to the norm | · |p and E−p
is the topological dual space of Ep . We denote by N = E + iE and Np = Ep + iEp ,
p ∈ Z, the complexifications of E and Ep , respectively.
Throughout the paper, we fix a Young function θ, i.e. a continuous, convex and
increasing function defined on R+ and satisfies the two conditions: θ(0) = 0 and
limx→∞ θ(x)/x = +∞. The polar function θ∗ of θ, defined by
θ∗ (x) = sup(tx − θ(t)),
x ≥ 0,
t≥0
is also a Young function. For more details , see Refs. [5], [12] and [18]. For a
complex Banach space (B, k · k), let H(B) denotes the space of all entire functions
on B, i.e. of all continuous C-valued functions on B whose restrictions to all affine
lines of B are entire on C. For each m > 0 we denote by Exp(B, θ, m) the space
of all entire functions on B with θ−exponential growth of finite type m, i.e.
n
o
Exp(B, θ, m) = f ∈ H(B); kf kθ,m := sup |f (z)|e−θ(mkzk) < ∞ .
z∈B
The projective system {Exp(N−p , θ, m); p ∈ N, m > 0} and the inductive system
{Exp(Np , θ, m); p ∈ N, m > 0} give the two spaces
Fθ (N ′ ) = proj lim
p→∞;m↓0
Exp(N−p , θ, m) ,
Gθ (N ) = indlim
p→∞;m→0
Exp(Np , θ, m).
(2.1)
It is noteworthy that, for each ξ ∈ N , the exponential function
eξ (z) := ehz,ξi ,
z ∈ N ′,
belongs to Fθ (N ′ ) and the set of such test functions spans a dense subspace of
Fθ (N ′ ).
We are interested in continuous linear operators from Fθ (N ′ ) into its topological
dual space Fθ∗ (N ′ ). The space of such operators is denoted by L(Fθ (N ′ ), Fθ∗ (N ′ ))
and assumed to carry the bounded convergence topology. A typical examples of
elements in L(Fθ (N ′ ), Fθ∗ (N ′ )), that will play a key role in our development, are
Hida’s white noise operators at . For z ∈ N ′ and ϕ(x) with Taylor expansions
440
HABIB OUERDIANE AND HAFEDH RGUIGUI
P∞
⊗n
, fn i
n=0 hx
in Fθ (N ′ ), the holomorphic derivative of ϕ at x ∈ N ′ in the direction z is defined by
ϕ(x + λz) − ϕ(x)
.
λ→0
λ
(a(z)ϕ)(x) := lim
(2.2)
We can check that the limit always exists, a(z) ∈ L(Fθ (N ′ ), Fθ (N ′ )) and a∗ (z) ∈
L(Fθ∗ (N ′ ), Fθ∗ (N ′ )), where a∗ (z) is the adjoint of a(z), i.e., for Φ ∈ Fθ∗ (N ′ ) and
φ ∈ Fθ (N ′ ), hha∗ (z)Φ, φii = hhΦ, a(z)φii. If z = δt ∈ E ′ we simply write at instead
of a(δt ) and the pair {at , a∗t } will be referred to as the QWN-process. In quantum
field theory at and a∗t are called the annihilation operator and creation operator
at the point t ∈ R. By a straightforward computation we have
at eξ = ξ(t) eξ ,
ξ ∈ N.
(2.3)
P
Similarly as above, for ψ ∈ Gθ∗ (N ) with Taylor expansion ψ(ξ) = n hψn , ξ ⊗n i
where ψn ∈ N ′⊗n , we use the common notation a(z)ψ for the derivative (2.2).
The Wick symbol of Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )) is by definition [17] a C-valued
function on N × N defined by
σ(Ξ)(ξ, η) = hhΞeξ , eη iie−hξ,ηi ,
ξ, η ∈ N.
′
(2.4)
), Fθ∗ (N ′ ))
By a density argument, every operator in L(Fθ (N
is uniquely determined by its Wick symbol. Moreover, if Gθ∗ (N ⊕ N ) denotes the nuclear space
obtained as in (2.1) by replacing Np by Np ⊕ Np , see [12], we have the following
characterization theorem for operator Wick symbols.
Theorem 2.1. ( See Refs. [12]) The Wick symbol map yields a topological isomorphism between L(Fθ (N ′ ), Fθ∗ (N ′ )) and Gθ∗ (N ⊕ N ).
In the remainder of this paper, for the sake of readers convenience, we simply
use the name symbol for the transformation σ.
Let µ be the standard Gaussian measure on E ′ uniquely specified by its characteristic function
Z
1
2
e− 2 |ξ|0 =
eihx,ξi µ(dx),
ξ ∈ E.
E′
In all the remainder of this paper we assume that the Young function θ satisfies
the following condition
θ(x)
lim sup 2 < +∞.
(2.5)
x
x→∞
It is shown in Ref. [5] that, under this condition, we have the nuclear Gel’fand
triple (1.2). Moreover, we observe that L(Fθ (N ′ ), Fθ (N ′ )), L(Fθ∗ (N ′ ), Fθ (N ′ ))
and L(L2 (E ′ , µ), L2 (E ′ , µ)) can be considered as subspaces of L(Fθ (N ′ ), Fθ∗ (N ′ )).
Furthermore, identified with its restriction to Fθ (N ′ ), each operator Ξ in the space
L(Fθ∗ (N ′ ), Fθ∗ (N ′ )) will be considered as an element of L(Fθ (N ′ ), Fθ∗ (N ′ )), so that
we have the continuous inclusions
L(Fθ∗ (N ′ ), Fθ∗ (N ′ )) ⊂ L(Fθ (N ′ ), Fθ∗ (N ′ )),
L(Fθ∗ (N ′ ), Fθ (N ′ )) ⊂ L(Fθ (N ′ ), Fθ∗ (N ′ )).
QWN-CONSERVATION OPERATOR AND ASSOCIATED DIFFERENTIAL EQUATION
441
It is a fundamental fact in QWN theory [17] (see, also Ref. [12]) that every white
noise operator Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )) admits a unique Fock expansion
Ξ=
∞
X
Ξl,m (κl,m ),
(2.6)
l,m=0
where, for each pairing l, m ≥ 0, κl,m ∈ (N ⊗(l+m) )′sym(l,m) and Ξl,m (κl,m ) is the
integral kernel operator characterized via the symbol transform by
σ(Ξl,m (κl,m ))(ξ, η) = hκl,m , η ⊗l ⊗ ξ ⊗m i,
ξ, η ∈ N.
(2.7)
This can be formally reexpressed as
Ξl,m (κl,m ) =
R
Rl+m
κl,m (s1 , · · · , sl , t1 , · · · , tm )
a∗s1 · · · a∗sl at1 · · · atm ds1 · · · dsl dt1 · · · dtm .
In this way Ξl,m (κl,m ) can be considered as the operator polynomials of degree
l + m associated to the distribution κl,m ∈ (N ⊗(l+m) )′sym(l,m) as coefficient; and
therefore every white noise operator is a “function” of the QWN. This gives a natural
idea for defining the derivatives of an operator Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )) with respect
to the QWN coordinate system {at , a∗t ; t ∈ R}.
From Refs. [7] and [8], (see also Refs. [9] and [1]), we summarize the novel
formalism of QWN-derivatives. For ζ ∈ N , then a(ζ) extends to a continuous
linear operator from Fθ∗ (N ′ ) into itself (denoted by the same symbol) and a∗ (ζ)
(restricted to Fθ (N ′ )) is a continuous linear operator from Fθ (N ′ ) into itself. Thus,
for any white noise operator Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )), the commutators
[a(ζ), Ξ] = a(ζ)Ξ − Ξ a(ζ),
[a∗ (ζ), Ξ] = a∗ (ζ)Ξ − Ξ a∗ (ζ),
are well defined white noise operators in L(Fθ (N ′ ), Fθ∗ (N ′ )). The QWN-derivatives
are defined by
Dζ+ Ξ = [a(ζ), Ξ] , Dζ− Ξ = −[a∗ (ζ), Ξ].
(2.8)
These are called the creation derivative and annihilation derivative of Ξ, respectively.
3. QWN-Conservation Operator
In the following technical lemma, by using the symbol transform σ, we reformulate the QWN-derivatives Dz± as natural QWN counterparts of the partial derivatives
∂
∂1,x1 ≡ ∂x∂ 1 and ∂2,x2 ≡ ∂x
on the space of entire functions with two variables
2
g(x1 , x2 ) in Gθ∗ (N ⊕ N ). More precisely, for x1 ,x2 ,z ∈ N ,
g(x1 + λz, x2 ) − g(x1 , x2 )
,
λ→0
λ
(∂1,z g)(x1 , x2 ) := lim
(3.1)
g(x1 , x2 + λz) − g(x1 , x2 )
.
(3.2)
λ
Then, in view of Theorem 2.1 and using the same technic of calculus used in [3],
we have the following
(∂2,z g)(x1 , x2 ) := lim
λ→0
442
HABIB OUERDIANE AND HAFEDH RGUIGUI
Lemma 3.1. Let be given z ∈ N . The creation derivative and annihilation derivative of Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )) are given by
Dz− Ξ = σ −1 ∂1,z σ(Ξ)
and
Dz+ Ξ = σ −1 ∂2,z σ(Ξ).
Moreover, their dual adjoints are given by
∗
(Dz− )∗ Ξ = σ −1 ∂1,z
σ(Ξ)
and
∗
(Dz+ )∗ Ξ = σ −1 ∂2,z
σ(Ξ).
In the remainder of this paper we need to use the action of Dz± on the operator
Ξl,m (κl,m ) for a given l, m ≥ 0 and κl,m in (N ⊗(l+m) )′sym(l,m) . Therefore, for
z ∈ N , by direct computation, the partial derivatives of the identity (2.7) in the
direction z are given by
∂1,z σ(Ξl,m (κl,m ))(ξ, η)
b
= mhκl,m , η ⊗l ⊗ (ξ ⊗(m−1) ⊗z)i
= σ(mΞl,m−1 (κl,m ⊗1 z))(ξ, η)
(3.3)
b ⊗(l−1) ) ⊗ ξ ⊗m i
= lhκl,m , (z ⊗η
(3.4)
and
∂2,z σ(Ξl,m (κl,m ))(ξ, η)
= σ(lΞl−1,m (z ⊗1 κl,m ))(ξ, η),
where, for zp ∈ (N ⊗p )′ , and ξl+m−p ∈ N ⊗(l+m−p) , p ≤ l + m, the contractions
zp ⊗p κl,m and κl,m ⊗p zp are defined by
hzp ⊗p κl,m , ξl−p+m i = hκl,m , zp ⊗ ξl−p+m i
hκl,m ⊗p zp , ξl+m−p i = hκl,m , ξl+m−p ⊗ zp i.
∗
∗
Similarly, if we denote ∂1,z
and ∂2,z
the adjoint operators of ∂1,z and ∂2,z respectively, we get
∗
∂1,z
σ(Ξl,m (κl,m ))(ξ, η) = σ(Ξl,m+1 (κl,m ⊗ z))(ξ, η)
(3.5)
∗
∂2,z
σ(Ξl,m (κl,m ))(ξ, η) = σ(Ξl+1,m (z ⊗ κl,m ))(ξ, η).
(3.6)
Note that, from [3] and the above discussion, for z ∈ N , the QWN-derivatives Dz±
and (Dz± )∗ are continuous linear operators from L(Fθ∗ (N ′ ), Fθ (N ′ )) into itself and
from L(Fθ (N ′ ), Fθ∗ (N ′ )) into itself, i.e., Dz± , (Dz± )∗ ∈ L(L(Fθ∗ (N ′ ), Fθ (N ′ ))) ∩
L(L(Fθ (N ′ ), Fθ∗ (N ′ ))).
Theorem 3.2. For z ∈ N and Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )), we have
(Dz+ )∗ Ξ = a∗ (z) ⋄ Ξ,
(Dz− )∗ Ξ = a(z) ⋄ Ξ.
P
Proof. Let Ξ = l,m=0 Ξl,m (κl,m ) ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )). Using (3.5), (3.6) and
Lemma 3.1, we get
∞
X
(Dz+ )∗ Ξ =
Ξl+1,m (z ⊗ κl,m )
(3.7)
l,m=0
and
(Dz− )∗ Ξ =
∞
X
l,m=0
Ξl,m+1 (κl,m ⊗ z).
(3.8)
QWN-CONSERVATION OPERATOR AND ASSOCIATED DIFFERENTIAL EQUATION
443
On the other hand,
σ(a∗ (z) ⋄ Ξ)(ξ, η)
=
=
σ(a∗ (z))(ξ, η).σ(Ξ)(ξ, η)
∞
X
hz, ηi
hκl,m , η ⊗l ⊗ ξ ⊗m i
l,m=0
=
∞
X
hz ⊗ κl,m , η ⊗l+1 ⊗ ξ ⊗m i
l,m=0
=
Then, for Ξ =
P∞
l,m=0

σ
∞
X

l,m=0
Ξl+1,m (z ⊗ κl,m ) (ξ, η).
Ξl,m (κl,m ) ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )), we get
a∗ (z) ⋄ Ξ =
∞
X
Ξl+1,m (z ⊗ κl,m ).
(3.9)
l,m=0
Similarly, we obtain
σ(a(z) ⋄ Ξ)(ξ, η)
=
=
σ(a(z))(ξ, η)σ(Ξ)(ξ, η)
∞
X
hz, ξi
hκl,m , η ⊗l ⊗ ξ ⊗m i
l,m=0
=
∞
X
hκl,m ⊗ z, η ⊗l ⊗ ξ ⊗m+1 i
l,m=0
=

σ
∞
X
l,m=0

Ξl,m+1 (κl,m ⊗ z) (ξ, η).
From which, we get
a(z) ⋄ Ξ =
∞
X
Ξl,m+1 (κl,m ⊗ z).
(3.10)
l,m=0
Hence, by (3.7), (3.8), (3.9) and (3.10) we get the desired statement.
3.1. Representation of the QWN-Conservation operator. For Locally convex
spaces X and Y we denote by L(X, Y) the set of all continuous linear operators
from X into Y. Let B1 and B2 in L(N ′ , N ′ ). For Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )), define
eQ
N
B1 ,B2 (σ(Ξ)) to be
eQ
N
B1 ,B2 (σ(Ξ))(ξ, η) =
∞
X
j=0
∗
∂1,e
∂ ∗ σ(Ξ)(ξ, η)
j 1,B2 ej
+
∞
X
∗
∂2,e
∂ ∗ σ(Ξ)(ξ, η).
j 2,B1 ej
j=0
(3.11)
Q
e
Using a technic of calculus used in [2, 3, 12, 13] one can show that NB1 ,B2 (σ(Ξ))
belongs to Gθ∗ (N ⊕ N ) which gives us an essence to the following
444
HABIB OUERDIANE AND HAFEDH RGUIGUI
Definition 3.3. For Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )), we define the QWN-conservation operator at Ξ by
eQ
NBQ1 ,B2 Ξ = σ −1 (N
(3.12)
B1 ,B2 (σ(Ξ))).
As a straightforward fact, the QWN-conservation operator is a continuous linear
operator from L(Fθ (N ′ ), Fθ∗ (N ′ )) into itself.
For a later use, define the operator Ξa,b for a, b ∈ N ′ by
Ξa,b ≡
∞
X
Ξl,m (κl,m (a, b)) ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )),
l,m=0
1 ⊗l
a ⊗ b⊗m . It is noteworthy that {Ξa,b ; a, b ∈ N ′ } spans
l!m!
a dense subspace of L(Fθ (N ′ ), Fθ∗ (N ′ )).
where κl,m (a, b) =
Proposition 3.4. The QWN-conservation operator admits on L(Fθ (N ′ ), Fθ∗ (N ′ ))
the following representation
NBQ1 ,B2
=
∞
X
−
(De−j )∗ DB
∗e +
j
2
j=1
∞
X
+
(De+j )∗ DB
∗e ·
j
1
(3.13)
j=1
Proof. From the fact
σ(Ξa,b )(ξ, η) = exp{ha, ηi + hb, ξi} ,
ξ, η ∈ N, a, b ∈ N ′
by using (3.11), we compute
a,b
eQ
N
B1 ,B2 σ Ξ
(ξ, η)
=
∞ X
hej , B2 bihej , ξi + hej , B1 aihej , ηi
eha,ηi+hb,ξi
j=0
=
=
(hB2 b, ξi + hB1 a, ηi) eha,ηi+hb,ξi
(hB2 b, ξi + hB1 a, ηi) σ Ξa,b (ξ, η).
(3.14)
On the other hand, we get


∞
∞
X
X
−
a,b 
(ξ,
η)
=
hej , ξihB2∗ ej , bieha,ηi+hb,ξi
σ  (De−j )∗ DB
∗e Ξ
j
2
j=1
j=1


∞
∞
X
X
+
a,b 
σ  (De+j )∗ DB
(ξ,
η)
=
hej , ηihB1∗ ej , aieha,ηi+hb,ξi
∗e Ξ
j
1
j=1
j=1
which gives that
P
P∞
∞
+
− ∗ −
a,b
a,b
(ξ, η)
+ j=1 (De+j )∗ DB
σ
∗e Ξ
j=1 (Dej ) DB ∗ ej Ξ
j
1
2
a,b
= (hB1 a, ηi + hB2 b, ξi)σ(Ξ
)(ξ, η)·
Hence, the representation (3.13) follows by (3.17) and density argument.
QWN-CONSERVATION OPERATOR AND ASSOCIATED DIFFERENTIAL EQUATION
445
Remark 3.5. Note that, by a straightforward calculus using the symbol map we
see that the QWN-conservation operator defined in this paper on L(Fθ (N ′ ), Fθ∗ (N ′ ))
coincides on L(Fθ∗ (N ′ ), Fθ (N ′ )) with the QWN-conservation operator defined in [3]
and coincides with its adjoint on L(Fθ (N ′ ), Fθ∗ (N ′ )), which shows that the QWNconservation operator is symmetric.
3.2. QWN-Conservation operator as a Wick derivation. It is shown (see Refs.
[12]) that Gθ∗ (N ⊕ N ) is closed under pointwise multiplication. Then, for any
Ξ1 , Ξ2 ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )), there exists a unique Ξ ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )) such
that σ(Ξ) = σ(Ξ1 )σ(Ξ2 ). The operator Ξ will be denoted Ξ1 ⋄ Ξ2 and it will be
referred to as the Wick product of Ξ1 and Ξ2 . It is noteworthy that, endowed with
the Wick product ⋄, L(Fθ (N ′ ), Fθ∗ (N ′ )) becomes a commutative algebra.
Since (L (Fθ (N ′ ), Fθ∗ (N ′ )) , ⋄) is a topological algebra, each white noise operator
Ξ0 in L (Fθ (N ′ ), Fθ∗ (N ′ )) gives rise to an operator-valued Wick operator
Ξ 7→ Ξ0 ⋄ Ξ ∈ L (Fθ (N ′ ), Fθ∗ (N ′ )) ,
Ξ ∈ L (Fθ (N ′ ), Fθ∗ (N ′ )) .
In fact this is a continuous operator. We then adopt the following slightly general
definition: a linear operator D from L (Fθ (N ′ ), Fθ∗ (N ′ )) into itself is called a Wick
derivation (see [11]) if
Ξ1 , Ξ2 ∈ L(Fθ (N ′ ), Fθ∗ (N ′ )).
D(Ξ1 ⋄ Ξ2 ) = D(Ξ1 ) ⋄ Ξ2 + Ξ1 ⋄ D(Ξ2 ),
As a non trivial example, we study, in this paper the QWN-conservation operator NBQ1 ,B2 on an appropriate subset of L (Fθ (N ′ ), Fθ∗ (N ′ )); more precisely, from
L(Fθ∗ (N ′ ), Fθ (N ′ )) into itself. As in [11], we can prove that D is a continuous
Wick derivation from L(Fθ (N ′ ), Fθ∗ (N ′ )) into itself if and only if there exist a
white noise operator coefficients F, G ∈ N ⊗ L (Fθ (N ′ ), Fθ∗ (N ′ )) such that
Z
Z
+
D=
F(t) ⋄ Dt dt + G(t) ⋄ Dt− dt,
(3.15)
R
R
where F(t), G(t) ∈ L (Fθ (N ′ ), Fθ∗ (N ′ )) are identified with QWN Wick operators with
parameter t. In fact t 7→ F(t) and t 7→ G(t) are L (Fθ (N ′ ), Fθ∗ (N ′ )) −valued processes on R, namely, F(t, x) and G(t, x) are elements in N ⊗ L (Fθ (N ′ ), Fθ∗ (N ′ )) ∼
=
N ⊗ Fθ∗ (N ′ ) ⊗ Fθ∗ (N ′ ).
Theorem 3.6. For B1 , B2 ∈ L(N ′ , N ′ ), the QWN-conservation operator is a Wick
derivation with coefficients
Z
Z
∗
F=
τB1 (s, .)as ds and G =
τB2 (s, .)as ds,
R
R
i.e., the QWN-conservation operator admits, on L(Fθ (N ′ ), Fθ∗ (N ′ )), the following
integral representation
Z
Z
NBQ1 ,B2 =
τB1 (s, t)a∗s ⋄ Dt+ dsdt +
τB2 (s, t)as ⋄ Dt− dsdt.
