Submitted to the Brazilian Journal of Probability and Statistics
cr
ip
t
Generalized backward stochastic variational
inequalities driven by a fractional Brownian
motion
Dariusz Borkowskia and Katarzyna Ja´
nczak-Borkowskab
a
b
Nicolaus Copernicus University
University of Technology and Life Sciences
an
us
Abstract. We study the existence and uniqueness of the generalized
reflected backward stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H greater than 1/2. The
stochastic integral used throughout the paper is the divergence type
integral.
M
1 Introduction
B
JP
S
-A
cc
ep
te
d
The backward stochastic differential equations (BSDEs) were first studied
´ and Peng, S. [1990]. Since then many papers have been dein Pardoux, E.
voted to the study of BSDEs, mainly due to their applications. The main
aim of studying BSDEs was to give a probabilistic interpretation for solutions of partial differential equations (PDEs for short). Pardoux and Zhang
´ and Zhang, S. [1998] introduced the generalized BSDEs, i.e.
in Pardoux, E.
BSDEs with an additional term - an integral with respect to an increasing
´ and R˘
process. Pardoux and R˘
a¸scanu in Pardoux, E.
a¸scanu, A. [1998] put
some constrains on the solution of the BSDE (or more precisely they put
some additional assumptions on the first component of the solution) and the
problem was called the backward stochastic variational inequality (BSVI)
(or in some special cases the reflected BSDE). In Ja´
nczak, K. [2009] and
Ja´
nczak-Borkowska, K. [2011] the existence and uniqueness of the generalized reflected BSDE was shown.
BSDEs driven by a fractional Brownian motion (fBm) were first studied
in Biagini, F. et al. [2002] (with Hurst parameter H > 1/2) and in Bender,
C. [2005] (with Hurst parameter H ∈ (0, 1)). Nonlinear BSDEs with respect
to a fractional Brownian motion (fBm) with Hurst parameter H > 1/2 were
first considered by Hu and Peng in Hu, Y. and Peng, S. [2009], but the
existence and uniqueness of the solution of the BSDE driven by a fBm was
MSC 2010 subject classifications: Primary 60H10, 60H05, 60Gxx; secondary 47J20
Keywords and phrases. Backward stochastic differential equation, fractional Brownian
motion, backward stochastic variational inequalities, subdifferential operator
1
2
D. Borkowski, K. Ja´
nczak-Borkowska
cr
ip
t
obtained with some restrictive assumption. Maticiuc and Nie, Maticiuc, L.
and Nie, T. [2013] improved their result and omitted this assumption. They
also developed a theory of backward stochastic variational inequalities, i.e.
they proved the existence and uniqueness of the solution of the reflected
BSDEs driven by a fBm. In the paper Ja´
nczak-Borkowska, K. [2013] the
existence and uniqueness of the generalized BSDEs driven by a fBm with
Hurst parameter H greater than 1/2 were shown.
In this paper we study the generalized BSVI driven by a fBm with Hurst
parameter H greater than 1/2. We prove that that kind of equation has a
unique solution.
RH (s, t) = E BsH BtH =
an
us
Let us now recall that a fBm with Hurst parameter H ∈ (0, 1) is a zero
mean Gaussian process B H = {BtH , t ≥ 0} with the covariance function
1 2H
t + s2H − |t − s|2H .
2
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d
M
H has the same law as aH B H for any
This process is a self-similar, i.e. Bat
t
a > 0, it has homogeneous increments. For H = 1/2 we obtain a standard
Wiener process, but for H 6= 1/2, the process B H is not a semimartingale.
These properties make this process a useful tool in models arising in physics,
telecommunication networks, finance, signal processing and other fields.
S
-A
Since B H is not a semimartingale when H 6= 1/2, we cannot use the classical theory of stochastic calculus to define the fractional stochastic integral.
Essentially, two different types of integrals with respect to a fBm have been
defined and studied. The first one is the pathwise Riemann-Stieltjes integral
(see Young, L.C. [1936]). This integral has the properties of Stratonovich
integral, which leads to difficulties in the applications. The second one, in¨ unel, A.S. [1998] is the divergence
troduced in Decreusefond, L. and Ust¨
operator (Skorokhod integral), defined as the adjoint of the derivative operator in the framework of the Malliavin calculus. Since this stochastic integral
satisfies the zero mean property and it can be expressed as the limit of Riemann sums defined using Wick products, it was later developed by many
authors.