(3.16)
R2
R2
446
HABIB OUERDIANE AND HAFEDH RGUIGUI
Proof. By straightforward computation, by using (3.11), we obtain
∞ X
a,b
ha,ηi+hb,ξi
eQ
N
σ
Ξ
(ξ,
η)
=
e
he
,
B
bihe
,
ξi
+
he
,
B
aihe
,
ηi
j
2
j
j
1
j
B1 ,B2
j=0
(hB2 b, ξi + hB1 a, ηi) eha,ηi+hb,ξi
(hB2 b, ξi + hB1 a, ηi) σ Ξa,b (ξ, η).
=
=
On the other hand, denote
N Q− (B2 ) ≡
N Q+ (B1 ) ≡
Z
Z
R2
R2
(3.17)
τB2 (s, t)as ⋄ Dt− dsdt
τB1 (s, t)a∗s ⋄ Dt+ dsdt.
Then, from the fact
σ(Ξa,b )(ξ, η) = exp{ha, ηi + hb, ξi} ,
ξ, η ∈ N, a, b ∈ N ′
and using (3.3) and (3.5) we compute
σ N Q− (B2 )Ξa,b (ξ, η)
XZ
=
τB2 (s, t)σ as )(ξ, η)σ(Dt− Ξl,m (κl,m (a, b)) (ξ, η)dsdt
l,m
=
=
R2
eha,ηi+hb,ξi
Z
τB2 (s, t)b(t)ξ(s)dsdt
R2
a,b
hB2 b, ξiσ(Ξ
)(ξ, η).
(3.18)
Similarly, using (3.4) and (3.6) we obtain
σ (N Q+ (B1 )Ξa,b (ξ, η) = hB1 a, ηiσ(Ξa,b )(ξ, η).
(3.19)
Hence, by Theorem 2.1 and a density argument, we complete the proof.
4. Application to Wick Differential Equation
In this section we give an important example of the differential equation associated with the QWN-conservation operator where the solution of (1.7) is given
explicitly in terms of the solution of the associated Cauchy problem (1.6). Let us
n
start by studying the Cauchy problem. Let B1 , B2 ∈ L(N ′ , N ′ ) such
P∞that {B1 ; n =
n
1, 2, · · · } and {B2 ; n = 1, 2, · · · } are equi-continuous. For Ξ = l,m=0 Ξl,m (κl,m )
e Q is
in L(Fθ (N ′ ), F ∗ (N ′ )) and κl,m ∈ (N ⊗(l+m) )′
, the transformation G
θ
sym(l,m)
t
defined by
eQ
G
t σ(Ξ)(ξ, η) :=
∞
X
l,m=0
(etB1 )⊗l ⊗ (etB2 )⊗m κl,m , η ⊗l ⊗ ξ ⊗m .
(4.1)
eQ
Note that it is easy to show that G
t σ(Ξ) belongs to Gθ ∗ (N ⊕ N ), see [2]. Then
using Theorem 2.1, there exists a continuous linear operator GQ
t acting on the
nuclear algebra L(Fθ (N ′ ), Fθ∗ (N ′ )) such that
−1 e Q
GQ
Gt σ(Ξ).
t Ξ = σ
QWN-CONSERVATION OPERATOR AND ASSOCIATED DIFFERENTIAL EQUATION
447
Similarly to the classical case studied in [2] and the scalar case studied in [6], we
have the following
Lemma 4.1. The solution of the Cauchy problem associated with the QWN- conservation operator (1.6) is given by Ut = GQ
t U0 .
∗
Let β be a Young function satisfying the condition (2.5) and put θ = (eβ − 1)∗ .
For Υ ∈ L(Fβ (N ′ ), Fβ∗ (N ′ )) the exponential Wick wexp(Υ) defined by
wexp(Υ) =
∞
X
1 ⋄n
Υ ,
n!
n=0
belongs to L(Fθ (N ′ ), Fθ∗ (N ′ )), see [12]. In the following we study the system of
Wick differential equations for white noise operators of the form
Dt Ξt = Πt ⋄ Ξt ,
Ξ0 ∈ L(Fβ (N ′ ), Fβ∗ (N ′ ))
(4.2)
where Πt ∈ L(Fβ (N ′ ), Fβ∗ (N ′ )) and
Dt :=
∂
− NBQ1 ,B2 .
∂t
Eq. (4.2) is referred to as a Wick differential equation associated to the QWNconservation operator.
Theorem 4.2. The unique solution of the Wick differential equation (4.2) is given
by
Z t
Z t
Q
Q
Q
Ξt = GQ
Ξ
⋄
wexp
G
(Π
)ds
=
G
(Ξ
)
⋄
wexp
G
(Π
)ds
.
0
s
0
s
t
t
−s
t−s
0
Proof. Applying the operator
∂
Ξt
∂t
=
0
(4.3)
∂
∂t
to the right hand side of Eq. (4.3), we get
Z t
∂ Q
Gt (Ξ0 ) ⋄ wexp
GQ
(Π
)ds
s
t−s
∂t
0
Z t
Z t
Q
Q
Q
Q
+ Gt (Ξ0 ) ⋄ NB1 ,B2
Gt−s (Πs )ds ⋄ wexp
Gt−s (Πs ) ds
0
0
Z t
+ GQ
GQ
t (Ξ0 ) ⋄ Πt ⋄ wexp
t−s (Πs ) ds.
0
Then, using Lemma 4.1 and Theorem 3.6 , we get
∂
Ξt = NBQ1 ,B2 (Ξt ) + Πt ⋄ Ξt .
∂t
Which shows that Ξt is solution of (4.2). Let Ξt be an arbitrary solution of (4.2)
and put
Z t
Ft = Ξt ⋄ wexp −
GQ
(Π
)ds
.
s
t−s
0
448
HABIB OUERDIANE AND HAFEDH RGUIGUI
Then, we have
∂
Ft
∂t
=
=
Z t
∂
Ξt ⋄ wexp −
GQ
(Π
)ds
s
t−s
∂t
0
Z t
+ NBQ1 ,B2 −
GQ
(Π
)ds
⋄ Ft − Πt ⋄ Ft
s
t−s
0
Z t
∂
Q
Ξt − Πt ⋄ Ξt ⋄ wexp −
Gt−s (Πs )ds
∂t
0
Z t
Q
Q
+ NB1 ,B2 wexp −
Gt−s (Πs )ds ⋄ Ξt .
0
Using Eq. (4.2) and the fact that NBQ1 ,B2 is a Wick derivation, we obtain
∂
Ft
∂t
Z t
= NBQ1 ,B2 (Ξt ) ⋄ wexp −
GQ
(Π
)ds
s
t−s
0
Z t
+ Ξt ⋄ NBQ1 ,B2 wexp −
GQ
(Π
)ds
s
t−s
0
= NBQ1 ,B2 (Ft ).
From which we deduce that Dt Ft = 0. Then, by Lemma 4.1, we get
Q
Ft = GQ
t (F0 ) = Gt (Ξ0 ).
Therefore, we deduce that
Ξt = GQ
t (Ξ0 ) ⋄ wexp
Z
0
t
(Π
)ds
.
GQ
s
t−s
Now, using (4.1), we obtain
tB2 ∗
(e ) ξ, (etB1 )∗ η .
σ GQ
t (Ξ) (ξ, η) = σ Ξ
Then, using the definition of the Wick product of two operators, for every Ξ1 ,
Ξ2 ∈ L(Fβ (N ′ ), Fβ∗ (N ′ )), we get
σ GQ
t (Ξ1 ⋄ Ξ2 ) (ξ, η)
= σ Ξ1 ⋄ Ξ2 (etB2 )∗ ξ, (etB1 )∗ η
= σ Ξ1 (etB2 )∗ ξ, (etB1 )∗ η . σ Ξ2 (etB2 )∗ ξ, (etB1 )∗ η
Q
= σ GQ
(Ξ
)
(ξ,
η)
.
σ
G
(Ξ
)
(ξ, η).
1
2
t
t
From which we deduce that
Q
Q
GQ
t (Ξ1 ⋄ Ξ2 ) = Gt (Ξ1 ) ⋄ Gt (Ξ2 ).
QWN-CONSERVATION OPERATOR AND ASSOCIATED DIFFERENTIAL EQUATION
Hence, we get
449
R
t
GQ
Ξ0 ⋄ wexp 0 GQ
t
−s (Πs )ds
Z t
Q
Q Q
= Gt (Ξ0 ) ⋄ wexp
Gt G−s (Πs )ds
0
Z t
Q
Q
= Gt (Ξ0 ) ⋄ wexp
Gt−s (Πs )ds .
0
Which completes the proof.
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Habib Ouerdiane: Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El-Manar, 1060 Tunis, Tunisia
E-mail address: [email protected]
Hafedh Rguigui: Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El-Manar, 1060 Tunis, Tunisia
E-mail address: [email protected]
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 451-470
www.serialspublications.com
SDE SOLUTIONS IN THE SPACE OF SMOOTH RANDOM
VARIABLES
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
Abstract. In this paper we analyze properties of a dual pair (G, G ∗ ) of spaces
of smooth and generalized random variables on a L´
evy white noise space. We
show that G ⊂ L2 (µ) which shares properties with a Fr´
echet algebra contains
a larger class of solutions of Itˆ
o equations driven by pure jump L´
evy processes.
Further a characterization of (G, G ∗ ) in terms of the S-transform is given.
We propose (G, G ∗ ) as an attractive alternative to the Meyer-Watanabe test
function and distribution space (D∞ , D−∞ ) [30] to study strong solutions of
SDE’s.
1. Introduction
Gel’fand triples or dual pairs of spaces of random variables have proved to be
very useful in the study of various problems of stochastic analysis. Important
applications pertain e.g. to the analysis of the regularity of the solutions of the
Zakai equation in non-linear filtering theory, positive distributions in potential
theory, the construction of local time of L´evy processes or the Clark-Ocone formula
for the hedging of contingent claims in mathematical finance. See e.g. [4, 6, 8, 28]
and the references therein.
The most prominent examples of dual pairs in stochastic and infinite dimensional analysis are ((S), (S)∗ ) of Hida and (D∞ , D−∞ ) of Meyer and Watanabe.
See [6], [8] and [30]. The Hida test function and distribution space ((S), (S)∗ )
has been e.g. successfully applied to quantum field theory, the theory of stochastic partial differential equations or the construction of Feynman integrals ([6, 7]).
One of the most interesting properties of the distribution space (S)∗ is that it accommodates the singular white noise which can be viewed as the time-derivative
of the Brownian motion. The latter provides a favorable setting for the study of
stochastic differential equations (see [24]). See also [15], where the authors derived
an explicit representation for strong solutions of Itˆo equations. From an analytic
point of view the pair ((S), (S)∗ ) also exhibits the nice feature that it can be characterized by the powerful tool of S−transform [6]. It is also worth mentioning
that test functions in (S) admit continuous versions on the white noise probability
space. However the Brownian motion is not contained in (S) since elements in (S)
have chaos expansions with kernels in the Schwartz test function space. Therefore
Received 2012-1-31; Communicated by Hui-Hsiung Kuo.
2000 Mathematics Subject Classification. Primary 60H07 60J75; Secondary 65C30 60H40.
Key words and phrases. Strong solutions of SDE’s with jumps; Malliavin calculus; white noise
analysis.
451
452
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
(S) does not seem to be suitable for the study of SDE’s. It turns out that the test
function space D∞ is more appropriate for the investigation of solutions of SDE’s
than (S), since it carries a larger class of solutions of Itˆo equations. However a
severe deficiency of the pair (D∞ , D−∞ ) compared to ((S), (S)∗ ) is that it lacks
the availability of characterization-type theorems.
In this paper we propose a dual pair (G, G ∗ ) of smooth and generalized random
variables on a L´evy white noise space which meets the following two important
requirements: A richer class of solutions of (pure jump) L´evy noise driven Itˆo
equations belongs to the test function space G. On the other hand (G, G ∗ ) allows
for a characterization-type theorem.
The pair (G, G ∗ ) has been studied in the Gaussian case by [2, 13, 22, 29]. See
also [4] and the references therein for the case of L´evy processes. Similarly to the
Gaussian case, G is defined by means of exponential weights of the number operator
on a L´evy white noise space. The space G comprises the test functions in (S) and
is included in the space D∞,2 ⊃ D∞ . The important question whether G contains
a bigger class of Itˆ
o jump diffusions has not been addressed so far in the literature.
We will give an affirmative answer to this problem. For example, one can more
or less directly show (Section 4) that solutions of compound Poisson driven SDE‘s
are contained in G. Furthermore we will discuss a characterization of Levy noise
functionals in terms of the S−transform by using the concept Bargmann-Segal
spaces (see [5]). We believe that the pair (G, G ∗ ) could serve as an alternative
tool to (D∞ , D−∞ ) for the study of L´evy noise functionals. By approximating
general Levy measures by finite ones and by using the “nice” topologies of G and
G ∗ it is conceivable that one can construct solutions to SDE’s (with discontinuous
coefficients) driven by more general Levy processes (as e.g. the variance gamma
process or even Levy processes of unbounded variation). See e.g. [17] in the
Brownian motion case. See also [24].
The paper is organized as follows: In Section 2 we introduce the framework of
our paper, that is we briefly elaborate some basic concepts of a white noise analysis
for L´evy processes and give the definitions of the pairs (D∞ , D−∞ ), (G, G ∗ ). In
Section 3 we discuss some properties of (G, G ∗ ) and provide a characterization
theorem. In Section 4 we verify that a bigger class of SDE solutions actually lives
in G.
2. Framework
In this section, we concisely recall some concepts of white noise analysis of pure
jump L´evy processes which was developed in [14, 15]. This theory presents a
framework which is suitable for all pure jump L´evy processes. For general information about white noise theory, see [6, 10, 11] and [19]. We conclude this section
with a discussion of the dual pairs (D∞ , D−∞ ), (G, G ∗ ) and ((S), (S)∗ ).
2.1. White noise analysis of L´
evy processes. A L´evy process L(t) is defined
as a stochastic process on R+ which starts in zero and has stationary and independent increments. It is a canonical example of a semimartingale, which is uniquely
determined by the characteristic triplet
(Bt , Ct , µ
ˆ) = (a · t, σ · t, dtν(dz)),
(2.1)
SDE SOLUTIONS
453
where a, σ are constants and ν is the L´evy measure on R0 = R \ {0}. We denote
by π the product measure π(dt, dz) := dtν(dz). For more information about such
processes, see e.g. [1, 3, 9, 25, 26]. In this paper, we are only dealing with the case
of pure jump L´evy processes without drift, i.e. (2.1) with a = σ = 0.
We want to work with a white noise measure, which is constructed on the
nuclear algebra Sep (X) as introduced in [15]. Here X := R × R0 . For that purpose,
recall that S(R) is the Schwartz space of test functions on R and the space S p (R)
e
is its dual space, which is the space of tempered distributions. The space S(X)
which is a variation of the Schwartz space on the space X is then defined as the
quotient algebra
e
S(X)
= S(X)/Nπ ,
(2.2)
where S(X) is a closed subspace of S(R2 ), given by
∂
2
S(X) := ϕ(t, z) ∈ S(R ) : ϕ(t, 0) = ( ϕ)(t, 0) = 0
∂z
(2.3)
and the closed ideal Nπ in S(X) is defined as
Nπ := {φ ∈ S(X) : kφkL2 (π) = 0}.
(2.4)
e
The space S(X)
is a nuclear algebra with a compatible system of norms given by
k φˆ kp,π := inf k φ + ψ kp ,
ψ∈Nπ
p ≥ 0,
(2.5)
where k · kp , p ≥ 0 are the norms of S(R2 ). Moreover the Cauchy-Bunjakowski
ˆ ψˆ ∈
inequality holds, that is for all p ∈ N there exists an Mp such that for all φ,
˜
S(X)
we have
w w
w w w w
wφˆψˆw ≤ Mp wφˆw wψˆw .
p,π
p,π
p,π
We indicate Sep (X) as its dual. For further information, see [15].
Next, we define the (pure jump) L´evy white noise probability measure µ on the
Borel sets of Ω = Sep (X), by means of Bochner-Minlos-Sazonov theorem
Z
Z
ihω,φi
iφ
e
dµ(ω) = exp
(e − 1)dπ
(2.6)
Sep (X)
X
e
for all φ ∈ S(X),
where hω, φi := ω(φ) denotes the action of ω ∈ Sep (X) on
e
e
φ ∈ S(X).
For ω ∈ Sep (X) and φ ∈ S(X),
define the exponential functional
Z
e˜(φ, ω) := (e(·, ω) ◦ l) (φ) = exp hω, ln(1 + φ)i −
φ(x)λ ⊗ ν(dx)
X
as a function of φ ∈ Seq0 (X) for functions φ ∈ Seq0 (X) satisfying φ(x) > −1 and
l(x) = ln(1 + x) is analytic in a neighborhood of zero with l(0) = 0 for all x ∈ X.
See [15].
ˆ
⊗n
e
e
Denote by S(X)
the n-th completed symmetric tensor product of S(X)
with
itself. Since e˜(φ, ω) is holomorphic in φ around zero for φ(x) > −1, it can be
454
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
expanded into a power series. Furthermore, there exist generalized Charlier polyˆ p
⊗n
e
nomials Cn (ω) ∈ (S(X)
) such that
e˜(φ, ω) =
X 1
hCn (ω), φ⊗n i
n!
(2.7)
n≥0
for φ in a certain neighborhood of zero. One shows that
ˆ
⊗n
e
{hCn (·), φ(n) i : φ(n) ∈ S(X)
, n ∈ N0 }
(2.8)
ˆ
⊗n
e
is a total set of L2 (µ). Further, one observes that for all n, m, φ(n) ∈ S(X)
,
ˆ
⊗m
e
ψ (m) ∈ S(X)
the orthogonality relation
Z
hCn (ω), φ(n) ihCm (ω), ψ (m) iµ(dω) = δn,m n! (φ(n) , ψ (m) )L2 (X n ,πn ) (2.9)
ep (X)
S
holds, where
δn,m =
0, n 6= m
1, else
is the Kronecker symbol. Using (2.9) and a density argument we can extend
hCn (ω), φ(n) i to act on φ(n) ∈ L2 (X n , π n ) for ω a.e. The functionals hCn (ω), φ(n) i
can be regarded as an n-fold iterated stochastic integral of functions φ(n) ∈
L2 (X n , π n ) with respect to the compensated Poisson random measure
e (dt, dz) = N (dt, dz) − ν(dz)dt,
N
where N (Λ1 , Λ2 ) := hω, 1Λ1 ×Λ2 i for Λ1 ∈ R and Λ2 ∈ R such that zero is not in
the closure of Λ2 , defined on our white noise probability space
(Ω, F , P ) = Sep (X), B(Sep (X)), µ .
In this setting, a square integrable pure jump L´evy process L(t) can be represented
as
Z tZ
e (dt, dz).
L(t) =
zN
0
R0
ˆ 2 (X n , π n ) the space of square integrable functions φ(n) (t1 , z1 , . . . , tn , zn )
Denote L
being symmetric in the n-pairs (t1 , z1 ), . . . , (tn , zn ). Then one infers from (2.7) to
(2.9) the L´evy-Itˆ
o chaos representation property of square integrable L´evy functionals: For all F ∈ L2 (µ), there exists a unique sequence of φ(n) ∈ L2 (X n , π n )
such that
X
F (ω) =
hCn (ω), φ(n) i
n≥0
for ω a.e. Moreover, we have the Itˆo-isometry
X
k F k2L2 (µ) =
n! k φ(n) k2L2 (X n ,πn ) .
n≥0
(2.10)
SDE SOLUTIONS
455
2.2. The spaces (D∞ , D−∞ ), (G, G ∗ ) and ((S), (S)∗ ). In our search for appropriate candidates of subspaces of L2 (µ) in which strong solutions of SDE’s
live, we shall focus on the Meyer-Watanabe test function and distribution spaces
(D∞ , D−∞ ) and the dual pair (G, G ∗ ) of smooth and generalized random variables
on the L´evy white noise space.
The Meyer-Watanabe test function D∞ for pure jump L´evy process (see e.g.
[4, 31, 32]) is defined as a dense subspace of L2 (µ) endowed with the topology
given by the seminorms

1/p
k
X
p
j
k F kk,p = E[| F |p ] +
E[k D·,·
F kL2 (πn ) ] ,
(2.11)
j=1
k ∈ N, p ≥ 1, with
Dtj1 ,z1 ,...,tj ,zj F (ω) := Dt1 ,z1 Dt2 ,z2 . . . Dtj ,zj F (ω)
for F ∈ D∞ , where Dt,z stands for the Malliavin derivative in the direction of
the (square integrable) pure jump L´evy process L(t), t ≥ 0. D·,· is defined as a
mapping
D : D1,2 → L2 (µ × π)
given by
X
Dt,z F =
n · hCn−1 (·), φ(n) (·, t, z)i,
(2.12)
n≥1
2
if F ∈ L (µ) with chaos expansion
F =
X
hCn (·), φ(n) i
n≥0
satisfies
X
n≥1
n · n!kφ(n) k2L2 (πn ) < ∞.
(2.13)
The domain D1,2 of D·,· is the space of all F ∈ L2 (µ) such that inequality (2.13)
holds. See [4, 23, 31, 32] for further information.
The Meyer-Watanabe distribution space D−∞ is defined as the (topological)
dual of D∞ . If one combines the transfer principle from the Wiener space (or
Gaussian white noise space) to the Poisson space as devised in [23] with the results
of [30], one finds that solutions of non-degenerate jump SDE’s exist in D∞ . This
is a striking feature which pays off dividends in the analysis of L´evy functionals.
However it seems not that easy to set up a characterization-type theorem for
(D∞ , D−∞ ) in the sense of [21]. Consequently, other Gel’fand triples have been
studied to overcome this deficiency. In [22] the authors study the pair (G, G ∗ ) and
provide sufficient conditions in terms of the S-transform to characterize (G, G ∗ ).
Using Bargmann-Segal spaces, a complete characterization of this pair (and for a
scale of closely related pairs) is obtained by [5] in the Gaussian case.
We will show in Section 3 and 4 that (G, G ∗ ) can be characterized by means
of the S-transform on the L´evy noise space and that G contains a richer class
of solutions of jump SDE’s. These two properties make (G, G ∗ ) an interesting
alternative to (D∞ , D−∞ ) to analyze functionals of L´evy processes.
456
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
The test function space G is a subspace of L2 (µ) which is constructed by means
of exponential weights of the Ornstein-Uhlenbeck or number operator. Denoted
by N , this operator acts on the elements of L2 (µ) by multiplying the n-th homogeneous chaos with n ∈ N0 . The space of smooth random variables G is defined as
the collection of all
X
f=
hCn (·), φ(n) i ∈ L2 (µ)
(2.14)
n≥0
such that
k f k2q :=k eqN f k2L2 (µ) < ∞
for all q ≥ 0. The latter condition is equivalent to
X
k f k2q =
n!e2qn k φ(n) k2L2 (X n ,πn )
(2.15)
n≥0
for all q ≥ 0. The space G is endowed with the topology given by the family
of norms k · kq , q ≥ 0. Its topological dual is the space of generalized random
variables G ∗ .
Let us turn our attention to the S-transform which is a fundamental concept
of white noise distribution theory and serves as a tool to characterize elements of
the Hida test function space (S) and the Hida distribution space (S)∗ . See [6] or
[15] for a precise definition of the pair ((S), (S)∗ ). The S-transform of Φ ∈ (S)∗ ,
denoted by S(Φ), is defined as the dual pairing
S(Φ)(φ) := hΦ, e˜(φ, ·)i, φ ∈ SeC (X),
(2.16)
P
∞
e
e
where k e˜(φ, ·) k2 = n=0 k φ k2n
p,π and SC (X) is the complexification of S(X).
The S-transform is a monomorphism, that is, if
Φ, Ψ ∈ (S)∗
S(Φ) = S(Ψ) for
then
Φ = Ψ.
One verifies, e.g. that
˜˙ (t, z))(φ) = φ(t, z),
S(N
(2.17)
˜˙ (t, z) the white noise of the compensated Poisson random measure N
˜ (dt, dz) in
N
∗
˜
(S) and φ ∈ SC (X). We refer the reader to [6] or [15] for more information on
the Hida test function space (S) and Hida distribution space (S)∗ .
Finally, we give the important definition of the Wick or Wick-Grassmann product, which can be considered a tensor algebra multiplication on the Fock space.
The Wick product of two distributions Φ, Ψ ∈ (S)∗ , denoted by Φ⋄Ψ, is the unique
element in (S)∗ such that
S(Φ ⋄ Ψ)(φ) = S(Φ)(φ)S(Ψ)(φ)
(2.18)
ˆ (m) i
hCn (ω), φ(n) i ⋄ hCm (ω), ψ (m) i = hCn+m (ω), φ(n) ⊗ψ
(2.19)
for all φ ∈ SeC (X). As an example one finds that
ˆ
⊗n
e
for φ(n) ∈ (S(X))
ˆ
⊗m
e
and ψ (m) ∈ (S(X))
. The latter and (2.7) imply that
⋄
e˜(φ, ω) = exp (hω, φi)
(2.20)
SDE SOLUTIONS
457
e
for φ ∈ S(X).
The Wick exponential exp⋄ (X) of an X ∈ (S)∗ is defined as
X 1
X ⋄n
(2.21)
exp⋄ (X) =
n!
n≥0
∗
provided the sum converges in (S) , where X ⋄n = X ⋄ . . . ⋄ X. We mention that
the following chain of continuous inclusions is valid:
(S) ֒→ G ֒→ L2 (µ) ֒→ G ∗ ֒→ (S)∗ .
3. Properties of the Spaces G and G ∗
In the Gaussian case the space G has the nice feature to be stable in the sense
of pointwise multiplication of random variables. More precisely, G is a Fr´echet
algebra. See [13, 22]. In the L´evy setting, we can show the following:
Theorem 3.1. Suppose that our L´evy measure ν satisfies the moment condition
Z
| z |n ν(dz) < ∞
R0
P
for all n ∈ N. Let F, G be in G with chaos expansions F = n≥0 hCn (·), φ(n) i and
P
G = n≥0 hCn (·), ϕ(n) i. Define KR = {(t, z) ∈ R × R0 : k (t, z) k< R}, R > 0.
Assume that
√
sup n! k φ(n) kL∞ (X n ,πn ) < ∞
(3.1)
n≥0
and
sup
n≥0
√
n! k ϕ(n) kL∞ (X n ,πn ) < ∞.
(3.2)
In addition require that there exists a R > 0 such that the compact support of φ(n)
and ϕ(n) are in (KR )n , i.e.,
supp φ(n) , supp ϕ(n) ⊆ (KR )n
for all n ≥ 0. Then
F · G ∈ G.
p
√
In particular, let λ0 =
+ ln(4R) + ln( 2 + 2) and assume that for
λ > 2λ0 , F, G ∈ Gλ . Then for all ν > λ0 + λ2 , F · G ∈ Gλ−ν .
P
(n)
Proof. Let F , G ∈ Gλ ⊂ G for some λ ∈ R with F =
i and
n≥0 hCn (·), φ
P
(m)
G = m≥0 hCm (·), ϕ i. Then,
X
k F k2λ =
n!e2λn k φ(n) k2L2 (πn ) < ∞,
ln(π(KR ))
4
n≥0
kG
k2λ =
X
m≥0
m!e2λm k ϕ(m) k2L2 (πm ) < ∞.
By the product formula in [12], we obtain the following:
hCn (·), φ(n) i · hCm (·), ϕ(m) i
m∧n
X (m∧n)−k
X
m n m−k
n−k
ˆ rk ϕ(m) i,
=
k! r!
hCm+n−2k−r (·), φ(n) ⊗
k
k
r
r
r=0
k=0
458
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
ˆ 2 (X n ), ϕ ∈ L
ˆ 2 (X m ), 0 ≤ k ≤ m ∧ n, 0 ≤ r ≤ m ∧ n − k, is
ˆ rk ϕ for φ ∈ L
where φ⊗
the symmetrization of the function φ ⊗rk ϕ on X n−k−r × X m−k−r × X r given by
Y
Z
r
φ ⊗k ϕ(A, B, Z) :=
p2 (z)
φ(A, Z, Y )ϕ(Y, Z, B)dπ ⊗k (Y )
Xk
z∈Z
for (A, B, Z) ∈ X n−k−r × X m−k−r × X r . Here
p
(z)
:= z1 · z2 · · · zr
z∈Z 2
Q
when Z = ((t1 , z1 ), (t2 , z2 ), · · · , (tr , zr )). Because of Lemma 3.4 in [12], we know
that
q
q
ˆ rk ϕ(m) kL2 (Rm+n−2k−r ) ≤ Rr 4 (π(KR ))(m+n−2r) · k φ(n) kL∞
k φ(n) ⊗
q
q
q
· k ϕ(m) kL∞ · k φ(n) kL2 (Rn ) · k ϕ(m) kL2 (Rm ) .
Moreover, using the conditions (3.1) and (3.2), we obtain the following inequality:
k hCn (·), φ(n) i · hCm (·), ϕ(m) i kλ−ν ≤
m∧n
X (m∧n)−k
X
k! r!
r=0
k=0
m n m−k
k
k
r
n − k (λ−ν)(m+n−2k−r)
ˆ rk ϕ(m) i kL2 (µ)
e
k hCm+n−2k−r (w), φ(n) ⊗
r
m∧n
X (m∧n)−k
X
m n m−k
n − k λ(m+n) −ν(m+n) −(λ−ν)(2k+r)
≤
k! r!
e
e
e
k
k
r
r
r=0
k=0
p
ˆ rk ϕ(m) kL2 (Rm+n−2k−r )
(m + n − 2k − r)! k φ(n) ⊗
1/2
1/2
λ
≤ constant k hCn (·), φ(n) i kλ k hCm (·), ϕ(m) i kλ e−ν(m+n) e 2 (m+n) 2m+n
m∧n
X mnp
−(λ−ν)2k
k!
(m + n − 2k)!e
k
k
k=0
p
p
m∧n
X (m∧n)−k
X Rr 4 (π(KR ))(m+n−2r) r! (m + n − 2k − r)! √ √ p
,
n! m! (m + n − 2k)!
r=0
k=0
for ν < λ. From now, without loss of generality we assume that π(KR ) > e and
R ≥ 1. It is clear that
p
p
r! (m + n − 2k − r)!
(m + n − 2k − r)!
r!
√ √ p
p
≤
≤ 1,
(n
∧
m)!
n! m! (m + n − 2k)!
(m + n − 2k)!
m∧n
X
r=0
Rr
q
4
(π(KR ))(m+n−2r)
m+n
4
ln(π(KR ))
m∧n
X
≤
e
≤
(m ∧ n)Rm+n e
<
Rr
r=0
(2R)m+n e
m+n
4
m+n
4
ln(π(KR ))
ln(π(KR ))
,
SDE SOLUTIONS
459
and
√
m∧n
q
X mnp
√
2 1/2
−(λ−ν)2k
k!
(m + n − 2k)!e
≤ (√
) ( 2 + 2)m+n .
k
k
2−1
k=0
Therefore,
k hCn (·), φ(n) i · hCm (·), ϕ(m) i kλ−ν
1/2
1/2
≤k hCn (·), φ(n) i kλ k hCm (·), ϕ(m) i kλ
Hσn σm ,
p
√
where H is a constant and σ = 4Re−ν+λ/2+ln(π(KR ))/4 ( 2 + 2). Then,
k F · G kλ−ν
∞
X
=k
≤
hCn (·), φ(n) ihCm (·), ϕ(m) i kλ−ν
m,n=0
∞
X
k hCn (·), φ(n) ihCm (·), ϕ(m) i kλ−ν
m,n=0
X
∞
≤H
≤H
n=0
X
∞
(n)
σ k hCn (·), φ
σ 4/3n
n=0
X
∞
n=0
≤ H(
n
3/4 X
∞
n=0
i
3
1−σ
4
3
X
∞
n=0
n
1/4
1/2
(n)
σ k hCn (·), ϕ
k hCn (·), φ(n) i k2λ
k hCn (·), ϕ(n) i k2λ
1
1/2
kλ
1/4 X
∞
n=0
i
1/2
kλ
σ 4/3n
3/4
1/2
) 2 k F kλ k G kλ ,
p
√
+ ln(4R) + ln( 2 + 2).
√
Remark 3.2. Note that, in the conditions (3.1) and (3.2), n! can be replaced by
√
3
8 n!.
if σ < 1, i.e. ν >
λ
2
+ λ0 , where λ0 =
ln(π(KR ))
4
Define the space D∞,2 ⊃ D∞ as
D∞,2 = proj lim Dk,2
(3.3)
k→0
and denote by D−∞,2 its topological dual. Then it is apparent from the definition
of G that
G ⊂ D∞,2 ⊂ L2 (µ) ⊂ D−∞,2 ⊂ G ∗ .
(3.4)
If L(t) is a Poisson process, then a transfer principle to Poisson spaces based on
exponential distributions (see [23]) gives
G ⊂ D∞ ⊂ L2 (µ) ⊂ D−∞ ⊂ G ∗ .
Finally, we want to discuss the characterization of the spaces G and G ∗ in terms
of the S-transform. For this purpose assume a densely defined operator A on
L2 (X, π) such that
Aξj = λj ξj , j ≥ 1,
460
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
e
where 1 < λ1 ≤ λ2 ≤ ... and {ξj }j≥1 ⊂ S(X)
is an orthonormal basis of L2 (X, π).
Further we require that there exists a α > 0 such that A−α/2 is Hilbert-Schmidt.
Then let us denote by S the standard countably Hilbert space constructed from
A (see [19]). An application of the Bochner-Minlos theorem leads to a Gaussian
measure µG on S p (dual of S) such that
Z
− 1 kφk2
eihω,φi µG (dω) = e 2 L2 (X,π)
Sp
for all ξ ∈ S. It is well-known that each element f in L2 (µG ) has the chaos
representation
E
XD
f=
Hn (·), φ(n) ,
(3.5)
n≥0
b p
for unique φ ∈ L (X , π ), n ≥ 0, where Hn (ω) ∈ (S ⊗n
) are generalized Hermite polynomials. Comparing (2.14) with (3.5) we observe that the mapping
(n)
2
n
n
U : L2 (µ) −→ L2 (µG )
given by
(3.6)
E
E
XD
XD
Cn (ω), φ(n) 7−→
Hn (ω), φ(n)
n≥0
n≥0
is a unitary isomorphism between the spaces L2 (µ) and L2 (µG ). In the following
let us denote by SG the S−transform on the Gaussian Hida distribution space
(S)∗µG which is defined as
where
SG (φ) = hΦ, ee(φ, ω)i , φ ∈ (S)∗µG ,
e(φ, ω) = e
e
hω,φi−1/2kφk2L2 (X,π)
(3.7)
.
∗
See [6]. Our characterization of (G, G ) requires the concept of Bargmann-Segal
space (see [27], [5] and the references therein):
Definition 3.3. Let µG, 21 be the Gaussian measure on S p associated with the
− 1 kφk2
characteristic function C(φ) := e 4 L2 (X,π) . Introduce the measure ν on SCp
given by
ν(dz) = µG, 12 (dx) × µG, 21 (dy),
where z = x + iy. Further denote by P the collection of all projections P of the
form
m
X
Pz =
hz, ξj i ξj , z ∈ SCp .
j=1
2
The Bargmann-Segal space E (ν) is the space consisting of all entire functions
f : L2C (X, µG ) −→ C such that
Z
sup
|f (P z)| ν(dz) < ∞.
P ∈P
SCp
So we obtain from Theorem 7.1 and 7.3 in [5] the following result:
SDE SOLUTIONS
461
Theorem 3.4. (i) The smooth random variable ϕ belongs to G if and only if
SG (U(ϕ))(λ·) ∈ E 2 (ν)
for all λ > 0.
(ii) The generalized random variable Φ is an element of G ∗ if and only if there is
a λ > 0 such that
SG (U(ϕ))(λ·) ∈ E 2 (ν).
Remark 3.5. The connection between SG ◦ U and S in (2.16) is given by the
following relation: Since
D
E D
E
b k
b k
b 1
b 1
⊗n
⊗n
⊗n
b
b
b
b
U Cn (·), φ⊗n
⊗...
⊗φ
=
H
(·),
φ
⊗...
⊗φ
n
1
1
k
k
for φ1 , ..., φk ∈ L2 (X, π), ni ≥ 1 with n1 + ... + nk = n we find (see (2.19)) that
D
E
b k
b 1
⊗n
b
b
S Cn (·), φ⊗n
⊗...
⊗φ
= (S (hC1 (·), φ1 i))n1 · ... · (S (hC1 (·), φk i))nk
1
k
as well as
SG ◦ U
D
b
b
1 b
k
b ⊗n
Cn (·), φ⊗n
⊗...⊗φ
1
k
E
= (SG ◦ U (hC1 (·), φ1 i))n1 · ... · (SG ◦ U (hC1 (·), φk i))nk .
We conclude this section with a sufficient condition for a Hida distribution to
be an element of G:
Theorem 3.6. Let Q be a positive quadratic form on L2 (X, π) with finite trace.
Further let Φ be in (S)∗ . Assume that for every ǫ > 0 there exists a K(ǫ) > 0 such
that
2
|SG (U(Φ))(zφ)| ≤ K(ǫ)eǫ|z| Q(φ,φ)
holds for all φ ∈ S and z ∈ C for some constant K > 0. Then Φ ∈ G.
Proof. The proof is a direct consequence from the proof of Theorem 4.1 in [22]. Example 3.7. Let γ ∈ L2 (X, π) with γ > −1 and ǫ > 0. Then
Y (t) := exp⋄ ( C1 (ω), χ[0,t] γ )
is the solution of
−
dY (t) = Y (t )
Z tZ
0
So we get
|SG (U(Y (t)))(zφ)|
where K(ǫ) = e1/(4ǫ) and
Q(φ, φ) =
≤
≤
Z
X
Thus Y (t) ∈ G.
R0
e (dt, du).
γ(t, u)N
Z
exp( χ[0,t] zφ(x)γ(x)π(dx) )
X
K(ǫ) exp(ǫ |z|2 Q(φ, φ)),
2
χ[0,t] γ(x)φ(x)π(dx) .
462
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
4. Solutions of SDE’s in G
In this section, we deal with strong solutions of pure jump L´evy stochastic
differential equations of the type
Z tZ
˜ (ds, dz)
X(t) = x +
γ(s, X(s− ), z)N
(4.1)
0
R0
for X(0) = x ∈ R, where γ : [0, T ] × R × R0 → R satisfies the linear growth and
Lipschitz condition, i.e.,
Z
| γ(t, x, z) |2 ν(dz) ≤ C(1+ | x |2 ),
(4.2)
Z
R0
R0
| γ(t, x, z) − γ(t, y, z) |2 ν(dz) ≤ K | x − y |2 ,
(4.3)
where C, K and M are some constants such that | γ(t, x, z) |< M for all x, y ∈ R,
0 ≤ t ≤ T . Note that since γ satisfies the conditions (4.2) and (4.3), there exists
a unique solution X = {X(t), t ∈ [0, T ]} with the initial condition X(0) = x.
Moreover, the process is adapted and c´adl´ag [1].
If ν(R0 ) < ∞ (i.e. X(t), t ≥ 0 is compound Poissonian), we will prove that
X(t) ∈ G, t ≥ 0. To this end we need some auxiliary results:
Lemma 4.1. Let {Xn }∞
n=0 be a sequence of random variables converging to X in
L2 (µ). Suppose that
sup k | Xn | kk,2 < ∞
n
k
k
for some k ≥ 1. Then X ∈ Dk,2 and D·,·
Xn , n ≥ 0 converges to D·,·
X in the
2
k
sense of the weak topology of L ((λ × ν × µ) ).
Proof. First note that supn k | Xn | kk,2 < ∞ is equivalent to
k
sup k (1 + N ) 2 Xn kk,2 < ∞.
n
By weak compactness, there exists a subsequence {Xni }∞
i,n=1 such that (1 +
k
2
N ) 2 Xni ) converges weakly to some element α ∈ L (µ × (λ × ν)k ). Then for
k
any Y in the domain of (1 + N ) 2 , it follows from the self-adjointness of N that
k
k
E X(1 + N ) 2 Y
= lim E Xni (1 + N ) 2 Y
n→∞
k
= lim E (1 + N ) 2 Xni Y
n→∞
k
= E lim (1 + N ) 2 Xni Y
n→∞
= E αY .
k
k
Therefore α = ((1 + N ) 2 )∗ X = (1 + N ) 2 X. For the proof in Brownian motion
case, see e.g. [18].
For notational convenience, we shall identify from now on Malliavin derivatives
N
of the same order, that is we set Dr,z
X(t) = DrN1 ,z1 ,r2 ,z2 ,...,rN ,zN X(t).
SDE SOLUTIONS
463
Lemma 4.2. Let X(t), 0 ≤ t ≤ T be defined as in the Equation (4.1). Then
N
X(t) ∈ D∞,2 , i.e. D·,·
X(t) exists for all N ≥ 1.
We need the following results to prove this lemma:
Proposition 4.3. Let X ∈ D1,2 and f be a real continuous function on R. Then
f (X) ∈ D1,2 and
Dt,z f (X) = f (X + Dt,z X) − f (X).
(4.4)
Proof. See e.g. [4].
Lemma 4.4. Let X(t), 0 ≤ t ≤ T be defined as in the Equation (4.1). Then, the
N -th Malliavin derivative of X(t) can be written as
Z tZ X
N N
−k X
N
N −k
N
i
˜ (ds, dξ)
Dr,z
X(t) =
(−1)k γ(s,
Dr,z
X(s− ), ξ)N
i
k
r
R0 k=0
i=0
N
−1
NX
−k−1 X N −1
N −k−1
k
i
X(r− ), z),
+N
(−1) γ(r,
Dr,z
k
i
i=0
k=0
(4.5)
0
for N ≥ 1 and Dr,z
X(t) := X(t).
Proof. We will prove the equality (4.5) by induction methodology. Since the proof
is basically based on calculations, we include it in the Appendix.
Now, we are ready to prove Lemma 4.2.