B
JP
The paper is organized as follows. In Section 2 we give some definitions
and results about a fractional stochastic integral, which will be needed
throughout the paper. Section 3 contains the definition of the generalized
BSVI driven by a fBm, assumptions and the formulation of the main theorem of the paper. In Section 4 we prove some a priori estimates. Finally,
using the penalization method we prove the main theorem in Section 5.
3
GBSVI driven by fBm
2 Fractional calculus
Denote φ(x) = H(2H − 1)|x|2H−2 , x ∈ R. Let ξ and η be measurable functions on [0, T ]. Define
Z tZ
0
t
φ(u − v)ξ(u)η(v)dudv
0
cr
ip
t
hξ, ηit =
Z
F (ω) = f
T
0
ξ1 (t)dBtH , . . . ,
an
us
and kξk2t = hξ, ξit . Note that, for any t ∈ [0, T ], hξ, ηit is a Hilbert scalar
product. Let H be the completion of the measurable functions such that
kξkt < ∞. The elements of H may be distributions.
Let (ξn )n be a sequence in H such that hξi , ξj iT = δij . By PT denote
the set of all polynomials of a fractional Brownian motion, i.e. it contains
elements of the form
T
Z
0
ξk (t)dBtH
!
,
=
k
X
∂f
∂xi
Z
T
ξ1 (t)dBtH , . . . ,
Z
T
ξk (t)dBtH
cc
ep
te
d
DsH F
M
where f is a polynomial function of k variables. The Malliavin derivative
operator DsH of an element F ∈ PT is defined as follows:
i=1
0
0
!
ξi (s), s ∈ [0, T ].
The divergence operator D H is closable from L2 (Ω, F, P ) to L2 (Ω, F, P ; H).
By D1,2 denote the Banach space being a completition of PT with the following norm: kF k21,2 = E|F |2 +EkDsH F k2T . Now we introduce another derivative
Z
DH
t F =
T
φ(t − s)DsH F ds.
0
-A
Theorem 2.1. Let F : (Ω, F, P) → H be a stochastic process such that
E kF k2T +
Z
0
T
Z
T
0
2
|DH
s Ft | dsdt
!
S
Then, the Itˆ
o-type stochastic integral denoted by
B
JP
L2 (Ω, F). Moreover, E
Z
T
0
E
Z
0
T
Fs dBsH
!2
Fs dBsH
=E
!
kF k2T
< ∞.
Z
T
0
Fs dBsH
exists in
= 0 and
+
Z
0
T
Z
0
T
H
DH
s Ft Dt Fs dsdt
!
.
4
D. Borkowski, K. Ja´
nczak-Borkowska
Theorem 2.2. Let f ∈ L2 ([0, T ]) be a deterministic function, H > 1/2.
Suppose that kf kt is continuously differentiable as a function of t ∈ [0, T ].
Set
Z
Z
t
Xt = X0 +
0
t
gs ds +
0
fs dBsH ,
t ∈ [0, T ],
R
t ∂F
∂F
(s, Xs )ds +
(s, Xs )dXs
0 ∂s
0 ∂x
∂2F
d
(s, Xs ) (kf k2s )ds, t ∈ [0, T ].
∂x2
ds
+
1
2
Z
t
0
Z
Z
an
us
F (t, Xt ) = F (0, X0 ) +
t
cr
ip
t
where X0 is a constant and g is deterministic with 0T |gs |ds < ∞. Let
F be continuously differentiable with respect to t and twice continuously
differentiable with respect to x. Then
T
0
Denote
Z
T
2
|DH
t f2 (s)| dsdt
0
F (t) =
Z
t
Z
t
Z
t
0
and
G(t) =
0
-A
Then
S
F (t)G(t) =
B
JP
< ∞,
E
Z
T
T
Z
cc
ep
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d
E
Z
M
Theorem
R 2.3. Let T ∈ (0, ∞) and let f1 (s), f2 (s), g1 (s), g2 (s) be in D1,2
H
and E 0T (|fi (s)| + |gi (s)|) ds < ∞. Assume that DH
t f2 (s) and Dt g2 (s)
are continuously differentiable with respect to (s, t) ∈ [0, T ]×[0, T ] for almost
all ω ∈ Ω. Suppose that
0
+
+
0
f1 (s)ds +
Z
g1 (s)ds +
Z
t
0
0
t
Z
t
t ∈ [0, T ]
g2 (s)dBsH ,
t ∈ [0, T ].
t
0
Z
0
G(s)f1 (s)ds +
0
2
|DH
t g2 (s)| dsdt < ∞.
f2 (s)dBsH ,
F (s)g1 (s)ds +
Z
0
t
F (s)g2 (s)dBsH
Z
t
0
DH
s F (s)g2 (s)ds +
G(s)f2 (s)dBsH
Z
0
t
DH
s G(s)f2 (s)ds.
The above theorems can be found in Duncan, T.E., et al. [2000], Hu, Y.
[2005], Hu, Y. and Peng, S. [2009], Maticiuc, L. and Nie, T. [2013] and for
a deeper discussion we refer the reader to Hu, Y. [2005], Nualart, D. [2010].
5
GBSVI driven by fBm
3 Generalized BSVI with respect to fBm
Assume that
(H1 ) σ : [0, T ] → R is a deterministic continuous differentiable
function such
Z
where η0 is a given constant.
= H(2H − 1)
0
d kσk2t = 2H(2H − 1)
dt
where σ
ˆ (t) =
Z tZ
Z
t
t
|u − v|2H−2 σ(u)σ(v)dudv, we have
0
|t − u|2H−2 σ(u)σ(t)du = 2σ(t)ˆ
σ (t) > 0,
0
t
Z
σ(s)dBsH , t ∈ [0, T ],
an
us
Note that, since
kσk2t
0
cr
ip
t
t
that σ(t) 6= 0, for all t ∈ [0, T ] and ηt = η0 +
φ(t − u)σ(u)du.
0
dYt +f (t, ηt , Yt , Zt )dt + g(t, ηt , Yt )dΛt −Zt dBtH ∈∂ϕ(Yt )dt + ∂ψ(Yt )dΛt
YT = ξ
(3.1)
cc
ep
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d
(
M
We will consider the following generalized backward stochastic variational
inequality driven by a fBm:
where Λ is an adapted increasing process, Λ0 = 0.
We suppose that there exist positive constants L and ν > 2L + 2 and
(H2 ) ξ = h(ηT ) for some function h
! with bounded derivative and such that
Z
T
E eνΛT |ξ|2 +
0
eνΛt |ηt |2 dt
< ∞.
-A
(H3 ) f : [0, T ] × R × R × R → R and g : [0, T ] × R × R → R are continuous
functions such that for all t ∈ [0, T ], x, x′ , y, y ′ , z, z ′ ∈ R,
|f (t, x, y, z) − f (t, x′ , y ′ , z ′ ) ≤ L(|x − x′ | + |y − y ′ | + |z − z ′ |)
B
JP
S
|g(t, x, y) − g(t, x, y ′ ) ≤ L|y − y ′ |
E
Z
0
T
νΛt
e
2
|f (t, 0, 0, 0)| dt +
Z
T
νΛt
e
0
2
|g(t, ηt , 0)| dΛt
(H4 ) functions ϕ, ψ : R → (−∞, ∞] satisfy
• ϕ, ψ are proper, convex and lower semi–continuous
• ϕ(y) ≥ ϕ(0) = 0, ψ(y) ≥ ψ(0) = 0.
!
< ∞.
6
D. Borkowski, K. Ja´
nczak-Borkowska
We will denote
∂ϕ(y) = {ˆ
y ∈ R; yˆ · (v − y) + ϕ(y) ≤ ϕ(v), ∀v ∈ R}
Domϕ = {y ∈ R; ϕ(y) < ∞},
Dom(∂ϕ) = {y ∈ R; ∂ϕ(y) 6= ∅}
hy, yˆi ∈ ∂ϕ ⇔ y ∈ Dom(∂ϕ), yˆ ∈ ∂ϕ(y)
cr
ip
t
(analogously for ψ).
Remark 3.1. ∂ϕ and ∂ψ are maximal in this sense that
(ˆ
y −ˆ
u)(y−u) ≥ 0, (y, yˆ), (u, uˆ) ∈ ∂ϕ,
(ˆ
y −ˆ
v )(y−v) ≥ 0, (y, yˆ), (v, vˆ) ∈ ∂ψ.
an
us
Now consider the set
1,2
V[0,T ] = {Y = φ(·, η) : φ ∈ Cpol
([0, T ] × R) and
∂φ
is bounded }.