Proof. Let us consider the Picard approximations Xn (t) to X(t) given by
Z tZ
˜ (ds, dz),
Xn+1 (t) = x +
γ(s, Xn (s− ), z)N
0
(4.6)
R0
for n ≥ 0 and X0 (t) = x. We want to show by induction on n that Xn (t) belongs
to DN,2 and
N Z t
X
ϕn+1,N (t) ≤ k1 + k2
ϕn,j (u) du,
j=1
0
for all n ≥ 0, N ≥ 1 and t ∈ [0, T ] where
"Z
#
N
2
ϕn+1,N (t) := sup E
sup | Dr,z Xn+1 (s) | ν(dz) . . . ν(dz) < ∞.
0≤r≤t
r≤s≤t
RN
0
Note that
N
Dr,z
Xn+1 (t)
N −k
i
˜ (ds, dξ)
Dr,z
Xn (s− ), ξ)N
=
(−1) γ(s,
i
k
r
R0 k=0
i=0
N
−1 NX
−k−1 X
N −1
N −k−1
i
+N
(−1)k γ(r,
Dr,z
Xn (r− ), z),
k
i
i=0
k=0
(4.7)
Z tZ
N X
N
k
N
−k X
464
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
0
with Dr,z
Xn (s− ) := Xn (s− ). See Lemma 4.4 for a proof. Then, by Doob’s
maximal inequality, Fubini’s theorem, Itˆo isometry, Equation (4.2) and (4.3), we
get
Z
N
X
E
2
j
Dr,z
Xn+1 (s) (ν(dz))j
Rj0 r≤s≤t
j=1
=
sup
Z
N
X
E
j=1
sup
Z
Rj0 r≤s≤t
s
r
j X
j
(−1)k
k
R0
Z
k=0
X
j−k j−k
i
˜ (du, dξ)
Xn (u− ), ξ N
γ u,
Dr,z
i
i=0
j−k−1
2
j−1 X j − k − 1
X
j−1
i
Dr,z
Xn (r− ), z
(ν(dz))j
(−1)k γ r,
+j
k
i
i=0
k=0
j N Z
Z s Z X
X
j
≤2
E sup
(−1)k
j
k
r≤s≤t
R
r
R
0
0
j=1
k=0
X
j−k 2 j−k
i
˜ (du, dξ)
γ u,
Dr,z
Xn (u− ), ξ N
(ν(dz))j
i
i=0
X
j−1 N
X Z
j−1
+2
E
sup j
(−1)k
j
k
r≤s≤t
R
0
j=1
k=0
j−k−1
2
X j − k − 1
i
γ r,
Dr,z
Xn (r− ), z
(ν(dz))j
i
i=0
Z t Z X
j N Z
X
j
(−1)k
≤8
E
j
k
r
R0
j=1 R0
k=0
X
2 j−k j−k
i
˜ (du, dξ)
γ u,
Dr,z
(ν(dz))j
Xn (u− ), ξ N
i
i=0
j−1 N
X Z X
j−1
+2
E
j
(−1)k
j
k
R0
j=1
k=0
j−k−1
2
X j − k − 1
i
γ r,
Dr,z
Xn (r− ), z
(ν(dz))j
i
i=0
Z t Z X
j N Z
X
j
(−1)k
=8
E
j
k
r
R0
j=1 R0
k=0
X
2
j−k j−k
i
−
γ u,
Dr,z Xn (u ), ξ
ν(dξ)du (ν(dz))j
i
i=0
SDE SOLUTIONS
465
Z X
j−1 j−1
(−1)k
+2
j E
j
k
R
0
j=1
N
X
2
k=0
j−k−1
2
X j − k − 1
i
−
j
γ r,
Dr,z Xn (r ), z
(ν(dz))
i
i=0
j
N Z t Z
X
X
i
− 2
j
≤ k1 + k2
E
|Dr,z Xn (u )| (ν(dz)) du,
j=1
r
Rj0 k=0
(4.8)
for some constants k1 and k2 . Applying a discrete version of Gronwall’s inequality
to Equation (4.8) we get
sup k | Xn | kN,2< ∞,
n
for all N ≥ 1. Moreover, note that
2
E sup | Xn (s) − X(s) |
→0
0≤s≤T
as n goes to infinity by the Picard approximation. Hence, by Lemma 4.1 we
conclude that X(t) ∈ D∞,2 .
Theorem 4.5. Let X(t) be the strong solution of the SDE,
Z
˜ (dt, dz),
dX(t) =
γ(t, X(t− ), z)N
(4.9)
R0
with X(0) = x ∈ R. Assume that γ : [0, T ] × R × R0 → R satisfies the conditions
(4.2) and (4.3). Then,
X(t) ∈ Gq
for all q ∈ R and for all 0 ≤ t ≤ T .
Proof. Using the isometry U : L2 (µ) −→ L2 (µG ) in (3.6) and Meyer’s inequality
(see e.g. [20]), we obtain that
2n
k N n X(t) k2 ≤ Cn k D·,·
X(t) k2L2 ((λ×ν)2n ×µ) + k X(t) k2L2 (µ) ,
where Cn ≥ 0 is a constant depending on n. The proof of Meyer’s inequality in
[20] or Theorem 1.5.1 in [18] shows that Cn is given by
Cn = M n−1
n−1
Y
1 j
(1 + ) 2 ,
j
j=1
n≥1
for a universal constant M . We see that
Cn ≤ M n−1 e
n−1
2
,
n ≥ 1.
466
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
Thus we get
k X(t) kq
=
≤
≤
k eqN X(t) kL2 (µ)
X qn
k N n X(t) kL2 (µ)
n!
n≥0
X q n n−2 n−1
2n
M 2 e 4 k D·,·
X(t) kL2 ((λ×ν)2n ×µ) + k X(t) kL2 (µ)
n!
n≥0
On the other hand, it follows from Equation (4.5) that
k
2n
D·,·
X(t)
kL2 ((λ×ν)2n ×µ)
n X
n
≤
L·
≤
L · (n + 1)2 + n3 2n−k
≤
(n + 1)
k=0
n
k
+n
L · 23n+1
for a constant L ≥ 0. Hence we get
k X(t) kq ≤ (L + 1) · e16
√
4
e M·q
2
n−1
X
k=0
n−k
k
!
k X(t) kL2 (µ) < ∞.
Remark 4.6. We shall mention that the proof of Theorem 4.5 also carries over to
backward stochastic differential equations (BSDE’s) of the type
Z T
Z TZ
˜ (ds, dz),
Y (t) = x +
f (s, Y (s), Z(s, ·)) ds −
Z(s− , z) N
(4.10)
t
t
R0
provided e.g. that the driver f is bounded and fulfills a linear growth and Lipschitz
condition and ν(R0 ) < ∞.
Appendix: Proof of Lemma 4.4
For N = 1, we have
Dr,z X(t)
1−k
Z tZ X
1 X 1 − k 1
i
˜ (ds, dξ) + γ(r, X(r− ), z)
=
(−1)k γ s,
Dr,z
X(s− ), ξ N
i
r R0 k=0 k
i=0
Z tZ
i
˜ (ds, dξ) + γ(r, X(r− ), z).
=
γ(s, X(s− ) + Dr,z
X(s− ), ξ) − γ(s, X(s− ), ξ) N
r R0
Let us assume that it holds for N ≥ 1. Hence
N +1
Dr,z
X(t)
N
= Dr,z (Dr,z
X(t))
Z tZ X
NX
N −k N
N −k
i
˜ (ds, dξ)
= Dr,z
(−1)k γ s,
Dr,z
X(s− ), ξ N
k
i
r
R0
i=0
k=0
SDE SOLUTIONS
NX
−k−1 N −1
N −k−1
i
(−1)k γ r,
Dr,z
X(r− ), z
k
i
i=0
k=0
Z tZ X
N N
=
(−1)k
k
r
R0 k=0
NX
−k N
−k X
N −k
N −k
i
i+1
˜ (ds, dξ)
Dr,z
X(s− ) +
Dr,z
X(s− ), ξ N
γ s,
i
i
i=0
i=0
NX
Z tZ X
N −k N −k
N
−
k
i
˜ (ds, dξ)
−
(−1) γ s,
Dr,z X(s ), ξ N
k
i
r
R0 k=0
i=0
NX
N −k X
N
N −k
−
k
i
+
(−1) γ r,
Dr,z X(r ), z
k
i
i=0
k=0
NX
−1 N −1
+N
(−1)k
k
k=0
NX
−k−1 NX
−k−1 N −k−1
N −k−1
i
i+1
γ r,
Dr,z
X(r− ) +
Dr,z
X(r− ), z
i
i
i=0
i=0
N
−1
NX
−k−1
X
N −1
N −k−1
k
i
−
−
(−1) γ r,
Dr,z X(r ), z .
k
i
i=0
+N
N
−1 X
467
k=0
Note that
N
−k X
i=0
N −k
i
and
i
Dr,z
+
i+1
Dr,z
NX
−k+1 N −k+1
−
i
X(s ) =
Dr,z
X(s− )
i
i=0
NX
−k−1 i=0
Hence,
N
−k X
N −k−1
N −1
i
i+1
i
Dr,z
+ Dr,z
X(s− ) =
Dr,z
X(s− ).
i
i
i=0
N +1
Dr,z
X(t)
NX
Z tZ X
N −k+1 N
N −k+1
k
i
−
˜ (ds, dξ)
=
(−1) γ s,
Dr,z X(s ), ξ N
k
i
r
R0 k=0
i=0
NX
Z tZ X
−k N N −k
N
i
−
k+1
˜ (ds, dξ)
Dr,z X(s ), ξ N
(−1) γ s,
+
i
k
r
R0 k=0
i=0
NX
N −k X
N
N −k
i
+
(−1)k γ r,
Dr,z
X(r− ), z
k
i
i=0
k=0
468
YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH
NX
NX
−1 −k N −1
N −k
k
i
−
+N
(−1) γ r,
Dr,z X(r ), z
k
i
i=0
k=0
NX
N
−1 −k−1 X
N −1
N −k−1
i
+
(−1)k γ r,
Dr,z
X(r− ), z
k
i
i=0
k=0
NX
Z tZ X
N −k+1 N
N −k+1
i
˜ (ds, dξ)
=
(−1)k γ s,
Dr,z
X(s− ), ξ N
k
i
r
R0 k=0
i=0
NX
Z tZ N
+1
−k+1
X
N
N −k+1
i
˜ (ds, dξ)
+
(−1)k γ s,
Dr,z
X(s− ), ξ N
i
r R0 k=1 k − 1
i=0
N
N
−k
X N
X N − k
−
k
i
+
(−1) γ r,
Dr,z X(r ), z
k
i
i=0
k=0
NX
NX
−1 −k N −1
N −k
k
i
−
+N
(−1) γ r,
Dr,z X(r ), z
k
i
i=0
k=0
NX
N −k X
N −k
N −1
i
+
(−1)k γ r,
Dr,z
X(r− ), z
k−1
i
i=0
k=1
Z tZ
N
+1 X
N +1
i
˜ (ds, dξ)
=
γ(s,
Dr,z
X(s− ), ξ)N
i
r
R0
i=0
Z tZ X
N N
N
+
(−1)k
+
k
k−1
r
R0 k=1
NX
−k+1 N −k+1
i
−
˜ (ds, dξ)
γ s,
Dr,z X(s ), ξ N
i
i=0
Z tZ
˜ (ds, dξ)
+
(−1)N +1 γ(s, X(s− ), ξ)N
r
R0
N X
NX
−k N
N −k
k
i
−
+
(−1) γ r,
Dr,z X(r ), z
k
i
i=0
k=0
X
N N
i
−
+ N γ r,
Dr,z X(r ), z + (−1)N γ(r, X(r− ), z)
i
i=0
NX
N
−1 −k X
N −1
N −1
N −k
k
i
−
+
+
(−1) γ r,
Dr,z X(r ), z
k−1
k
i
i=0
k=1
Z tZ
N
+1 X
N +1
i
˜ (ds, dξ)
=
γ(s,
Dr,z
X(s− ), ξ)N
i
r
R0
i=0
Z tZ
˜ (ds, dξ)
+
(−1)k γ(s, X(s− ), ξ)N
r
R0
SDE SOLUTIONS
469
NX
N −k+1 X
N +1
N −k+1
k
i
−
˜ (ds, dξ)
+
(−1) γ s,
Dr,z X(s ), ξ N
k
i
r
R0 k=1
i=0
NX
N −k X
N −k
N
k
i
−
+
(−1) γ r,
Dr,z X(r ), z
k
i
i=0
k=0
NX
N −k X
N
N −k
i
Dr,z
X(r− ), z
+N
(−1)k γ r,
k
i
i=0
k=0
NX
Z t Z NX
+1 −k+1 N +1
N −k+1
i
˜ (ds, dξ)
=
(−1)k γ s,
Dr,z
X(s− ), ξ N
k
i
r
R0 k=0
i=0
NX
N −k X
N
N −k
i
+ (N + 1)
(−1)k γ r,
Dr,z
X(r− ), z .
k
i
i=0
Z tZ
k=0
Acknowledgment. The authors would like to thank N. Privault and C. Scheid
for their useful comments.
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Yeliz Yolcu Okur: Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey
E-mail address: [email protected]
Frank Proske: Centre of Mathematics for Applications (CMA), University of Oslo,
Norway
E-mail address: [email protected]
Hasslah Binti Salleh: Department of Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Malaysia
E-mail address: [email protected]
Serials Publications
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 471-486
www.serialspublications.com
SOLUTIONS OF LINEAR ELLIPTIC EQUATIONS IN
GAUSS-SOBOLEV SPACES
PAO-LIU CHOW
Abstract. The paper is concerned with a class of linear elliptic equations
in a Gauss-Sobolev space setting. They arise from the stationary solutions
of the corresponding parabolic equations. For nonhomogeneous elliptic equations, under appropriate conditions, the existence and uniqueness theorem
for strong solutions is given. Then it is shown that the associated resolvent
operator is compact. Based on this result, we shall prove a Fredholm Alternative theorem for the elliptic equation and a Sturm-Liouville type of theorem
for the eigenvalue problem of a symmetric elliptic operator.
1. Introduction
The subject of parabolic equations in infinite dimensions has been studied by
many authors, (see e.g., the papers [6, 7, 17, 3] and in the books [8, 9]). As in
finite dimensions, an elliptic equation may be regarded as the equation for the
stationary solution of some parabolic equation, if exists, when the time goes to
infinity. For early works, in the abstract Wiener space, the infinite-dimensional
Laplace equation was treated in the context of potential theory by Gross [13]
and a nice exposition of the connection between the infinite-dimensional elliptic
and parabolic equations was given by Daleskii [6]. More recently, in the book
[9] by Da Prato and Zabczyk, the authors gave a detailed treatment of infinitedimensional elliptic equations in the spaces of continuous functions, where the
solutions are considered as the stationary solutions of the corresponding parabolic
equations. Similarly, in [4], we considered a class of semilinear parabolic equations
in an L2 -Gauss-Sobolev space and showed that, under suitable conditions, their
stationary solutions are the mild solutions of the related elliptic equations. So
far, in studying the elliptic problem, most results rely on its connection to the
parabolic equation which is the Kolmogorov equation of some diffusion process in
a Hilbert space. However, for partial differential equations in finite dimensions, the
theory of elliptic equations is considered in its own rights, independent of related
parabolic equations [12]. Therefore it is worthwhile to generalize this approach
to elliptic equations in infinite dimensions. In the present paper, similar to the
finite-dimensional case, we shall begin with a class of linear elliptic equations in a
L2 -Sobolev space setting with respect to a suitable Gaussian measure. It will be
Received 2012-8-26; Communicated by the editors.
2000 Mathematics Subject Classification. Primary 60H; Secondary 60G, 35K55, 35K99, 93E.
Key words and phrases. Elliptic equation in infinite dimensions, Gauss-Sobolev space, strong
solutions, compact resolvent, eigenvalue problem.
471
472
PAO-LIU CHOW
shown that several basic results for linear elliptic equations in finite dimensions
can be extended to the infinite-dimensional counter parts. In passing it is worth
noting that the infinite-dimensional Laplacians on a L´
evy - Gel’fand triple was
treated by Barhoumi, Kuo and Ouerdian [1].
To be specific, the paper is organized as follows. In Section 2, we recall some
basic results in Gauss-Sobolev spaces to be needed in the subsequent sections.
Section 3 pertains to the strong solutions of some linear elliptic equations in a
Gauss-Sobolev space, where the existence and uniqueness Theorem 3.2 is proved.
Section 4 contains a key result (Theorem 4.1) showing that the resolvent of the
elliptic operator is compact. Based on this result, the Fredholm Alternative Theorem 4.4 is proved. In Section 5, we first characterize the spectral properties of the
elliptic operator in Theorem 5.1. Then the eigenvalue problem for the symmetric
part of the elliptic operator is studied and the results are summarized in Theorem
5.2 and Theorem 5.3. They show that the eigenvalues are positive, nondecreasing
with finite multiplicity, and the set of normalized eigenfunctions forms a complete
orthonormal basis in the Hilbert space H consisting of all L2 (µ)−functions, where
µ is an invariant measure defined in Theorem 2.1. Moreover the principal eigenvalue is shown to be simple and can be characterized by a variational principle.
2. Preliminaries
Let H be a real separable Hilbert space with inner product (·, ·) and norm | · |.
Let V ⊂ H be a Hilbert subspace with norm k·k. Denote the dual space of V by V ′
and their duality pairing by h·, ·i. Assume that the inclusions V ⊂ H ∼
= H′ ⊂ V ′
are dense and continuous [15].
Suppose that A : V → V ′ is a continuous closed linear operator with domain
D(A) dense in H, and Wt is a H-valued Wiener process with the covariance operator R. Consider the linear stochastic equation in a distributional sense:
dut
u0
= Aut dt + d Wt ,
t > 0,
(2.1)
= h ∈ H.
Assume that the following conditions (A) hold:
(A.1) Let A : V → V ′ be a self-adjoint, coercive operator such that
hAv, vi ≤ −βkvk2 ,
for some β > 0, and (−A) has positive eigenvalues 0 < α1 ≤ α2 ≤ · · · ≤
αn ≤ · · · , counting the finite multiplicity, with αn ↑ ∞ as n → ∞. The
corresponding orthonormal set of eigenfunctions {en } is complete.
(A.2) The resolvent operator Rλ (A) and covariance operator R commute, so
that Rλ (A)R = R Rλ (A), where Rλ (A) = (λI − A)−1 , λ ≥ 0, with I
being the identity operator in H.
(A.3) The covariance operator R : H → H is a self-adjoint operator with a finite
trace such that T r R < ∞.
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
473
It follows from (A.2) and (A.3) that {en } is also the set of eigenfunctions of R
with eigenvalues {ρn } such that
R e n = ρn e n ,
where ρn > 0 and
∞
X
n = 1, 2, · · · , n, · · · ,
(2.2)
ρn < ∞.
n=1
By applying Theorem 4.1 in [5] for invariant measures and a direct calculation,
we have the following theorem.
Theorem 2.1. Under conditions (A), the stochastic equation (2.1) has a unique
invariant measure µ on H, which is a centered Gaussian measure with covariance
1
operator Γ = − A−1 R.
2
Remark 2.2. We make the following two remarks:
(1) It is easy to check that en ′ s are also eigenfunctions of Γ so that
Γ en = γn en , n = 1, 2, · · · , n, · · · ,
(2.3)
ρn
where γn =
.
2αn
tA
(2) Let e , t ≥ 0, denote the semigroup of operators on H generated by A.
Without condition R(A.2), the covariance operator of the invariant measure
∞
µ is given by Γ = 0 etA RetA dt, which cannot be evaluated in a closed
2
form. Though an L (µ)− theory can be developed in the subsequent analysis, one needs to impose some conditions which are not easily verifiable.
Let H = L2 (H, µ) be a Hilbert space consisting of real-valued functions Φ on
H with norm defined by
Z
|||Φ||| = { |Φ(v)|2 µ(dv)}1/2 ,
H
and the inner product [·, ·] given by
Z
[Θ, Φ] =
Θ(v)Φ(v)µ(dv),
for Θ, Φ ∈ H.
H
Let n = (n1 , n2 , · · · , nk , · · · ), where nk ∈ Z+ , be a sequence of nonnegative
∞
X
integers, and let Z = {n : n = |n| =
nk < ∞}, so that nk = 0 except for a
k=1
finite number of n′k s. Let hm (r) be the normalized
one-dimensional Hermite polynomial of degree m. For v ∈ H, define a Hermite
(polynomial) functional of degree n by
∞
Y
Hn (v) =
hnk [ℓk (v)],
k=1
−1/2
where we set ℓk (v) = (v, Γ
ek ) and Γ−1/2 denotes a pseudo-inverse of Γ1/2 . For
a smooth functional Φ on H, let DΦ and D2 Φ denote the Fr´echet derivatives of
the first and second orders, respectively. The differential operator
1
AΦ(v) = T r[RD2 Φ(v)] + hAv, DΦ(v)i
(2.4)
2
474
PAO-LIU CHOW
is well defined for a polynomial functional Φ with DΦ(v) lies in the domain D(A)
of A. However this condition is rather restrictive on Φ. For ease of calculations, in
place of Hermite functionals, introduce an exponential family EA (H) of functionals
as follows [8]:
.
EA (H) = Span{Re Φh , Im Φh : h ∈ D(A)},
(2.5)
.
where Φh (v) = exp{i(h, v)}. It is known that EA (H) ⊂ D(A) is dense in H. For
v ∈ EA (H), the equation (2.4) is well defined.