∂t
H
By V˜[0,T
] denote the completition of the set of processes from V[0,T ] with the
following norm
=E
Z
T
t
2H−1 νΛt
e
0
2
Z
T
t2H−1 eνΛt |φ(t, ηt )|2 dt
M
kY
k2H
|Yt | dt = E
0
cc
ep
te
d
H,Λ
and by V˜[0,T
] – the completition of the set of processes from V[0,T ] with a
norm
T
Z
kY k2H,Λ = E
0
t2H−1 eνΛt |Yt |2 dΛt = E
Z
T
0
t2H−1 eνΛt |φ(t, ηt )|2 dΛt .
Definition 3.2. A solution of a generalized backward stochastic variational
inequality (GBSVI) driven by a fBm (3.1) associated with data (ξ, f, g, Λ)
is a quadruple (Y, Z, U, V ) = (Yt , Zt , Ut , Vt )t∈[0,T ] of processes satisfying
Z
T
f (s, ηs , Ys , Zs )ds +
-A
Yt = ξ +
−
t
Z
t
T
Us ds −
Z
t
T
Vs dΛs ,
Z
t
T
g(s, ηs , Ys )dΛs −
Z
t
T
Zs dBsH
t ∈ [0, T ]
(3.2)
B
JP
S
and such that
(Yt , Ut ) ∈ ∂ϕ,
(Yt , Vt ) ∈ ∂ψ,
t ∈ [0, T ]
1/2
1/2,Λ
H , U, V ∈ V
˜ H ∩ V˜ H,Λ .
and Y ∈ V˜[0,T ] ∩ V˜[0,T ] , Z ∈ V˜[0,T
]
[0,T ]
[0,T ]
Theorem 3.3. Assume (H1 )−(H4 ). There exists a unique solution of (3.2).
The proof of the above theorem is deferred to the Section 5.
7
GBSVI driven by fBm
4 A priori estimates
Theorem 4.1. Assume (H1 ) − (H4 ) and let (Y, Z, U, V ) be a solution of
(3.2). Then for all t ∈ [0, T ],
|Yt | +
T
Z
νΛs 2H−1
e
s
t
≤ CE eνΛT |ξ|2 +
+
Z
T
T
Z
T
Z
2
|Zs | ds +
νΛs
e
t
2
|Ys | dΛs
eνΛs |f (s, 0, 0, 0)|2 ds
t
νΛs
e
t
2
|g(s, ηs , 0)| dΛs +
Z
T
νΛs
e
2
!
cr
ip
t
E e
2
|ηs | ds = CΘ(t, T ).
an
us
νΛt
t
Proof. By C we will denote a constant which may vary from line to line.
From the Itˆo formula,
Z
T
t
Z
eνΛs d|Ys |2 −
T
T
t
−ν
Z
T
eνΛs Ys dYs − 2
=e
2
|ξ| + 2
Z
t
+2
-A
−2
−2
Z
Z
Z
Z
T
t
eνΛs DH
s Ys Zs ds
eνΛs |Ys |2 dΛs
t
νΛT
eνΛs |Ys |2 νdΛs
t
cc
ep
te
d
= eνΛT |ξ|2 − 2
Z
M
eνΛt |Yt |2 = eνΛT |ξ|2 −
T
νΛs
e
t
T
t
T
t
T
eνΛs Ys f (s, ηs , Ys , Zs )ds
Ys g(s, ηs , Ys )dΛs − 2
eνΛs Ys Us ds − 2
Z
T
t
eνΛs DH
s Ys Zs ds − ν
Z
T
t
eνΛs Ys Zs dBsH
eνΛs Ys Vs dΛs
Z
t
T
eνΛs |Ys |2 dΛs .
B
JP
S
It is known (see ex. Hu, Y. and Peng, S. [2009], Maticiuc, L. and Nie, T.
[2013]) that
DH
s Ys =
Z
0
T
φ(s − r)DrH Ys dr =
σ
ˆ (s)
Zs .