Returning to the Hermite functionals, it is known that the following holds [2]:
Proposition 2.3. The set of all Hermite functionals {Hn : n ∈ Z} forms a
complete orthonormal system in H. Moreover we have
AHn (v) = −λn Hn (v),
where λn =
∞
X
∀ n ∈ Z,
nk αnk .
k=1
We now introduce the L2 −Gauss-Sobolev spaces. For Φ ∈ H, by Proposition
2.2, it can be expressed as
Φ=
where Φn = [Φ, Hn ] and |||Φ|||2 =
X
X
Φ n Hn ,
n∈Z
|Φn |2 < ∞.
n
Let Hm denote the Gauss-Sobolev space of order m defined by
Hm = {Φ ∈ H : k|Φk|m < ∞}
for any integer m, where the norm
|||Φ|||m = |||(I − A)m/2 Φ||| = {
X
(1 + λn )m |Φn |2 }1/2 ,
(2.6)
n
′
with I being the identity operator in H = H0 . For m ≥ 1, the dual space Hm
of
Hm is denoted by H−m , and the duality pairing between them will be denoted by
hh·, ·iim with hh·, ·ii1 = hh·, ·ii. Clearly, the sequence of norms {|||Φ|||m } is increasing,
that is,
|||Φ|||m < |||Φ|||m+1 ,
for any integer m, and, by identify H with its dual H′ , we have
Hm ⊂ Hm−1 ⊂ · · · ⊂ H1 ⊂ H ⊂ H−1 ⊂ · · · ⊂ H−m+1 ⊂ H−m ,
for m ≥ 1,
and the inclusions are dense and continuous. Of course the spaces Hm can be
defined for any real number m, but they are not needed in this paper.
Owing to the use of the invariant measure µ, it is possible to develop a L2 -theory
of infinite-dimensional parabolic and elliptic equations connected to stochastic
PDEs. To do so, similar to the finite-dimensional case, the integration by parts is
an indispensable technique. In the abstract Wiener Space, the integration by parts
with respect the Wiener measure was obtained by Kuo [14]. As a generalization
to the Gaussian invariant measure µ, the following integration by parts formula
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
475
holds (see Lemma 9.2.3 [9]). In this case, instead of the usual derivative DΦ, it is
more natural to use the R−derivative
DR Φ = R1/2 DΦ,
which can be regarded as a Gross derivative or the derivative of Φ in the direction
.
of HR = R1/2 H.
Proposition 2.4. Let g ∈ HR and Φ, Ψ ∈ H1 . Then we have
Z
Z
Z
(DR Φ, g) Ψ dµ = −
Φ(DR Ψ, g) dµ +
(v, Γ−1/2 g)Φ Ψ dµ.
H
H
(2.7)
H
The following properties of A are crucial in the subsequent analysis. For now
let the differential operator A given by (2.4) be defined in the set of Hermite
polynomial functionals. In fact it can be extended to a self-adjoint linear operator
in H. To this end, let PN be a projection operator in H onto its subspace spanned
by the Hermite polynomial functionals of degree N and define AN = PN A. Then
the following theorem holds (Theorem 3.1, [2]).
Theorem 2.5. The sequence {AN } converges strongly to a linear symmetric operator A : H2 → H, so that, for Φ, Ψ ∈ H2 , the second integration by parts formula
holds:
Z
Z
Z
1
(AΦ, Ψ) dµ =
(AΨ)Φ dµ = −
(DR Φ, DR Ψ) dµ,
(2.8)
2 H
H
H
Moreover A has a self-adjoint extension, still denoted by A with domain dense in
H.
In particular, for m = 2, it follows from (2.6) and (2.8) that
Corollary 2.6. The H1 -norm can be defined as
1
|||Φ|||1 = {|||Φ|||2 + |||DR Φ|||2 }1/2 ,
2
. 1/2
for all Φ ∈ H1 , where DR Φ = R DΦ.
(2.9)
Remark 2.7. In (2.9) the factor 12 was not deleted for convenience. Also it becomes
clear that the space H1 consists of all L2 (µ)-functions whose R-derivatives are
µ−square-integrable.
Let the functions F : H → H and G : H → R be bounded and continuous. For
Q ∈ L2 ((0, T ); H) and Θ ∈ H, consider the initial-value problem for the parabolic
equation:
∂
Ψt (v) =
∂t
Ψ0 (v) =
A Ψt (v) − (F (v), DR Ψt (v)) − G(v)Ψt (v) + Qt (v),
(2.10)
Θ(v),
for 0 < t < T, v ∈ H, where A is given by (2.4). Suppose that the conditions for
Theorem 4.2 in [3] are met. Then the following proposition holds.
476
PAO-LIU CHOW
Proposition 2.8. The initial-value problem for the parabolic equation (2.10) has
a unique solution Ψ ∈ C([0, T ]; H) ∩ L2 ((0, T ); H1 ) such that
Z T
Z T
sup |||Ψt |||2 +
|||Ψs |||21 ds ≤ K(T ){1 + |||Θ|||2 +
|||Qs |||2 ds},
(2.11)
0≤t≤T
0
0
where K(T ) is a positive constant depending on T .
Moreover, when Qt = Q independent of t, it was shown that, as t → ∞, the
solution Φt of (2.10) approaches the mild solution Φ of the linear elliptic equation
−A Φ(v) + (F (v), DR Φ(v)) + G(v)Φ(v) = Q(v),
(2.12)
or, for α > 0 and Aα = A + α, Φ satisfies the equation
Φ(v) = A−1
α {(F (v), DR Φ(v)) + G(v)Φ(v) − Q(v)},
v ∈ H.
In what follows, we shall study the strong solutions (to be defined) of equation
(2.12) in an L2 -Gauss-Sobolev space setting and the related eigenvalue problems.
3. Solutions of Linear Elliptic Equations
Let L denote an linear elliptic operator defined by
L Φ = −A Φ + F Φ + G Φ,
Φ ∈ EA (H),
(3.1)
where A is given by (2.4) and
F Φ = (F (·), DR Φ(·)).
(3.2)
Then, for Φ ∈ H1 and Q ∈ H, the elliptic equation (2.12) can be written as
L Φ = Q,
(3.3)
in a generalized sense. Multiplying the equation (3.1) by Ψ ∈ H1 and integrating
the resulting equation with respect to µ, we obtain
Z
Z
1
(LΦ) Ψ dµ =
{ (DR Φ, DR Ψ) + (F Φ) Ψ + (G Φ) Ψ }dµ,
(3.4)
2
H
H
where the second integration by parts formula (2.8) was used.
Associated with L, we define a bilinear form B(·, ·) : H1 × H1 → R as follows
Z
1
B(Φ, Ψ) =
{ (DR Φ, DR Ψ) + (F Φ) Ψ + (G Φ) Ψ } dµ
2
H
(3.5)
1
=
[DR Φ, DR Ψ] + [F Φ, Ψ] + [GΦ, Ψ],
2
for Φ, Ψ ∈ H1 .
Now consider a generalized solution of the elliptic equation (3.3). There are
several versions of generalized solutions, such as mild solution, strict solution and
so on (see [9]). Here, for Q ∈ H−1 , a generalized solution Φ is said to be a strong
(or variational) solution of problem (3.3) if Φ ∈ H1 and it satisfies the following
equation
B(Φ, Ψ) = hhQ, Ψii,
for all Ψ ∈ H1 .
(3.6)
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
477
Lemma 3.1. (Energy inequalities) Suppose that F : H → H and G : H → R are
bounded and continuous. Then the following inequalities hold.
There exists a constant b > 0 such that
|B(Φ, Ψ)| ≤ b |||Φ|||1 |||Ψ|||1 ,
for Φ, Ψ ∈ H1 ,
(3.7)
and, for any ε ∈ (0, 1/2), B satisfies the coercivity condition:
1
β2
B(Φ, Φ) ≥ ( − ε)|||DR Φ|||2 + (δ −
)|||Φ|||2 ,
2
4ε
for Φ ∈ H1 ,
(3.8)
where β = sup |F (v)| and δ = inf G(v).
v∈H
v∈H
Proof. From the equations (3.2) and (3.5), we have
Z
1
|B(Φ, Ψ)| = |
{ (DR Φ, DR Ψ) + (F, DR Φ) Ψ + (G Φ) Ψ }dµ|
2
H
1
≤ |||DR Φ||| |||DR Ψ||| + |||(F, DR Φ)||| |||Ψ||| + |||GΦ||| |||Ψ|||
2
1
≤ |||DR Φ||| |||DR Ψ||| + β |||DR Φ||| |||Ψ||| + γ |||Φ||| |||Ψ|||,
2
(3.9)
where β = sup |F (v)| and γ = sup |G(v)|. It follows from (3.9) that
v∈H
v∈H
|B(Φ, Ψ)| ≤ b |||Φ|||1 |||Ψ|||1 ,
for some suitable constant b > 0.
By setting Ψ = Φ in (3.2) and (3.5), we obtain
Z
1
B(Φ, Φ) =
{ (DR Φ, DR Φ) + (F, DR Φ) Φ + (G Φ) Φ }dµ
2
H
1
=
|||DR Φ|||2 + [(F, DR Φ), Φ] + [GΦ, Φ]
2
1
≥
|||DR Φ|||2 − β |||DR Φ||| |||Φ||| + δ |||Φ|||2 ,
2
(3.10)
where δ = inf G(v).
v∈H
For any ε > 0, we have
β |||DR Φ||| |||Φ||| ≤ ε |||DR Φ|||2 +
β2
|||Φ|||2 .
4ε
(3.11)
By making use of (3.11) in (3.10), we can get the desired inequality (3.8):
1
β2
B(Φ, Φ) ≥ ( − ε)|||DR Φ|||2 + (δ −
)|||Φ|||2 ,
2
4ε
which completes the proof.
With the aid of the energy estimates, under suitable conditions on F and G,
the following existence theorem can be established.
478
PAO-LIU CHOW
Theorem 3.2. (Existence of strong solutions) Suppose the functions F : H → H
and G : H → R are bounded and continuous. Then there is a constant α0 ≥ 0 such
that for each α > α0 and for any Q ∈ H−1 , the following elliptic problem
.
Lα Φ = L Φ + α Φ = Q
(3.12)
has a unique strong solution Φ ∈ H1 .
Proof. By definition of a strong solution, we have to show the that there exists a
unique solution Φ ∈ H1 satisfying the variational equation
.
Bα (Φ, Ψ) = B(Φ, Ψ) + α [Φ, Ψ] = hhQ, Ψii,
(3.13)
for all Ψ ∈ H1 .
To this end we will apply the Lax-Milgram Theorem [18] in the real separable
Hilbert space H1 . By Lemma 3.1, the inequality (3.7) holds similarly for Bα with
a different constant b1 > 0,
|Bα (Φ, Ψ)| ≤ b1 |||Φ|||1 |||Ψ|||1 ,
In particular we take ε =
then used in (3.13) to give
for Φ, Ψ ∈ H1 .
(3.14)
1
and α0 = |δ − β 2 | in the inequality (3.11), which is
4
1
|||DR Φ|||2 + η |||Φ|||2 ,
4
where η = α − α0 > 0 by assumption. It follows that
Bα (Φ, Φ) ≥
Bα (Φ, Φ) ≥ κ |||Φ|||21 ,
(3.15)
(3.16)
1
for κ = min{ , η}. In view of (3.14) and (3.15), the bilinear form Bα (·, ·) satisfies
4
the hypotheses for the Lax-Milgram Theorem.
For Q ∈ H−1 , hhQ, ·ii defines a bounded linear functional on H1 . Hence there
exists a function Φ ∈ H1 which is the unique solution of the equation
Bα (Φ, Ψ) = hhQ, Ψii
for all Ψ ∈ H1 .
Remark 3.3. By writing
Bα (Φ, Ψ) = hhLα Φ , Ψii,
it follows from Theorem 3.2 that the mapping Lα : H1 → H−1 is an isomorphism.
Corollary 3.4. Suppose that F ∈ Cb (H; H) and G ∈ Cb (H) such that
inf G(v) > sup |F (v)|2 .
v∈H
(3.17)
v∈H
Then there exists a unique strong solution Φ ∈ H1 of the equation
L Φ = Q.
Proof. This follows from the fact that, under condition (3.17), the inequality (3.16)
holds with α = 0.
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
479
4. Compact Resolvent and Fredholm Alternative
For U ∈ H, consider the elliptic problem:
.
Lλ Φ = L Φ + λ Φ = U,
(4.1)
where λ > α0 is a real parameter. By Theorem 3.2, the problem (4.1) has a unique
strong solution Φ ∈ H1 satisfying
Bλ (Φ, Ψ) = hhLλ Φ , Ψii = hhU, Ψii,
(4.2)
for all Ψ ∈ H1 . For each U ∈ H, let us express the solution of (4.1) as
Φ = L−1
λ U.
(4.3)
Denote the resolvent operator Kλ of L on H by
.
Kλ U = L−1
λ U
(4.4)
for all U ∈ H.
In the following theorem, we will show that the resolvent operator Kλ : H → H
is compact.
Theorem 4.1. (Compact Resolvent) Under the conditions of Theorem 3.2 with
λ = α, the resolvent operator Kα : H → H is bounded, linear and compact.
Proof. By the estimate (3.16) and equation (3.13), we have
κ |||Φ|||21
≤
Bα (Φ, Φ) = [U , Φ]
≤
|||U ||| |||Φ|||1 ,
which, in view of (4.3) and (4.4), implies
|||Φ|||1 = ||| Kα U |||1 ≤ C ||| U |||,
(4.5)
1
. Hence the linear operator Kα : H → H is bounded.
κ
To show compactness, let {Un } be a bounded sequence in H with ||| Un ||| ≤ C0
for some C0 > 0 and for each n ≥ 1. Define Φn = Kα Un . Then, by (4.5), we
obtain
1
|||Φn |||1 ≤
||| Un ||| ≤ C1 ,
(4.6)
κ
C0
where C1 =
. It follows that {Φn } is a bounded sequence in the separable
κ
Hilbert space H1 and, hence, there exists a subsequence, to be denoted by {Φk }
for simplicity, which converges weakly to Φ or Φk ⇀ Φ in H1 . To show that the
subsequence will converge strongly in H, by Proposition 2.2, we can express
X
X
Φ=
φn Hn and Φk =
φn,k Hn ,
(4.7)
for C =
n∈Z
n∈Z
where φn = [Φ, Hn ] and φn,k = [Φk , Hn ]. For any integer N > 0, let ZN = {n ∈
Z : 1 ≤ |n| ≤ N } and Z+
N = {n ∈ Z : |n| > N }. By the orthogonality of Hermite
480
PAO-LIU CHOW
functionals Hn′ s, we can get
|||Φ − Φk |||2
=
X
(φn − φn,k )2 +
X
(φn − φn,k )2 +
X
[Φn − Φn,k , Hn ]2 +
n∈ZN
≤
(φn − φn,k )2
n∈Z+
N
n∈ZN
≤
X
1 X
(1 + λn )(φn − φn,k )2
N
+
(4.8)
n∈ZN
n∈ZN
α1
|||Φ − Φk |||21 .
N
Since Φk ⇀ Φ, by a theorem on the weak convergence in H1 (p.120, [18]), the
subsequence {Φk } is bounded such that
sup |||Φk ||| ≤ C, and |||Φ||| ≤ C, some constant C > 0.
k≥1
For the last term in the inequality (4.8), given any ε > 0, there is an integer N0 > 0
such that
α1
2α1 C 2
ε
|||Φ − Φk |||21 ≤
< , for N > N0 .
(4.9)
N
N
2
Again, by the weak convergence of {Φk }, for the given N0 , we have
X
X
[Φn − Φn,k , Hn ]2 = 0.
lim
(φn − φn,k )2 = lim
k→∞
k→∞
n∈ZN0
n∈ZN0
Therefore there is an integer m > 0
X
ε
[Φn − Φn,k , Hn ]2 <
2
(4.10)
n∈ZN
for k > m. Now the estimates (4.8)–(4.10) implies
lim |||Φ − Φk ||| = 0,
k→∞
which proves the compactness of the resolvent Kγ.
Due to the coercivity condition (3.16), the compactness of the resolvent operator
implies the following fact.
Theorem 4.2. The embedding of H1 into H is compact.
In the Wiener space, a direct proof of this important result was given in [10]
and [16].
To define the adjoint operator of L, we first introduce the divergence operator
D⋆ . Let F ∈ C1b (H; H) be expanded in terms of the eigenfunctions {ek } of A:
F =
∞
X
fk ek ,
k=1
where fk = (F, ek ). Then the divergence D⋆ F of F is defined as
X
.
D⋆ F = T r (DF ) =
(D fk , ek ).
∞
k=1
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
481
Recall the covariant operator Γ = (−1)A−1 R for µ. We shall need the following
integration by parts formula.
Lemma 4.3. Suppose that the function F : H → Γ(H) ⊂ H is bounded continuous
and differentiable such that
sup |T rDF (v)| < ∞.
(4.11)
v∈H
and
sup |(Γ−1 F (v), v)| < ∞.
(4.12)
v∈H
Then, for Φ, Ψ ∈ H1 , the following equation holds
Z
Z
Z
(DΦ, F )Ψ dµ = −
Φ D⋆ (Ψ F ) dµ +
h Γ−1 F (v), v iΦ Ψ dµ.
H
H
(4.13)
H
Proof. The proof is similar to that of Lemma 9.2.3 in [8], it will only be sketched.
Let
n
. X
fk ek ,
(4.14)
Fn =
k=1
with fk = (F, ek ). Then the sequence {Fn } converges strongly to F in H. In view
of (4.14), we have
Z
n Z
X
(DΦ, Fn )Ψ dµ =
(DΦ, ek )fk Ψ dµ.
(4.15)
H
H
k=1
By invoking the first integral-by-parts formula (2.7),
Z
Z
Z
(DΦ, ek )fk Ψ = −
Φ(D(Ψfk ), ek ) dµ +
(v, Γ−1 ek ))fk ΦΨ dµ,
H
H
H
so that (4.15) yields
Z
n Z
X
(DΦ, Fn )Ψ dµ = −
ΦD⋆ (Ψfk ek ) dµ
H
k=1 H
n Z
X
+
=−
Z
H
k=1
ΦD⋆ (ΨFn ) dµ +
Z
(v, Γ−1 ek )fk ΦΨ dµ
(4.16)
H
(v, Γ−1 Fn (v)) ΦΨdµ(v).
H
Now the formula (4.13) follows from (4.15) by taking the limit termwise as n →
∞.
Let Φ, Ψ ∈ EA (H) and let F, G be given as in by Lemma 4.3. we can write
[LΦ, Ψ] = [Φ, L⋆ Ψ],
where L⋆ is the formal adjoint of L defined by
.
L⋆ Ψ = −AΨ + D⋆ (Ψ F ) − hΓ−1 F (v), vi Ψ − G Ψ.
The associated bilinear form B⋆ : H1 × H1 → R is given by
B⋆ (Φ, Ψ) = B(Ψ, Φ),
(4.17)
482
PAO-LIU CHOW
for all Φ, Ψ ∈ H1 . For Q ∈ H, consider the adjoint problem
L⋆ Ψ = Q.
(4.18)
A function Ψ ∈ H1 is said to be a strong solution of (4.16) provided that
B⋆ (Ψ, Φ) = [Q, Φ]
for all Φ ∈ H1 .
Now, for U ∈ H, consider the nonhomogeneous problem
L Φ = U,
(4.19)
and the related homogeneous problems
L Φ = 0,
(4.20)
and
L⋆ Ψ = 0.
(4.21)
Let N and N denote, respectively, the subspaces of solutions of (4.20) and (4.21)
in H1 . Then, by applying the Fredholm theory of compact operators [18], we can
prove the following theorem.
⋆
Theorem 4.4. (Fredholm Alternative) Let L and L⋆ be defined by (3.1) and (4.17)
respectively, in which F satisfies the conditions (4.11) and (4.12) in Lemma 4.3.
(1) Exactly one of the following statements is true:
(a) For each Q ∈ H, the nonhomogeneous problem (4.19) has a unique
strong solution.
(b) The homogeneous problem (4.20) has a nontrivial solution.
(2) If case (b) holds, the dimension of null space N is finite and equals to the
dimension of N ⋆ .
(3) The nonhomogeneous problem (4.19) has a solution if and only if
[U, Ψ] = 0,
for all Ψ ∈ N ⋆ .
Proof. To prove the theorem, we shall convert the differential equations into equivalent Fredholm type of equations involving a compact operator. To proceed let α
be given as in Theorem 4.1 and rewrite equation (4.19) as
.
Lα Φ = LΦ + αΦ = αΦ + U.
By theorem 4.1, the equation (4.19) is equivalent to the equation
Φ = Kα (αΦ + U ),
which can be rewritten as the Fredholm equation
(I − T ) Φ = Q,
(4.22)
where I is the identity operator on H,
T = α Kα
and
Q = Kα U.
Since Kα : H → H is compact, T is also compact and Q belongs to H. By
applying the Fredholm Alternative Theorem [11] to equation (4.22), the equivalent
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
483
statements (1)–(3) hold for the Fredholm operator (I − T ). Due to the equivalence
of the problems (4.19) and (4.22), the theorem is thus proved.
5. Spectrum and Eigenvalue Problem
For λ ∈ R, consider strong solutions of the eigenvalue problem:
L Φ = λΦ.