σ(s)
Moreover by Remark 6 in Maticiuc, L. and Nie, T. [2013], there exists M > 0
such that for all t ∈ [0, T ], t2H−1 /M ≤ σ
ˆ (t)/σ(t) ≤ M t2H−1 .
8
D. Borkowski, K. Ja´
nczak-Borkowska
By Lipschitz continuity of f and g we have
2yf (s, η, y, z) ≤ 2L|y| (|η| + |y| + |z|) + 2|y||f (s, 0, 0, 0)|
!
cr
ip
t
M L2
≤ L2 + 2L + 2H−1 + 1 |y|2 + |η|2
s
1 2H−1 2
+
s
|z| + |f (s, 0, 0, 0)|2
M
2yg(s, η, y) ≤ 2L|y|2 + 2|y||g(s, η, 0)| ≤ (2L + 1) |y|2 + |g(s, η, 0)|2 .
t
νΛT
≤E e
+E
eνΛs |Ys |2 dΛs +
2
|ξ| +
T
νΛs
e
T
Z
t
eνΛs s2H−1 |Zs |2 ds
2
|f (s, 0, 0, 0)| ds +
t
T
Z
νΛs
e
t
1
E
M
Z
2
|ηs | ds + E
T
t
T
Z
νΛs
e
t
Z
2
|g(s, ηs , 0)| dΛs
!
M L2
L + 2L + 2H−1 + 1 eνΛs |Ys |2 ds
s
T
2
t
eνΛs s2H−1 |Zs |2 ds + (2L + 1) E
cc
ep
te
d
+
Z
2
M
M
T
Z
E eνΛt |Yt |2 + ν
an
us
By the above and by Remark 3.1,
T
Z
t
eνΛs |Ys |2 dΛs .
Since ν ≥ (2L + 2) we can write
νΛt
E e
Z
2
|Yt | +
T
νΛs
e
t
1
|Ys | dΛs +
M
2
≤ Θ(t, T ) + E
Z
T
t
Z
T
νΛs 2H−1
e
s
t
2
|Zs | ds
!
M L2
L + 2L + 2H−1 + 1 eνΛs |Ys |2 ds.
s
2
(4.1)
-A
By the Gronwall inequality,
νΛt
Ee
(
2
2
|Yt | ≤ Θ(t, T ) exp (L + 2L + 1)(T − t) + M L
2T
2−2H
− t2−2H
2 − 2H
B
JP
S
and by (4.1) also
E
Z
T
νΛs 2H−1
e
t
s
2
|Zs | ds +
Z
T
νΛs
e
t
2
|Ys | dΛs
!
≤ CΘ(t, T ).
)
9
GBSVI driven by fBm
˜ U
˜ , V˜ ) be two solutions of (3.2)
Proposition 4.2. Let (Y, Z, U, V ) and (Y˜ , Z,
˜
˜
with data (ξ, f, g, Λ) and (ξ, f , g˜, Λ), respectively. Then
T
Z
t
eνΛs |Ys − Y˜s |2 dΛs +
˜2 +
≤ CE eνΛT |ξ − ξ|
+
Z
Z
t
T
νΛs
e
t
T
T
Z
s2H−1 eνΛs |Zs − Z˜s |2 ds
t
eνΛs |f (s, ηs , Ys , Zs ) − f˜(s, ηs , Ys , Zs )|2 ds
2
|g(s, ηs , Ys ) − g˜(s, ηs , Ys )| dΛs .
cr
ip
t
E eνΛt |Yt − Y˜t |2 +
eνΛt |Yt − Y˜t |2 + ν
Z
˜2 + 2
≤ eνΛT |ξ − ξ|
T
t
Z
t
−2
t
T
t
Z
t
Z
T
t
T
eνΛs s2H−1 |Zs − Z˜s |2 ds
eνΛs (Ys − Y˜s ) f (s, ηs , Ys , Zs ) − f˜(s, ηs , Y˜s , Z˜s ) ds
eνΛs (Ys − Y˜s ) g(s, ηs , Ys ) − g˜(s, ηs , Y˜s ) dΛs
M
−2
Z
T
T
2
M
eνΛs (Ys − Y˜s )(Zs − Z˜s )dBsH − 2
Z
t
T
cc
ep
te
d
+2
Z
eνΛs |Ys − Y˜s |2 dΛs +
an
us
Proof. By the Itˆo formula, computing similarly as in the previous theorem
˜s )ds
eνΛs (Ys − Y˜s )(Us − U
eνΛs (Ys − Y˜s )(Vs − V˜s )dΛs .
From assumptions we get
2(y − y˜)(f (s, η, y, z) − f˜(s, η, y˜, z˜)) ≤ 2(y − y˜)(f (s, η, y, z) − f˜(s, η, y, z))
!
|y − y˜|2 +
s2H−1
|z − z˜|2
M
-A
L2 M
+ 2L + 2H−1
s
and
S
2(y − y˜)(g(s, η, y)− g˜(s, η, y˜)) ≤ 2(y − y˜)(g(s, η, y)− g˜(s, η, y)) + 2L|y − y˜|2 .
B
JP
˜t ∈ ∂ϕ(Y˜t ),
Since Ut ∈ ∂ϕ(Yt ) and U
˜t )(Yt − Y˜t ) = Ut (Yt − Y˜t ) + U
˜t (Y˜t − Yt )
(Ut − U
≥ ϕ(Yt ) − ϕ(Y˜t ) + ϕ(Y˜t ) − ϕ(Yt ) = 0.
Similarly (Vt − V˜t )(Yt − Y˜t ) ≥ 0.
10
D. Borkowski, K. Ja´
nczak-Borkowska
Since ν ≥ 2L + 2, we obtain
E e
|Yt − Y˜t |2 +
Z
T
e
t
˜2 + 2
≤ E eνΛT |ξ − ξ|
+2
νΛs
Z
t
Z
+E
T
νΛs
e
t
Z
T
1
|Ys − Y˜s |2 dΛs +
M
T
Z
T
νΛs 2H−1
e
e
t
eνΛs (Ys − Y˜s ) f (s, ηs , Ys , Zs )− f˜(s, ηs , Ys , Zs ) ds
(Ys − Y˜s ) g(s, ηs , Ys ) − g˜(s, ηs , Ys ) dΛs
νΛs
s
t
|Zs − Z˜s |2 ds
L2 M
2L + 2H−1
s
!
|Ys − Y˜s |2 ds.
cr
ip
t
νΛt
an
us
Using the Gronwall lemma we get the required inequality.
5 Penalization scheme
1
1
|y − v|2 + ϕ(v); v ∈ R = |y − Jε (y)|2 + ϕ(Jε (y)), (5.1)
2ε
2ε
cc
ep
te
d
ϕε (y) = inf
M
We will approximate the function ϕ by a sequence of convex, C 1 class functions ϕε , ε > 0, defined by
where Jε (y) = y − ε∇ϕε (y).
Here are some properties of ϕε (see Barbu, V. [1976] or Br´ezis, H. [1973]):
y − Jε (y)
∈ ∂ϕ(Jε (y));
ε
|Jε (y) − Jε (v)| ≤ |y − v| and lim Jε (y) = πDomϕ (y);
(5.3)
0 ≤ ϕε (y) ≤ y∇ϕε (y),
(5.4)
∇ϕε (y) =
-A
εց0
(5.2)
B
JP
S
where by πDomϕ (y) we denote the projection of y on the closure of the set
Domϕ. Moreover consider analogous approximation ψε for the function ψ
(with J˜ε (y) = y − ε∇ψε (y)).
We introduce some compatibility assumptions:
for all ε > and all t ∈ [0, T ], η, y, z ∈ R
(i) ∇ϕε (y) · g(t, η, y) ≤ (∇ψε (y) · g(t, η, y))+
(5.5)
(ii) ∇ψε (y) · f (t, η, y, z) ≤ (∇ϕε (y) · f (t, η, y, z))+ .
11
GBSVI driven by fBm
Note that if y · g(t, η, y) ≤ 0 and y · f (t, η, y, z) ≤ 0 for all η, y, z ∈ R and
t ∈ [0, T ] then the compatibility assumptions are satisfied (it follows from
(5.4)). Moreover if for some a ≤ 0 ≤ b we define convex indicator functions
(
∞,
ψ(y) =
y<a
(
0,
y≤b
∞,
y>b
cr
ip
t
ϕ(y) =
y≥a
0,
Ytε
=ξ+
−
Z
T
Z
t
T
f (s, ηs , Ysε , Zsε )ds
∇ϕε (Ysε )ds −
T
g(s, ηs , Ysε )dΛs
t
∇ψε (Ysε )dΛs ,
t
−
Z
T
t
Zsε dBsH
t ∈ [0, T ].