(5.1)
Here, for simplicity, we only treat the case of real solutions. As usual a nontrivial
solution Φ of (5.1) is called an eigenfunction and the corresponding real number
λ, an eigenvalue of L. The (real) spectrum Σ of L consists of all of its eigenvalues.
Theorem 5.1. (Spectral Property) The spectrum Σ of L is at most countable. If
the set Σ is infinite, then Σ = {λk ∈ R : k ≥ 1} with λk ≤ λk+1 , each with a finite
multiplicity, for k ≥ 1, and λk → ∞ as k → ∞.
Proof. By taking a real number α, rewrite the equation (5.1) as
.
Lα = L Φ + αΦ = (λ + α) Φ.
(5.2)
By taking α > α0 as in Theorem 4.2, the equation (5.2) can be converted into the
eigenvalue problem for the resolvent operator
Kα Φ = ρ Φ,
(5.3)
where
1
.
(5.4)
λ+α
By Theorem 4.1, the resolvent operator Kα on H is compact. Therefore its
spectrum Σα is discrete. If the spectrum is infinite, then Σα = {ρk ∈ R : k ≥ 1}
with ρk ≥ ρk+1 , each of a finite multiplicity, and lim ρk = 0. Now it follows
ρ=
k→∞
from equation (5.4) that spectrum Σ of L has the asserted property with λk =
1
− α.
ρk
Remark 5.2. As in finite dimensions, the eigenvalue problem (5.1) may be generalized to the case of complex-valued solutions in a complex Hilbert space. In this
case the eigenvalues λk may be complex.
As a special case, set F ≡ 0 in L and the reduced operator L0 is given by
L0 Φ = (−A) Φ + G Φ.
Clearly L0 is a formal self-adjoint operator, or L0 = L⋆0 . The corresponding
bilinear form
1
B0 (Φ, Ψ) = hhL0 Φ , Ψii = [RDΦ, DΨ] + [GΦ, Ψ],
(5.5)
2
for Φ, Ψ ∈ H1
Consider the eigenvalue problem:
L0 Φ = λ Φ.
(5.6)
For the special case when G = 0, L0 = −A. The eigenvalues and the eigenfunctions of (−A) were given explicitly in Proposition 2.2. The results show that
484
PAO-LIU CHOW
the eigenvalues can be ordered as a non-increasing sequence {λk } and the corresponding eigenfunctions are orthonormal Hermite polynomial functionals. With
a smooth perturbation by G, similar results will hold for the eigenvalue problem
(5.6) as stated in the following theorem.
Theorem 5.3. (Symmetric Eigenvalue Problem) Suppose that G : H → R+ be a
bounded, continuous and positive function, and there is a constant δ > 0 such that
G(v) ≥ δ, ∀ v ∈ H. Then the following statements hold:
(1) Each eigenvalue of L0 is positive with finite multiplicity. The set Σ0 of
eigenvalues forms a nondecreasing sequence (counting multiplicity)
0 < λ1 ≤ λ2 ≤ · · · λk ≤ · · ·
such that λk → ∞ as k → ∞.
(2) There exists an orthonormal basis Φk , k = 1, 2, · · · of H, where Φk is an
eigenfunction as a strong solution of
L0 Φk = λk Φk ,
for k = 1, 2, · · · .
Proof. By the assumption on G, it is easy to check that the bilinear form B0 :
.
H1 × H1 → R satisfies the conditions of Theorem 4.1. Hence the inverse K0 = L−1
0
is a self-adjoint compact operator in H. Similar to the proof of Theorem 5.1,
by converting the problem (5.6) into an equivalent eigenvalue problem for K0 ,
the statements (1) and (2) are well-known spectral properties of a self-adjoint,
compact operator in a separable Hilbert space.
As in finite dimensions the smallest or the principal eigenvalue λ1 can be characterized by a variational principle.
Theorem 5.4. (The Principal Eigenvalue) The principal eigenvalue can be obtained by the variational formula
λ1 =
inf
Φ∈H1 ,|||Φ|||6=0
B0 (Φ, Φ)
.
|||Φ|||2
(5.7)
Proof. To verify the variation principle, a strong solution Φ of equation (5.6) must
satisfy
B0 (Φ, Φ) = λ [Φ, Φ]
or
. B0 (Φ, Φ)
λ = J(Φ) =
,
|||Φ|||2
provided that |||Φ||| 6= 0. In view of the coercivity condition
B0 (Φ, Φ) ≥ κ |||Φ|||21
(5.8)
(5.9)
for some κ > 0, J is bounded from below so that the minimal value of J is given
by
λ⋆ =
inf
J(Ψ),
(5.10)
Ψ∈H1 ,|||Ψ|||6=0
ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES
485
which can also be written as
λ⋆ =
inf
Ψ∈H1 ,|||Ψ|||=1
Q(Ψ),
(5.11)
where we set Q(Φ) = B0 (Φ, Φ).
To show that λ⋆ = λ1 and the minimizer of Q gives rise to the principal eigenfunction Φ1 , choose a minimizing sequence {Ψn } in H1 with |||Ψn |||1 = 1 such that
Q(Ψn ) → λ1 as n → ∞. By (5.9) and the boundedness of the sequence {Ψn } in
H1 , the compact embedding Theorem 4-2 implies the existence of a subsequence,
to be denoted by {Ψk }, which converges to a function Ψ ∈ H1 with |||Ψ|||1 = 1.
Since Q is a quadratic functional, by the Parallelogram Law and equation (5.11),
we have
Ψj − Ψk
1
Ψj + Ψk
Q(
) = (Q(Ψj ) + Q(Ψk )) − Q(
)
2
2
2
1
Ψj + Ψk
≤ (Q(Ψj ) + Q(Ψk ) − λ⋆ |||(
)||| → 0
2
2
as j, k → ∞. Again, by (5.9), we deduce that {Ψk } is a Cauchy sequence in H1
which converges to Ψ ∈ H1 and Q(Ψ) = λ⋆ . Now, for Θ ∈ H1 and t ∈ R, let
f (t) = J(Ψ + t Θ).
As well-known in the calculus of variations, for J to attain its minimum at Ψ, it
is necessary that
f ′ (0) = 2{B0 (Ψ, Θ) − λ⋆ [Φ, Θ]} = 0,
which shows Ψ is the strong solution of
L0 (Ψ) = λ⋆ Ψ.
Hence, in view of (5.8), we conclude that λ⋆ = λ1 and Ψ is the eigenfunction
associated with the principal eigenvalue.
Acknowledgement. This work was done to be presented at the Special Session
on Stochastic Analysis in honor of Professor Hui-Hsiung Kuo in the 2012 AMS
Annual Meeting in Boston.
References
1. Barhoumi, A., Kuo, H-H., Ouerdian, H.: Infinite dimensional Laplacians on a L´
evy - Gel’fand
triple; Comm. Stoch. Analy. 1 (2007) 163–174
2. Chow, P. L.: Infinte-dimensional Kolmogorov equations in Gauss-Sobolev spaces; Stoch.
Analy. and Applic. 14 (1996) 257–282
3. Chow, P. L.: Infinte-dimensional parabolic equations in Gauss-Sobolev spaces; Comm. Stoch.
Analy. 1 (2007) 71–86
4. Chow, P. L.: Singular perturbation and stationary solutions of parabolic equations in GaussSobolev spaces; Comm. Stoch. Analy. 2 (2008) 289–306
5. Chow, P. L., Khasminski, R. Z.: Stationary solutions of nonlinear stochastic evolution equations; Stoch. Analy. and Applic. 15 (1997) 677–699
6. Daleskii, Yu. L.: Infinite dimensional elliptic operators and parabolic equations connected
with them; Russian Math. Rev. 22 (1967) 1–53
7. Da Prato, G.: Some results on elliptic and parabolic equations in Hilbert space; Rend. Mat.
Accad Lencei. 9 (1996) 181–199
8. Da Prato, G., Zabczyk,J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, UK, 1992
486
PAO-LIU CHOW
9. Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces.
Cambridge University Press, Cambridge, 2002
10. Da Prato, G., Malliavin, P., Nualart, D.: Compact families of Wiener functionals; C.R.
Acad. Sci. Paris. 315, 1287–1291, 1992.
11. Dunford, N., Schwartz, J.: Linear Operators Part I. Interscience Pub., New York, 1958
12. Gilbarg, D., Truginger, N. S.: Elliptic Partial Differential Equations of Second Order.
Springer, Berlin, 1998
13. Gross, L., Potential theory in Hilbert spaces; J. Funct. Analy. 1 (1967) 123–181
14. Kuo, H.H.: Integration by parts for abstract Wiener measures; Duke Math. J. 41 (1974)
373–379
15. Lions, J. L., Mangenes, E.: Nonhomogeneous Boundary-value Problems and Applications.
Springer-Verlag, New York, 1972
16. Peszat, S.: On a Sobolev space of functions of infinite numbers of variables; Bull. Pol. Acad.
Sci. 98 (1993) 55–60
17. Piech, M. A.: The Ornstein-Uhleneck semigroup in an infinite dimensional L2 setting; J.
Funct. Analy. 18 (1975) 271–285
18. Yosida, K.: Functional Analysis. Springer-Verlag, New York, 1967.
P.L. Chow: Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
E-mail address: [email protected]
Communications on Stochastic Analysis
Vol. 6, No. 3 (2012) 487-511
Serials Publications
www.serialspublications.com
TEMPORAL CORRELATION OF DEFAULTS IN SUBPRIME
SECURITIZATION
ERIC HILLEBRAND, AMBAR N. SENGUPTA, AND JUNYUE XU
A BSTRACT. We examine the subprime market beginning with a subprime mortgage, followed by a portfolio of such mortgages and then a series of such portfolios. We obtain
an explicit formula for the relationship between loss distribution and seniority-based interest rates. We establish a link between the dynamics of house price changes and the
dynamics of default rates in the Gaussian copula framework by specifying a time series
model for a common risk factor. We show analytically and in simulations that serial correlation propagates from the common risk factor to default rates. We simulate prices of
mortgage-backed securities using a waterfall structure and find that subsequent vintages
of these securities inherit temporal correlation from the common risk factor.
1. Introduction
In this paper we (i) derive closed-form mathematical formulas (4.3) and (4.12) connecting interest rates paid by tranches of Collateralized Debt Obligations (CDOs) and the
corresponding loss distributions, (ii) present a two-step Gaussian copula model (Proposition 6.1) governing correlated CDOs, and (iii) study the behavior of correlated CDOs both
mathematically and through simulations. The context and motivation for this study is the
investigation of mortgage backed securitized structures built out of subprime mortgages
that were at the center of the crisis that began in 2007. Our investigation demonstrates,
both theoretically and numerically, how the serial correlation in the evolution of the common factor, reflecting the general level of home prices, propagates into a correlated accumulation of losses in tranches of securitized structures based on subprime mortgages
of specific vintages. The key feature of these mortgages is the short time horizon to default/prepayment that makes it possible to model the corresponding residential mortgage
backed securities (RMBS) as forming one-period CDOs. We explain the difference in behavior between RMBS based on subprime mortgages and those based on prime mortgages
in Table 1 and related discussions.
During the subprime crisis, beginning in 2007, subprime mortgages created at different times have defaulted one after another. Figure 1, lower panel, shows the time series
of serious delinquency rates of subprime mortgages from 2002 to 2009. (By definition
of the Mortgage Banker Association, seriously delinquent mortgages refer to mortgages
that have either been delinquent for more than 90 days or are in the process of foreclosure.) Defaults of subprime mortgages are closely connected to house price fluctuations,
as suggested, among others, by [26] (see also [4, 16, 29].) Most subprime mortgages
Received 2012-5-22; Communicated by the editors.
2000 Mathematics Subject Classification. 62M10; 91G40.
Key words and phrases. Mortgage-backed securities, CDO, vintage correlation Gaussian copula .
487
488
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
F IGURE 1. Two-Year Changes in U.S. House Price and Subprime
ARM Serious Delinquency Rates
US Home Price Index Changes (Two−Year Rolling Window)
50
0
−50
−100
2002
2003
2004
2005
2006
2007
2008
2009
US Subprime Adjustable−Rate−Mortgage Serious Delinquency Rates (%)
40
30
20
10
0
2002
2003
2004
2005
2006
2007
2008
2009
“U.S. home price two-year rolling changes” are two-year overlapping changes in the S&P Case-Shiller U.S.
National Home Price index. “Subprime ARM Serious Delinquency Rates” are obtained from the Mortgage
Banker Association. Both series cover the first quarter in 2002 to the second quarter in 2009.
are Adjustable-Rate Mortgages (ARM). This means that the interest rate on a subprime
mortgage is fixed at a relatively low level for a “teaser” period, usually two to three years,
after which it increases substantially. Gorton [26] points out that the interest rate usually
resets to such a high level that it “essentially forces” a mortgage borrower to refinance or
default after the teaser period. Therefore, whether the mortgage defaults or not is largely
determined by the borrower’s access to refinancing. At the end of the teaser period, if
the value of the house is much greater than the outstanding principal of the loan, the borrower is likely to be approved for a new loan since the house serves as collateral. On the
other hand, if the value of the house is less than the outstanding principal of the loan, the
borrower is unlikely to be able to refinance and has to default.
We analyze how the dynamics of housing prices propagate, through the dynamics of
defaults, to the dynamics of tranche losses in securitized structures based on subprime
mortgages. To this end, we introduce the notion of vintage correlation, which captures
the correlation of default rates in mortgage pools issued at different times. Under certain assumptions, vintage correlation is the same as serial correlation. After showing that
changes in a housing index can be regarded as a common risk factor of individual subprime mortgages, we specify a time series model for the common risk factor in the Gaussian copula framework. We show analytically and in simulations that the serial correlation
of the common risk factor introduces vintage correlation into default rates of pools of
TEMPORAL CORRELATION OF DEFAULTS
489
subprime mortgages of subsequent vintages. In this sense, serial correlation propagates
from the common risk factor to default rates. In simulations of the price behavior of
Mortgage-Backed Securities (MBS) over different cohorts, we find that the price of MBS
also exhibits vintage correlation, which is inherited from the common risk factor.
One of our objectives in this paper is to provide a formal examination of one of the important causes of the current crisis. (For different perspectives on the causes and effects
of the subprime crisis, see also [12, 15, 20, 27, 39, 42, 43].) Vintage correlation in default
rates and MBS prices also has implications for asset pricing. To price some derivatives,
for example forward starting CDO, it is necessary to predict default rates of credit assets
created at some future time. Knowing the serial correlation of default probabilities can
improve the quality of prediction. For risk management in general, some credit asset portfolios may consist of credit derivatives of different cohorts. Vintage correlation of credit
asset performance affects these portfolios’ risks. For instance, suppose there is a portfolio
consisting of two subsequent vintages of the same MBS. If the vintage correlation of the
MBS price is close to one, for example, the payoff of the portfolio has a variance almost
twice as big as if there were no vintage correlation.
2. The Subprime Structure
In a typical subprime mortgage, the loan is amortized over a long period, usually 30
years, but at the end of the first two (or three) years the interest rate is reset to a significantly higher level; a substantial prepayment fee is charged at this time if the loan is paid
off. The aim is to force the borrower to repay the loan (and obtain a new one), and the
prepayment fee essentially represents extraction of equity from the property, assuming
the property has increased in value. If there is sufficient appreciation in the price of the
property then both lender and borrower win. However, if the property value decreases
then the borrower is likely to default.
Let us make a simple and idealized model of the subprime mortgage cashflow. Let
P0 = 1 be the price of the property at time 0, when a loan of the same amount is taken
to purchase the property (or against the property as collateral). At time T the price of
the property is PT , and the loan is terminated, resulting either in a prepayment fee k
plus outstanding loan amount or default, in which case the lender recovers an amount R.
For simplicity of analysis at this stage we assume 0 interest rate up to time T ; we can
view the interest payments as being built into k or R, ignoring, as a first approximation,
defaults prior to time T (for more on early defaults see [7]). The borrower refinances if
PT is above a threshold P∗ (say, the present value of future payments on a new loan) and
defaults otherwise. Thus the net cashflow to the lender is
k1[PT >P∗ ] − (1 − R)1[PT ≤P∗ ] ,
(2.1)
with all payments and values normalized to time-0 money. The expected earning is therefore
(k + 1 − R)P(PT > P∗ ) − (1 − R),
for the probability measure P being used. We will not need this expected value but observe simply that a default occurs when PT < P∗ , and so, if log PT is Gaussian then
default occurs for a particular mortgage if a suitable standard Gaussian variable takes a
value below some threshold.
490
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
It is clear that nothing like the above model would apply to prime mortgages. The
main risk (for the lender) associated to a long-term prime mortgage is that of prepayment,
though, of course, default risk is also present. A random prepayment time embedded
into the amortization schedule makes it a different problem to value a prime mortgage.
In contrast, for the subprime mortgage the lender is relying on the prepayment fee and
even the borrower hopes to extract equity on the property through refinancing under the
assumption that the property value goes up in the time span [0, T ]. (The prepayment
fee feature has been controversial; see, for example, [14, page 50-51].) We refer to the
studies [7, 14, 26] for details on the economic background, evolution and ramifications of
the subprime mortgage market, which went through a major expansion in the mid 1990s.
3. Portfolio Default Model
Securitization makes it possible to have a much larger pool of potential investors in
a given market. For mortgages the securitization structure has two sides: (i) assets are
mortgages; (ii) liabilities are debts tranched into seniority levels. In this section we briefly
examine the default behavior in a portfolio of subprime mortgages (or any assets that have
default risk at the end of a given time period). In Section 4 we will examine a model
structure for distributing the losses resulting from defaults across tranches.
For our purposes consider N subprime mortgages, issued at time 0 and (p)repaid or
defaulting at time T , each of amount 1. In this section, for the sake of a qualitative
understanding we assume 0 recovery, and that a default translates into a loss of 1 unit (we
neglect interest rates, which can be built in for a more quantitatively accurate analysis).
Current models of home price indices go back to the work of Wyngarden [49] , where
indices were constructed by using prices from repeated sales of the same property at
different times (from which property price changes were calculated). Bailey et al. [3]
examined repeated sales data and developed a regression-based method for constructing
an index of home prices. This was further refined by Case and Shiller [13] into a form that,
in extensions and reformulations, has become an industry-wide standard. The method in
[13] is based on the following model for the price PiT of house i at time T :
log PiT = CT + HiT + NiT ,
(3.1)
where CT is the log-price at time T across a region (city, in their formulation), HiT is
a mean-zero Gaussian random walk (with variance same for all i), and NiT is a housespecific random error of zero mean and constant variance (not dependent on i). The three
terms on the right in equation (3.1) are independent and (NiT ) is a sale-specific fluctuation
that is serially uncorrelated; a variety of correlation structures could be introduced in
modifications of the model. We will return to this later in equation (5.7) (with slightly
different notation) where we will consider different values of T . For now we focus on a
portfolio of N subprime mortgages i ∈ {1, . . . , N } with a fixed value of T .
Let Xi be the random variable
Xi =
log PiT − mi
,
si
(3.2)
where mi is the mean and si is the standard devation of log PiT with respect to some
probability measure of interest (for example, the market pricing risk-neutral measure).
TEMPORAL CORRELATION OF DEFAULTS
491
Keeping in mind (3.1) we assume that
Xi =
p
√
ρZ + 1 − ρ i
(3.3)
for some ρ > 0, where (Z, 1 , . . . , N ) is a standard Gaussian in RN +1 , with independent
components. Mortgage i defaults when Xi crosses below a threshold X∗ , so that the
assumed common default probability for the mortgages is
(3.4)
P[Xi < X∗ ] = E 1[Xi <X∗ ] .
The total number of defaults, or portfolio loss (with our assumptions), is
L=
N
X
1[Xj <X∗ ] .
(3.5)
j=1
The cash inflow at time T is the random variable
S(T ) =
N
X
1[Xj ≥X∗ ] .
(3.6)
j=1
Pooling of investment funds and lending them for property mortgages is natural and
has long been in practice (see Bogue [9, page 73]). In the modern era Ginnie Mae issued
the first MBS in 1970 in “pass through” form which did not protect against prepayment
risk. In 1983 Freddie Mac issued Collateralized Mortgage Obligations (CMOs) that had
a waterfall-like structure and seniority classes with different maturities. The literature on
securitization is vast (see, for instance, [19, 36, 41]).
4. Tranche Securitization: Loss Distribution and Tranche Rates
In this section we derive a relation between the loss distribution in a cashflow CDO and
the interest rates paid by the tranches. We make the simplifying assumption that all losses
and payments occur at the end of one period. This assumption is not unreasonable for
subprime mortgages that have a short interest-rate reset period, which we take effectively
as the lifetime of the mortgage (at the end of which it either pays back in full with interest
or defaults). We refer to the constituents of the portfolio as “loans”, though they could
be other instruments. Figure 2 illustrates the structure of the portfolio and cashflows. As
pointed out by [10, page xvii] there is “very little research or literature” available on cash
CDOs; the complex waterfall structures that govern cashflows of such CDOs are difficult
to model in a mathematically sound way. For technical descriptions of cashflow waterfall
structures, see [23, Chapter 14].
Consider a portfolio of N loans, each with face value of one unit. Let S(T ) be the cash
inflow from the investments made by the portfolio at time T , the end of the investment
period. Next consider investors named 1, 2, . . . , M , with investor j investing amount Ij .