(5.6)
M
t
T
Z
+
Z
an
us
then ∇ϕε (y) = − 1ε (y−a)− and ∇ψε (y) = 1ε (y−b)+ , where x− = max(−x, 0),
x+ = max(x, 0), and the compatibility assumptions become g(t, η, y) ≥ 0 for
y ≤ a and f (t, η, y, z) ≤ 0 for y ≥ b (compare Remark 2. from Maticiuc, L.
and R˘
a¸scanu, A. [2010]).
Consider a sequence of generalized BSDEs
Since ∇ϕε and ∇ψε are Lipschitz continuous functions, then by Ja´
nczakε
ε
Borkowska, K. [2013] (5.6) has a unique solution (Y , Z ).
cc
ep
te
d
Proposition 5.1. Let assumptions (H1 ) − (H4 ) hold. Then
νΛt
E e
|Ytε |2
+
Z
T
νΛs
e
t
≤ CE eνΛT |ξ|2 +
T
Z
+
νΛs
e
t
|Ysε |2 dΛs
Z
T
t
T
Z
+
νΛs 2H−1
e
s
t
|Zsε |2 ds
eνΛs |f (s, 0, 0, 0)|2 + |ηs |2 ds
2
|g(s, ηs , 0)| dΛs
= CΘ(t, T ).
-A
Proof. Similarly as in the proof of Theorem 4.1 we have
νΛt
B
JP
S
E e
|Ytε |2
+
Z
T
νΛs
e
t
|Ysε |2 dΛs
≤ E eνΛT |ξ|2 +
+E
Z
1
+
M
T
Z
T
νΛs 2H−1
e
s
t
eνΛs |f (s, 0, 0, 0)|2 ds +
t
Z
T
νΛs
e
t
− 2E
Z
t
T
|ηs | ds + E
νΛs
e
2
Z
t
Z
t
T
|Zsε |2 ds
T
eνΛs |g(s, ηs , 0)|2 dΛs
!
M L2
L + 2L + 2H−1 + 1 eνΛs |Ysε |2 ds
s
Ysε ∇ϕε (Ysε )ds
2
+
Z
t
T
νΛs
e
Ysε ∇ψε (Ysε )dΛs
!
.
12
D. Borkowski, K. Ja´
nczak-Borkowska
By (5.4) and analogous inequality for ψε , we obtain
Z
E eνΛt |Ytε |2 +
+2
Z
t
T
t
T
νΛs
e
1
M
eνΛs |Ysε |2 dΛs +
Ysε (∇ϕε (Ysε )ds
≤ Θ(t, T ) + E
Z
T
t
Z
T
t
eνΛs s2H−1 |Zsε |2 ds
+ ∇ψε (Ysε )dΛs )
!
cr
ip
t
M L2
L + 2L + 2H−1 + 1 eνΛs |Ysε |2 ds.
s
2
an
us
Now using similar arguments as in the proof of Theorem 4.1 we finish the
proof.
Proposition 5.2. Under assumptions (H1 ) − (H4 ) and (5.5) there exists a
positive constant C such that for any t ∈ [0, T ]
T
Z
t
(b) E
Z
T
t
eνΛs s2H−1 ϕ (Jε (Ysε )) + ψ J˜ε (Ysε )
dΛs ≤ CΘ2 (t, T )
2 cc
ep
te
d
(c) EeνΛt t2H−1 |Ytε − Jε (Ytε )|2 + Ytε − J˜ε (Ytε )
(d) EeνΛt t2H−1 ϕ (Jε (Ytε )) + ψ J˜ε (Ytε )
(e)E
Z
t
where
T
eνΛs s2H−1 |∇ϕε (Ysε )|2 ds + |∇ψε (Ysε )|2 dΛs ≤ CΘ2 (t, T )
M
(a) E
eνΛs s2H−1
≤ ε · CΘ2 (t, T )
≤ CΘ2 (t, T )
2
|Ysε − Jε (Ysε )|2 ds + Ysε − J˜ε (Ysε ) dΛs ≤ ε2 CΘ2 (t, T ).