The most senior investor, labeled 1, receives an interest rate r1 (return per unit investment
over the full investment period) if at all possible; this investor’s cash inflow at time T is
Y1 (T ) = min {S(T ), (1 + r1 )I1 } .
Proceeding in this way, investor j has payoff
(4.1)
492
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
F IGURE 1. Illustration of schematic structure of MBS
F IGURE 2. Illustration of schematic structure of MBS
Mortgage
Pool
A Typical Mortgage
Principal: $1;
Annual interest rate:
9%;
Maturity: 15 years.
1
2
3
4
MBS
M1
$$$
M2
$$$
M3
$$$
M4
$$$
Interest
Rate
Senior
70%
6%
Mezzanine
25%
15%
M99
$$$
Subordinate
4%
20%
M100
$$$
Equity
1%
N/A
v
V = 120 T = 144




X
Yj (T ) = min S(T ) −
Yi (T ), (1 + rj )Ij .


(4.2)
1≤i<j
Using the market pricing measure (risk-neutral measure) Q we should have
EQ [Yj (T )] = (1 + R0 )Ij ,
(4.3)
where R0 is the risk-free interest rate for the period of investment.
Given a model for S(T ), the rates rj can be worked out, in principle, recursively from
equation (4.2) as follows. Using the distribution of X(1) we can back out the value of the
supersenior rate r1 from
EQ [min {S(T ), (1 + r1 )I1 }] = EQ [Y1 (T )] = (1 + R0 )I1 .
(4.4)
Now we use this value of r1 in the equation for Y2 (T ):
EQ [min {S(T ) − Y1 (T ), (1 + r2 )I2 }] = EQ [Y2 (T )] = (1 + R0 )I2 ,
(4.5)
and (numerically) invert this to obtain the value of r2 implied by the market model. Note
that in equation (4.5) the random variable Y1 (T ) on the left is given by equation (4.1)
using the already computed value of r1 . Proceeding in this way yields the full spectrum
of tranche rates rj .
1
Now we turn to a continuum model for tranches,
again with one time period. Consider
an idealized securitization structure ABS. Investors are subordinatized by a seniority parameter y ∈ [0, 1]. An investor in a thin “tranchelet” [y, y + δy] invests the amount δy and
is promised an interest rate of r(y) (return on unit investment for the entire investment
period) if there is no default. In this section we consider only one time period, at the end
of which the investment vehicle closes.
TEMPORAL CORRELATION OF DEFAULTS
493
Thus, if there is sufficient return on the investment made by the ABS, a tranche [a, b] ⊂
[0, 1] will be returned the amount
Z b
1 + r(y) dy.
a
In particular, assuming that the total initial investment in the portfolio is normalized to
R1
one, the maximum promised possible return to all the investors is 0 1 + r(y) dy. The
portfolio loss is
Z 1
1 + r(y) dy − S(T ),
(4.6)
L=
0
where S(T ) is the total cash inflow, all assumed to occur at time T , from investments
made by the ABS. Note that L is a random variable, since S(T ) is random.
Consider a thin tranche [y, y + δy]. If S(T ) is greater than the maximum amount
promised to investors in the tranche [y, 1], that is if
Z 1
S(T ) >
1 + r(s) ds,
(4.7)
y
then the tranche [y, y + δy] receives its maximum promised amount 1 + r(y) δy. (If
S(T ) is insufficient to cover the more senior investors, the tranchelet [y, y + δy] receives
nothing.) The condition (4.7) is equivalent to
Z y
L<
1 + r(s) ds,
(4.8)
0
as can be seen from the relation (4.6). Thus, the thin tranche receives the amount
i 1 + r(y) δy
1h R y
L<
0
1+r(s) ds
Using the risk-neutral probability measure Q, we have then
Z y
Q L<
1 + r(s) ds 1 + r(y) δy = (1 + R0 ) δy,
(4.9)
0
where R0 is the risk-free interest rate for the period of investment. Thus,
Z y
1 + r(y) FL
1 + r(s) ds = 1 + R0
(4.10)
0
where FL is the distribution function of the loss L with respect to the measure Q.
Let λ(·) be the function given by
Z y
λ(y) =
1 + r(s) ds,
(4.11)
0
which is strictly increasing as a function of y, with slope > 1 (assuming the rates r(·) are
positive). Hence λ(·) is invertible. Then the loss distribution function is obtained as
FL (l) =
1 + R0
.
1 + r λ−1 (l)
(4.12)
If r(y) are the market rates then the market-implied loss distribution function FL is given
by (4.12). On the other hand, if we have a prior model for the loss distribution FL then
the implied rates r(y) can be computed numerically using (4.12).
494
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
A real tranche is a “thick” segment [a, b] ⊂ [0, 1] and offers investors some rate r[a,b] .
This rate could be viewed as obtained from the balance equation:
Z b
(1 + r[a,b] )(b − a) =
(1 + r(y) dy,
a
which means that the tranche rate is the average of the rates over the tranche:
Z b
1
r[a,b] =
r(y) dy.
b−a a
(4.13)
5. Modeling Temporal Correlation in Subprime Securitization
We turn now to the study of a portfolio consisting of several CDOs (each homogeneous) belonging to different vintages. We model the loss by a “multi-stage” copula,
one operating within each CDO and the other across the different CDOs. The motivation comes from the subprime context. Each CDO is comprised of subprime mortgages
of a certain vintage, all with a common default/no-default decision horizon (typically two
years). It is important to note that we do not compare losses at different times for the same
CDO; we thus avoid problems in using a copula model across different time horizons.
Definition 5.1 (Vintage Correlation). Suppose we have a pool of mortgages created
at each time v = 1, 2, · · · , V . Denote the default rates of each vintage observed at
a fixed time T > V as p1 , p2 , · · · , pV , respectively. We define vintage correlation
φj := Corr(p1 , pj ) for j = 2, 3, · · · , V as the default correlation between the j − th
vintage and the first vintage.
As an example of vintage correlation, consider wines of different vintages. Suppose
there are several wine producers that have produced wines of ten vintages from 2011
to 2020. The wines are packaged according to vintages and producers, that is, one box
contains one vintage by one producer. In the year 2022, all boxes are opened and the percentage of wines that have gone bad is obtained for each box. Consider the correlation of
fractions of bad wines between the first vintage and subsequent vintages. This correlation
is what we call vintage correlation.
The definition of vintage correlation can be extended easily to the case where the base
vintage is not the first vintage but any one of the other vintages. Obviously, vintage correlation is very similar to serial correlation. There are two main differences. First, the
consideration is at a specific time in the future. Second, in calculating the correlation
between any two vintages, the expected values are averages over the cross-section. That
is, in the wine example, expected values are averages over producers. In mortgage pools,
they are averages over different mortgage pools. Only if we assume the same stochastic
structure for the cross-section and for the time series of default rates, vintage correlation
and serial correlation are equivalent. We do not have to make this assumption to obtain
our main results. Making this assumption, however, does not invalidate any of the results either. Therefore, we use the terms “vintage correlation” and “serial correlation”
interchangeably in our paper.
To model vintage correlation in subprime securitization, we use the Gaussian copula
approach of Li [34], widely used in industry to model default correlation across names.
The literature on credit risk pricing with copulas and other models has grown substantially
TEMPORAL CORRELATION OF DEFAULTS
495
in recent years and an exhaustive review is beyond the scope of his paper; monographs include [8, 18, 32, 40, 44]. Other works include, for example, [1, 2, 5, 6, 10, 17, 22, 24, 30,
33, 35, 45, 46]. There are approaches to model default correlation other than default-time
copulas. One method relies on the so-called structural model, which goes back to Merton’s (1974) work on pricing corporate debt. An essential point of the structural model
is that it links the default event to some observable economic variables. The paper [31]
extends the model to a multi-issuer scenario, which can be applied to price corporate debt
CDO. It is assumed that a firm defaults if its credit index hits a certain barrier. Therefore,
correlation between credit indices determines the correlation of default events. The advantage of a structural model is that it gives economic meaning to underlying variables.
Other approaches to CDO pricing are found, for example, in [28] and in [47]. The work
[11] provides a comparison of common CDO pricing models.
In our model each mortgage i of vintage v has a default time τv,i , which is a random
variable representing the time at which the mortgage defaults. If the mortgage never
defaults, this value is infinity. If we assume that the distribution of τv,i is the same across
all mortgages of vintage v, we have
Fv (s) = P[τv,i < s], ∀i = 1, 2, ..., N,
(5.1)
where the index i denotes individual mortgages and the index v denotes vintages. We
assume that Fv is continuous and strictly increasing. Given this information, for each
vintage v the Gaussian copula approach provides a way to obtain the joint distribution of
the τv,i across i. Generally, a copula is a joint distribution function
C (u1 , u2 , ..., uN ) = P (U1 ≤ u1 , U2 ≤ u2 , ..., UN ≤ uN ) ,
where u1 , u2 , ..., uN are N uniformly distributed random variables that may be correlated.
It can be easily verified that the function
C [F1 (x1 ), F2 (x2 ), ..., FN (xN )] = G(x1 , x2 , ..., xN )
(5.2)
is a multivariate distribution function with marginals given by the distribution functions
F1 (x1 ), F2 (x2 ),..., FN (xN ). Sklar [48] proved the converse, showing that for an arbitrary multivariate distribution function G(x1 , x2 , ..., xN ) with continuous marginal distributions functions F1 (x1 ), F2 (x2 ),..., FN (xN ), there exists a unique C such that equation
(5.2) holds. Therefore, in the case of default times, there is a Cv for each vintage v such
that
Cv [Fv (t1 ), Fv (t2 ), ..., Fv (tN )] = Gv (t1 , t2 , ..., tN ),
(5.3)
where Gv on the right is the joint distribution function of (τv,1 , . . . , τv,N ). Since we
assume Fv to be continuous and strictly increasing, we can find a standard Gaussian
random variable Xv,i such that
Φ(Xv,i ) = Fv (τv,i ) ∀v = 1, 2, ..., V ; i = 1, 2, ..., N,
(5.4)
τv,i = Fv−1 (Φ(Xv,i )) ∀v = 1, 2, ..., V ; i = 1, 2, ..., N,
(5.5)
or equivalently,
496
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
where Φ is the standard normal distribution function. To see that this is correct, observe
that
P[τv,i ≤ s] = P [Φ(Xv,i ) ≤ Fv (s)] = P Xv,i ≤ Φ−1 (Fv (s))
= Φ Φ−1 (Fv (s)) = Fv (s).
The Gaussian copula approach assumes that the joint distribution of (Xv,1 , . . . , Xv,N ) is
a multivariate normal distribution function ΦN . Thus the joint distribution function of
default times τv,i is obtained once the correlation matrix of the Xv,i is known. A standard
simplification in practice is to assume that the pairwise correlations between different Xv,i
are the same across i. Suppose that the value of this correlation is ρv for each vintage v.
Consider the following definition
p
√
(5.6)
Xv,i := ρv Zv + 1 − ρv εi ∀i = 1, 2, . . . , N ; v = 1, 2, . . . , V,
where εv,i are i.i.d. standard Gaussian random variables and Zv is a Gaussian random
variable independent of the εv,i . It can be shown easily that in each vintage v, the variables
Xv,i defined in this way have the exact joint distribution function ΦN .
Using the information above, for each vintage v, the Gaussian copula approach obtains
the joint distribution function Gv for default times as follows. First, N Gaussian random
variables Xv,i are generated according to equation (5.6). Second, from equation (5.5) a set
of N default times τv,i is obtained, which has the desired joint distribution function Gv .
In equation (5.6), the common factor Zv can be viewed as a latent variable that captures
the default risk in the economy, and εi is the idiosyncratic risk for each mortgage. The
variable Xv,i can be viewed as a state variable for each mortgage. The parameter ρv is
the correlation between any two individual state variables. It is obvious that the higher the
value of ρv , the greater the correlation between the default times of different mortgages.
Assume that we have a pool of N mortgages i = 1, . . . , N for each vintage v =
1, . . . , V . Each individual mortgage within a pool has the same initiation date v and
interest adjustment date v 0 > v. Let Yv,i be the change in the logarithm of the price Pv,i
of borrower i’s (of vintage v) house during the teaser period [v, v 0 ]. From equation (3.1),
we can deduce that
Yv,i := log Pv0 ,i − log Pv,i = ∆Cv + ev,i ,
(5.7)
where ∆Cv := log Cv0 − log Cv is the change in the logarithm of a housing market index
Cv , and ev,i are i.i.d. normal random variables for all i = 1, 2, ..., N , and v = 1, 2, ..., V .
As outlined in the introduction, default rates of subprime ARM depend on house price
changes during the teaser period. If the house price fails to increase substantially or even
declines, the mortgage borrower cannot refinance, absent other substantial improvements
in income or asset position. They have to default shortly after the interest rate is reset to
a high level. We assume that the default, if it happens, occurs at time v 0 . Therefore, we
assume that a mortgage defaults if and only if Yv,i < Y ∗ , where Y ∗ is a predetermined
threshold.
We can now give a structural interpretation of the common risk factor Zv in the Gaussian copula framework. Define
∆Cv
Zv0 :=
,
(5.8)
σ∆C
TEMPORAL CORRELATION OF DEFAULTS
497
where σ∆C is the unconditional standard deviation of ∆Cv . Then we have
Yv,i = Zv0 σ∆C + ev,i .
Further standardizing Yv,i , we have
0
Xv,i
:=
Yv,i
σe
Z 0 σ∆C + ev,i
σ∆C
Zv0 + p 2
ε0v,i
= v
=p 2
2
σY
σY
σ∆C + σe
σ∆C + σe2
where σe is the standard deviation of ev,i , and ε0v,i := ev,i /σe . The third equality follows
from the fact that
q
2
σY = σ∆C
+ σe2 .
Define
ρ0 :=
Then
0
Xv,i
=
2
σ∆C
.
2
σ∆C
+ σe2
p
p
ρ0 Zv0 + 1 − ρ0 ε0v,i
∀i = 1, 2, . . . , N ;
t = 1, 2, . . . , T
(5.9)
Note that equation (5.9) has exactly the same form as equation (5.6). The default event is
0
defined as Xv,i
< X ∗0 where
Y∗
X ∗0 := p
2
σ∆C
+ σe2
.
Let
0
0
τv,i
:= Fv−1 Φ(Xv,i
) ,
and
τv∗0 := Fv−1 (Φ(X ∗0 )) ,
0
then the default event can be defined equivalently as τv,i
≤ τ ∗0 . The comparison between
equation (5.9) and (5.6) shows that the common risk factor Zv in the Gaussian copula
model for subprime mortgages can be interpreted as a standardized change in a house
price index. This is consistent with our remarks in the context of (3.2) that the CaseShiller model provides a direct justification for using the Gaussian copula, with common
risk factor being the housing price index.
In light of this structural interpretation, the common risk factor Zv is very likely to
be serially correlated across subsequent vintages. More specifically, we find that Zv0 is
proportional to a moving average of monthly log changes in a housing price index. To see
this, let v be the time of origination and v 0 be the end of the teaser period. Then,
Z v0
∆Cv =
d log Iτ ,
v
where I is the house price index. For example, if we measure house price index changes
quarterly, as in the case of the Case-Shiller housing index, we have
X
∆Cv =
(log Iτ − log Iτ −1 ),
(5.10)
τ ∈[v,v 0 ]
where the unit of τ is a quarter. If we model this index by some random shock arriving
each quarter, equation (5.10) is a moving average process. Therefore, from equation (5.8)
we know that Zv0 has positive serial correlation. Figure 1 shows that the time series of
498
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
Case-Shiller index changes exhibits strong autocorrelation, and is possibly integrated of
order one.
6. The Main Theorems: Vintage Correlation in Default Rates
Since the common risk factor is likely to be serially correlated, we examine the implications for the stochastic properties of mortgage default rates. We specify a time series
model for the common risk factor in the Gaussian copula and determine the relationship
between the serial correlation of the default rates and that of the common risk factor.
Proposition 6.1 (Default Probabilities and Numbers of Defaults). Let k = 1, 2, ..., N ,
p
p
p
√
Xk = ρZ + 1 − ρ εk ,
and
Xk0 = ρ0 Z 0 + 1 − ρ0 ε0k
(6.1)
with
Z 0 = φZ +
0
p
1 − φ2 u,
(6.2)
ε1 , ..., εN , ε01 , ..., ε0N ,
where ρ, ρ ∈ (0, 1), φ ∈ (−1, 1), and Z,
u are mutually independent standard Gaussians. Consider next the number of Xk that fall below some threshold
X∗ , and the number of Xk0 below X∗0 :
A=
N
X
1{Xk ≤X∗ } ,
and
A0 =
k=1
N
X
1{Xk0 ≤X∗0 } ,
(6.3)
k=1
where X∗ and X∗0 are constants. Then
Cov(A, A0 ) = N 2 Cov(p, p0 ),
where
p = p(Z) := P[Xk ≤ X∗ | Z] = Φ
0
p =
P[Xk0
≤
X∗0
0
0
(6.4)
√ X∗ − ρZ
√
,
1−ρ
and
(6.5)
0
| Z ] = p (Z ).
Moreover, the correlation between A and A0 equals the correlation between p and p0 , in
the limit as N → ∞.
Proof. We first show that
E[AA0 ] = E [E[A | Z]E[A0 | Z 0 ]] .
(6.6)
Note that A is a function of Z and ε = (ε1 , . . . , εN ), and A is a function (indeed, the
same function as it happens) of Z 0 and ε0 = (ε01 , . . . , ε0N ). Now for any non-negative
bounded Borel functions f and g on R, and any non-negative bounded Borel functions F
and G on R × RN , we have, on using self-evident notation,
0
E[f (Z)g(Z 0 )F (Z, )G(Z 0 , 0 )]
Z
p
0
= f (z)g(φz + 1 − φ2 x)F (z, y1 , ..., yN )G(z 0 , y10 , ..., yN
) dΦ(z, x, y, y 0 )
| {z }
|
{z
}
| {z }
z0
Z
=
0
f (z)g(z )
Z
y
y0
Z
F (z, y) dΦ(y)
G(z 0 , y 0 ) dΦ(y 0 )
dΦ(z, x)
= E [f (Z)g(Z 0 )E[F (Z, ) | Z]E[G(Z 0 , 0 ) | Z 0 ]] .
(6.7)
TEMPORAL CORRELATION OF DEFAULTS
499
This says that
E [F (Z, ε)G(Z 0 , ε0 ) | Z, Z 0 ] = E[F (Z, ε) | Z]E[G(Z 0 , ε0 ) | Z 0 ].
(6.8)
Taking expectation on both sides of equation (6.8) with respect to Z and Z , we obtain
0
E [F (Z, ε)G(Z 0 , ε0 )] = E [E[F (Z, ε) | Z]E[G(Z 0 , ε0 ) | Z 0 ]] .
(6.9)
Substituting F (Z, ε) = A, and G(Z 0 , ε0 ) = A0 , we have equation (6.6) and
E[AA0 ] = E [E[A | Z]E[A0 | Z 0 ]]
= E[N pN p0 ] = N 2 E[pp0 ],
(6.10)
The last line is due to the fact that conditional on Z, A is a sum of N independent indicator
variables and follows a binomial distribution with parameters N and Ep. Applying (6.9)
again with F (Z, ε) = A, and G(Z 0 , ε0 ) = 1, or indeed, much more directly by repeated
expectations, we have
E[A] = N E[p],
and
E[A0 ] = N E[p0 ].
(6.11)
Hence we conclude that
Cov(A, A0 ) = E(AA0 ) − E[A]E[A0 ]
= N 2 E[pp0 ] − N 2 E[p]E[p0 ]
= N 2 Cov(p, p0 ).
We have
Var(A) = E E[A2 | Z] − N 2 (E[p])2 = N E[p(1 − p)] + N 2 Var(p).
(6.12)
Similarly,
Var(A0 ) = N E[p0 (1 − p0 )] + N 2 Var(p0 ).
Putting everything together, we have for the correlations:
Corr(A, A0 ) = q
1+
Corr(p, p0 )
q
E[p(1−p)]
1+
0
N Var(p )
E[p0 (1−p0 )]
N Var(p)
(6.13)
0
= Corr(p, p ) as N → ∞.
Theorem 6.2 (Vintage Correlation in Default Rates). Consider a pool of N mortgages
created at each time v, where N is fixed. Suppose within each vintage v, defaults are
governed by a Gaussian copula model as in equations (5.1), (5.5), and (5.6) with common risk factor Zv being a zero-mean stationary Gaussian process. Assume further that
ρv = Corr(Xv,i , Xv,j ), the correlation parameter for state variables Xv,i of individual mortgages of vintage v, is positive. Then, Av and Av0 , the numbers of defaults
observed at time T within mortgage vintages v and v 0 are correlated if and only if
φv,v0 = Corr(Zv , Zv0 ) 6= 0, where Zv is the common Gaussian risk factor process.
Moreover, in the large portfolio limit, Corr(Av , Av0 ) approaches a limiting value determined by φv,v0 , ρv , and ρv0 .
500
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
Proof. Conditional on the common risk factor Zv , the number of defaults Av is a sum of
N independent indicator variables and follows a binomial distribution. More specifically,
N k
P(Av = k|Zv ) =
p (1 − pv )N −k
(6.14)
k v
where pv is the default probability conditional on Zv , i.e.,
pv = P(τv,i ≤ τ ∗ |Zv ) = P(Xv,i ≤ Xv∗ |Zv ),
with
Xv∗ = Φ−1 (Fv (T )),
where Fv (T ) is the probability of default before the time T . Then
pv = P (Xv,i ≤ Xv∗ |Zv ) = Φ (Zv∗ ) ,
where
Zv∗
(6.15)
√
Xv∗ − ρv Zv
√
=
.
1 − ρv
(6.16)
pv0 = Φ (Zv∗0 ) ,
(6.17)
Similarly,
where
√
Xv∗0 − ρv0 Zv0
√
.
(6.18)
1 − ρv 0
Note that if Zv and Zv0 are jointly Gaussian with correlation coefficient φv,v0 , we can
write
q
(6.19)
Zv = φv,v0 Zv0 + 1 − φ2v,v0 uv,v0 for t > j,
Zv∗0 =
where uv,v0 are standard Gaussians that are independent of Zv0 . Combining equation
(6.16), (6.18) and (6.19), we have
q
ρv (1 − φ2v,v0 )
∗
∗
X
−
bφ
X
j v0
√
−
uv,v0 ,
(6.20)
Zv∗ = aφv,v0 Zv∗0 + t√
1 − ρt
1 − ρv
where
s
r
ρv (1 − ρv0 )
ρv
a=
, b=
.
ρv0 (1 − ρv )
ρv 0
Cov(pv , pv0 ) = Cov (Φ(Zv∗ ), Φ(Zv∗0 ))
q
 


2 )
ρ
(1
−
φ
∗
∗
0
v
v,v
0
X
−
bφ
X
v,v
v0
√
= Cov Φ aφv,v0 Zv∗0 + v √
−
uv,v0  , Φ (Zv∗0 ) .
1 − ρv
1 − ρv
(6.21)
Since a > 0 as ρv ∈ (0, 1), we know that the covariance and the correlation between pv
and pv0 are determined by φv,v0 , ρv , and ρv0 . They are nonzero if and only if φv,v0 6= 0.
Applying Proposition 6.1, we know that
Corr(pv , pv0 )
q
v (1−pv )]
v 0 (1−pv 0 )]
1 + E[p
1 + E[p
N Var(p )
N Var(p )
Corr(Av , Av0 ) = q
v0
∀v 6= v 0 .
(6.22)
v
Therefore, Av and Av0 have nonzero correlation as long as pv and pv0 do.
TEMPORAL CORRELATION OF DEFAULTS
501
Equations (6.21) and (6.22) provide closed-form expressions for the serial correlation
of default rates pv of different vintages and the number of defaults Av . However, we
cannot directly read from equation (6.21) how the vintage correlation of default rates
depends on φv,v0 . The theorem below (whose proof extends an idea from [37]) shows that
this dependence is always positive.
Theorem 6.3 (Dependence on Common Risk Factor). Under the same settings as in
Theorem 6.2, assume that both the serial correlation φv,v0 of the common risk factor and
the individual state variable correlation ρv are always positive. Then the number Av of
defaults in the vintage-v cohort by time T is positively correlated with the number Av0 in
the vintage-(v 0 ) cohort. Moreover, this correlation is an increasing function of the serial
correlation parameter φv,v0 in the common risk factor.
Proof. We will use the notation established in Proposition 6.1. We can assume that
v 6= v 0 . Recall that in the Gaussian copula model, name i in the vintage-v cohort defaults by time T if the standard Gaussian variable Xv,i falls below a threshold Xv∗ . The
unconditional default probability is
P[Xv,i ≤ Xv∗ ] = Φ(Xv∗ ).
For the covariance, we have
Cov(Av , Av0 ) =
N
X
Cov(1[Xv,k ≤Xv∗ ] , 1[Xv0 ,l ≤X ∗0 ] )
v
k,l=1
(6.23)
2
= N Cov(1[X≤Xv∗ ] , 1[X 0 ≤X ∗0 ] ),
v
where X, X 0 are jointly Gaussian, each standard Gaussian, with mean zero and covariance
E[XX 0 ] = E[Xv,k Xv0 ,l ],
which is the same for all pairs k, l, since v 6= v 0 . This common value of the covariance
arises from the covariance between Zv and Zv0 along with the covariance between any
Xv,k with Zv ; it is
√
(6.24)
Cov(X, X 0 ) = φj ρv ρv0 .
Now since X, X 0 are jointly Gaussian, we can express them in terms of two independent
standard Gaussians:
W1 := X,
1
√
(6.25)
W2 := q
[X 0 − φv,v0 ρv ρv0 X].
1 − ρv ρv0 φ2v,v0
We can check readily that these are standard Gaussians with zero covariance, and
X = W1 ,
q
(6.26)
√
X 0 = φv,v0 ρv ρv0 W1 + 1 − ρv ρv0 φ2v,v0 W2 .
Let
The assumption that ρ and φv,v0
√
α = φv,v0 ρv ρv0 .
are positive (and, of course, less than 1) implies that
0 < α < 1.
502
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
Note that the covariance between pv and pv0 can be expressed as
Cov(pv , pv0 ) = E(pv pv0 ) − E(pv )E(pv0 )
h ii
h
= E E 1{Xv,i ≤Xv∗ } Zv E 1{X 0 ≤X ∗ } Zv0 − E(pv )E(pv0 )
v ,i
v0
h
i
= E 1{Xv,i ≤Xv∗ } 1{X 0 ≤X ∗ } − E(pv )E(pv0 )
v ,i
v0
= P [Xv,i ≤ Xv∗ , Xv0 ,i ≤ Xv∗ ] − E(pv )E(pv0 )
h
i
p
= P W1 ≤ Xv∗ , αW1 + 1 − α2 W2 ≤ Xv∗0 − E(pv )E(pv0 )
Z Xv∗ ∗
Xv0 − αw1
√
ϕ(w1 ) dw1 − E(pv )E(pv0 ),
=
Φ
1 − α2
−∞
where ϕ(·) is the probability density function of the standard normal distribution. The
third equality follows from equation (6.9). The fifth equality follows from equation (6.26).
The unconditional expectation of pv is independent of α, because
E(pv ) = E [ P(Xv,i ≤ Xv∗ |Zv ) ] = Φ(Xv∗ ).
(6.27)
It follows that
∂
Cov(pv , pv0 ) =
∂α
Z
Xv∗
ϕ
−∞
Z Xv∗
Xv∗0 − αw1
√
1 − α2
∂
ϕ(w1 )
Xv∗0 −αw1
√
1−α2
∂α
dw1
−w1 + αXv∗0
Xv∗0 − αw1
√
ϕ(w1 )
=
dw1
ϕ
3
1 − α2
−∞
(1 − α2 ) 2
∗
Z Xv∗
Xv0 − αw1
1
∗
√
ϕ(w1 ) dw1 .
=−
(w1 − αXv0 )ϕ
3
1 − α2
(1 − α2 ) 2 −∞
(6.28)
The last two terms in the integrand simplify to
"
#
∗
2
Xv0 − αw1
1
(w1 − αXv∗0 ) + Xv∗0 2 (1 − α2 )
√
ϕ
ϕ(w1 ) =
exp −
.
2π
2(1 − α2 )
1 − α2
(6.29)
Substituting equation (6.29) into (6.28), we have
"
#
Xv∗0 2
Z Xv∗
2
exp
−
2
∂
(w1 − αXv∗0 )
∗
dw1 .
Cov(pv , pv0 ) = −
(w1 − αXv0 ) exp −
3
∂α
2(1 − α2 )
2π (1 − α2 ) 2 −∞
Make a change of variable and let
w1 − αXv∗0
y := √
.
1 − α2
It follows, upon further simplification, that
1
∂
X ∗ 2 − 2αXv∗ Xv∗0 + Xv∗0 2
√
Cov(pv , pv0 ) =
> 0.
(6.30)
exp − v
∂α
2(1 − α2 )
2π 1 − α2
Thus, we have shown that the partial derivative of the covariance with respect to α is
positive. Since
√
α = ρv ρv0 φv,v0 ,
TEMPORAL CORRELATION OF DEFAULTS
503
TABLE 1. Default Probabilities Through Time (F (τ )).
Time (Month)
Default Probability
Subprime
12
24
0.04 0.10
36
0.12
72
0.13
144
0.14
Time (Month)
Default Probability
Prime
12
24
0.01 0.02
36
0.03
72
0.04
144
0.05
with ρs and φv,v0 assumed to be positive, we know that the partial derivatives of the covariance with respect to φv,v0 , ρv and ρv0 are also positive everywhere. Note that the
unconditional variance of pv is independent of φv,v0 (although dependent of ρs ), which
can be seen from equation (6.15). It follows that the serial correlation of pv has positive
partial derivative with respect to φv,v0 . Recall equation (6.21), which shows that the covariance of pv and pv0 is zero for any value of ρs when φv,v0 = 0. This result together
with the positive partial derivatives of the covariance with respect to φv,v0 ensure that
the covariance and thus the vintage correlation of pv and pv0 is always positive. From
equation (6.22), noticing the fact that both the expectation and variance of pv are independent of φv,v0 , we know that the correlation between Av and Av0 must also be positive
everywhere and monotonically increasing in φv,v0 .
7. Monte Carlo Simulations
In this section, we study the link between serial correlation in a common risk factor
and vintage correlation in pools of mortgages in two sets of simulations: First, a series
of mortgage pools is simulated to illustrate the analytical results of Section 6. Second, a
waterfall structure is simulated to study temporal correlation in MBS.
7.1. Vintage Correlation in Mortgage Pools. We conduct a Monte Carlo simulation to
study how serial correlation of a common risk factor propagates into vintage correlation in
default rates. We simulate default times for individual mortgages according to equations
(5.1), (5.5), and (5.6). From the simulated default times, the default rate of a pool of mortgages is calculated. In each simulation, we construct a cohort of N = 100 homogeneous
mortgages in every month v = 1, 2, . . . , 120. We simulate a monthly time series of the
common risk factor Zv , which is assumed to have an AR(1) structure with unconditional
mean zero and variance one,
p
Zv = φZv−1 + 1 − φ2 uv ∀v = 2, 3, . . . , 120.
(7.1)
The errors uv are i.i.d. standard Gaussian. The initial observation Z1 is a standard normal
random variable. We report the case where φ = 0.95. Each mortgage i issued at time v
has a state variable Xv,i assigned to it that determines its default time. The time series
properties of Xv,i follow equation (5.6). The error εi in equation (5.6) is independent of
uv .
504
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
F IGURE 3. Serial Correlation in Default Rates of Subprime Mortgages
Default Rates Across Vintages
0.12
0.1
0.08
0
20
40
60
80
100
120
Vintage Correlation
1
0.5
0
−0.5
0
2
4
2
4
6
8
10
12
14
16
Lag
Sample Partial Autocorrelation Function
6
8
18
20
18
20
1
0.5
0
−0.5
0
10
Lag
12
14
16
To simulate the actual default rates of mortgages, we need to specify the marginal distribution functions of default times F (·) as in equation (5.1). We define a function F(·),
which takes a time period as argument and returns the default probability of a mortgage
within that time period since its initiation. We assume that this F(·) is fixed across different vintages, which means that mortgages of different cohorts have a same unconditional
default probability in the next S periods from their initiation, where S = 1, 2, . . . . It is
easy to verify that Fv (T ) = F(T − v). The values of the function F(·) are specified in
Table 1, for both subprime and prime mortgages. Intermediate values of F(·) are linearly
interpolated from this table. While these values are in the same range as actual default
rates of subprime and prime mortgages in the last ten years, their specification is rather
arbitrary as it has little impact on the stochastic structure of the simulated default rates.
We set the observation time T to be 144, which is two years after the creation of the last
vintage, as we need to give the last vintage some time window to have possible default
events. For example, in each month from 1998 to 2007, 100 mortgages are created.
We need to consider two cases, subprime and prime. For the subprime case, every
vintage is given a two-year window to default, so the unconditional default probability
is constant across vintages. On the other hand, prime mortgages have decreasing default
probability through subsequent vintages. For example, in our simulation, the first vintage
has a time window of 144 months to default, the second vintage has 143 months, the
third has 142 months, and so on. Therefore, older vintages are more likely to default by
observation time T than newer vintages. This is why the fixed ex-post observation time
of defaults is one difference that distinguishes vintage correlation from serial correlation.
TEMPORAL CORRELATION OF DEFAULTS
505
F IGURE 4. Serial Correlation in Default Rates of Prime Mortgages
Default Rates Across Vintages
0.06
0.04
0.02
0
20
40
60
80
100
120
Vintage Correlation
1
0.5
0
−0.5
0
2
4
2
4
6
8
10
12
14
16
Lag
Sample Partial Autocorrelation Function
6
8
18
20
18
20
1
0.5
0
−0.5
0
10
Lag
12
14
16
We construct a time series τv,i of default times of mortgage i issued at time v according
to equation (5.5). (Note that this is not a time series of default times for a single mortgage,
since a single mortgage defaults only once or never. Rather, the index i is a placeholder
for a position in a mortgage pool. In this sense, τv,i is the time series of default times of
mortgages in position i in the pool over vintages v.) Time series of default rates A¯v are
computed as:
#{mortgages for which τv,i ≤ τv∗ }
A¯v (τv∗ ) =
.
N
In the subprime case, τv∗ = 24; in the prime case, τv∗ = T − v varies across vintages.
The simulation is repeated 1000 times. For the subprime case, the average simulated
default rates are plotted in Figure 3. For the prime case, average simulated default rates
are plotted in Figure 4. Note that because of the decreasing time window to default, the
default rates in Figure 4 have a decreasing trend.
In the subprime case, we can use the sample autocorrelation and partial autocorrelation
functions to estimate vintage correlation, because the unconditional default probability is
constant across vintages, so that averaging over different vintages and averaging over
different pools is the same. In the prime case, we have to calculate vintage correlation
proper. Since we have 1000 Monte Carlo observations of default rates for each vintage,
we can calculate the correlation between two vintages using those samples. For the partial
autocorrelation function, we simply demean the series of default rates and obtain the usual
partial autocorrelation function. We plot the estimated vintage correlation in the second
rows of Figure 3 and 4 for subprime and prime cases, respectively. As can be seen, the
506
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
correlation of the default rates of the first vintage with older vintages decreases geometrically. In both cases, the estimated first-order coefficient of default rates is close to but less
than φ = 0.95, the AR(1) coefficient of the common risk factor. The partial autocorrelation functions are plotted in the third rows of Figures 3 and 4. They are significant only
at lag one. This phenomenon is also observed when we set φ to other values. Both the
sample autocorrelation and partial autocorrelation functions indicate that the default rates
follow a first-order autoregressive process, similar to the specification of the common risk
factor. However, compared with the subprime case, the default rates of prime mortgages
seem to have longer memory.
The similarity between the magnitude of the autocorrelation coefficient of default rates
and common risk factor can be explained by the following Taylor expansion. Taylorexpanding equation (6.15) at Zv∗ = 0 to first order, we have
1
1
+ √ Zv∗ .
(7.2)
2
2π
Since pv is approximately linear in Zv∗ , which is a linear transformation of Zt , it follows
a stochastic process that has approximately the same serial correlation as Zt .
pv ≈
7.2. Vintage Correlation in Waterfall Structures. We have already shown using the
Gaussian copula approach that the time series of default rates in mortgage pools inherits
vintage correlation from the serial correlation of the common risk factor. We now study
how this affects the performance of assets such as MBS that are securitized from the
mortgage pool in a so-called waterfall. The basic elements of the simulation are: (i) A
time line of 120 months and an observation time T = 144. (ii) The mortgage contract
has a principal of $ 1, maturity of 15 years, and annual interest rate 9%. Fixed monthly
payments are received until the mortgage defaults or is paid in full. A pool of 100 such
mortgages is created every month. (iii) There is a pool of 100 units of MBS, each of
principal $1, securitized from each month’s mortgage cohort. There are four tranches: the
senior tranche, the mezzanine tranche, the subordinate tranche, and the equity tranche.
The senior tranche consists of the top 70% of the face value of all mortgages created
in each month; the mezzanine tranche consists of the next 25%; the subordinate tranche
consist of the next 4%; the equity tranche has the bottom 1%. Each senior MBS pays an
annual interest rate of 6%; each mezzanine MBS pays 15%; each subordinate MBS pays
20%. The equity tranche does not pay interest but retains residual profits, if any.
The basic setup of the simulation is illustrated in Figure 2. For a cohort of mortgages issued at time v and the MBS derived from it, the securitization process works
as follows. At the end of each month, each mortgage either defaults or makes a fixed
monthly payment. The method to determine default is the same that we have used before:
mortgage i issued at time v defaults at τv,i , which is generated by the Gaussian copula
approach according to equations (5.1), (5.5), and (5.6). We consider both subprime and
prime scenarios, as in the case of default rates. For subprime mortgages, we assume that
each individual mortgage receives a prepayment of the outstanding principal at the end
of the teaser period if it has not defaulted, so the default events and cash flows only happen within the teaser period. For the prime case, there is no such restriction. Again, we
assume the common risk factor to follow an AR(1) process with first-order autocorrelation coefficient φ = 0.95. The cross-name correlation coefficient ρ is set to be 0.5. The
unconditional default probabilities over time are obtained from Table 1.
TEMPORAL CORRELATION OF DEFAULTS
507
F IGURE 5. Serial Correlation in Principal Losses of Subprime MBS
Vintage Corr
Mezzanine Tranche
Equity Tranche
1
1
0.5
0.5
0.5
0
0
0
−0.5
0
Partial Autocorr
Subordinate Tranche
1
5
10
−0.5
0
5
10
−0.5
0
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
0
5
10
−0.5
0
5
10
−0.5
0
5
10
5
10
The first row plots the vintage correlation of the principal loss of each tranche. The correlation is estimated
using the sample autocorrelation function. The second row plots the partial autocorrelation functions.
If a mortgage has not defaulted, the interest payments received from it are used to
pay the interest specified on the MBS from top to bottom. Thus, the cash inflow is used
to pay the senior tranche first (6% of the remaining principal of the senior tranche at
the beginning of the month). The residual amount, if any, is used to pay the mezzanine
tranche, after that the subordinate tranche, and any still remaining funds are collected
in the equity tranche. If the cash inflow passes a tranche threshold but does not cover
the following tranche, it is prorated to the following tranche. Any residual funds after
all the non-equity tranches have been paid add to the principal of the equity tranche.
Principal payments are processed analogously. We assume a recovery rate of 50% on
the outstanding principal for defaulted mortgages. The 50% loss of principal is deducted
from the principal of the lowest ranked outstanding MBS.
Before we examine the vintage correlation of the present value of MBS tranches, we
look at the time series of total principal loss across MBS tranches. In our simulations, no
loss of principal occurred for the senior tranche. The series of expected principal losses
of other tranches and their sample autocorrelation and sample partial autocorrelation are
plotted in Figures 5 and 6 for subprime and prime scenarios respectively. We use the
same method to obtain the autocorrelation functions for prime mortgages as in the case of
default rates. The correlograms show that the expected loss of principal for each tranche
follows an AR(1) process.
The series of present values of cash flows for each tranche and their sample autocorrelation and partial autocorrelation functions are plotted in Figures 7 and 8 for subprime
508
E. HILLEBRAND, A. N. SENGUPTA, AND J. XU
F IGURE 6. Serial Correlation in Principal Losses of Prime MBS
Vintage Corr
Mezzanine Tranche
Equity Tranche
1
1
0.5
0.5
0.5
0
0
0
−0.5
0
Partial Autocorr
Subordinate Tranche
1
10
20
−0.5
0
10
20
−0.5
0
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
0
10
20
−0.5
0
10
20
−0.5
0
10
20
10
20
The first row plots the vintage correlation of the principal loss of each tranche. The correlation is estimated
using the correlation between the first and subsequent vintages, each of which has a Monte Carlo sample size
of 1000. The second row plots the partial autocorrelation functions of the demeaned series of principal losses.
and prime scenarios, respectively. The senior tranche displays a significant first-order autocorrelation coefficient due to losses in interest payments although there are no losses
in principal. The partial autocorrelation functions, which have significant positive values
for more than one lag, suggest that the cash flows may not follow an AR(1) process due
to the high non-linearity. However, the estimated vintage correlation still decreases over
vintages, same as in an AR(1) process, which indicates that our findings for default rates
can be extended to cash flows.
Acknowledgment. Discussions with Darrell Duffie and Jean-Pierre Fouque helped improve the paper. We thank seminar and conference participants at Stanford University,
LSU, UC Merced, UC Riverside, UC Santa Barbara, the 2010 Midwest Econometrics
Group Meetings in St. Louis, Maastricht University, and CREATES. EH acknowledges
support from the Danish National Research Foundation. ANS acknowledges research
support received as a Mercator Guest Professor at the University of Bonn in 2011 and
2012, and discussions with Sergio Albeverio and Claas Becker.
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509
F IGURE 7. Cashflow correlation in Subprime MBS
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F IGURE 8. Cashflow correlation in Prime MBS
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Partial Autocorr
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E RIC H ILLEBRAND : CREATES - C ENTER FOR R ESEARCH IN E CONOMETRIC A NALYSIS
S ERIES , D EPARTMENT OF E CONOMICS AND B USINESS , A ARHUS U NIVERSITY, D ENMARK
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T IME
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