-A
Θ2 (t, T ) = E T 2H−1 eνΛT (ϕ(ξ) + ψ(ξ))
+
Z
t
S
+
Z
t
T
eνΛs s2H−1 |ηs |2 + |Ysε |2 + |Zsε |2 + |f (s, 0, 0, 0)|2 ds
T
eνΛs s2H−1 |Ysε |2 + |g(s, ηs , 0)|2 dΛs .
B
JP
Proof. In the proof below we will use similar arguments as in the proof
´ and R˘
of Proposition 2.2 in Pardoux, E.
a¸scanu, A. [1998] and in the proof
of Proposition 11 in Maticiuc, L. and R˘
a¸scanu, A. [2010]. Since ∇ϕε (Yrε ) ∈
ε
∂ϕ(Jε (Yr )),
∇ϕε (Yrε ) · (Ysε − Yrε ) ≤ ϕε (Ysε ) − ϕε (Yrε ).
13
GBSVI driven by fBm
Now
eνΛr ϕε (Ysε ) − eνΛr ϕε (Yrε ) + eνΛs ϕε (Ysε ) − eνΛs ϕε (Ysε )
≥ eνΛr ∇ϕε (Yrε ) · (Ysε − Yrε )
cr
ip
t
and
eνΛs ϕε (Ysε ) ≥ eνΛr ϕε (Yrε ) + (eνΛs − eνΛr )ϕε (Ysε ) + eνΛr ∇ϕε (Yrε ) · (Ysε − Yrε ) .
an
us
Take s > r ≥ 0. Multiplying the above inequality by s2H−1 , using the fact
that eνΛr ϕε (Yrε ) ≥ 0 we get
s2H−1 eνΛs ϕε (Ysε ) ≥ r 2H−1 eνΛr ϕε (Yrε ) + s2H−1 (eνΛs − eνΛr )ϕε (Ysε )
+ s2H−1 eνΛr ∇ϕε (Yrε ) · (Ysε − Yrε ) .
M
Take s = ti+1 ∧ T , r = ti ∧ T , where 0 = t0 < t1 < . . . < t ∧ T and
ti+1 − ti = 1/n. Summing up over i and passing to the limit as n → ∞, we
deduce
Z
cc
ep
te
d
T 2H−1 eνΛT ϕε (YTε ) ≥ t2H−1 eνΛt ϕε (Ytε ) +
+
Z
t
T
t
T
νs2H−1 eνΛs ϕε (Ysε )dΛs
s2H−1 eνΛs ∇ϕε (Ysε )dYsε .
We have similar inequalities for function ψε . Summing these two inequalities
we get,
t2H−1 eνΛt (ϕε (Ytε ) + ψε (Ytε )) + ν
-A
≤ T 2H−1 eνΛT (ϕε (ξ) + ψε (ξ)) −
≤ T 2H−1 eνΛT(ϕε (ξ) + ψε (ξ))
+
S
B
JP
Z
T
Z
T
t
−
t
T
t
s2H−1 eνΛs (ϕε (Ysε ) + ψε (Ysε )) dΛs
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) dYsε
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) f (s, ηs , Ysε , Zsε )ds
t
−
Z
T
T
Z
t
+
Z
Z
t
T
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) g(s, ηs , Ysε )dΛs
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) Zsε dBsH
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) (∇ϕε (Ysε )ds + ∇ψε (Ysε )dΛs ) .
14
D. Borkowski, K. Ja´
nczak-Borkowska
Therefore,
+
T
Z
t
+
T
Z
t
Z
t
T
s2H−1 eνΛs (ϕε (Ysε ) + ψε (Ysε )) dΛs
s2H−1 eνΛs |∇ϕε (Ysε )|2 ds + |∇ψε (Ysε )|2 dΛs
s2H−1 eνΛs ∇ϕε (Ysε )∇ψε (Ysε ) (ds + dΛs )
T
Z
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε ))f (s, ηs , Ysε , Zsε )ds
t
T
Z
t
−
Z
T
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) Zsε dBsH .
cc
ep
te
d
t
s2H−1 eνΛs (∇ϕε (Ysε ) + ∇ψε (Ysε )) g(s, ηs , Ysε )dΛs
M
+
+
an
us
≤ T 2H−1 eνΛT (ϕε (ξ) + ψε (ξ))
cr
ip
t
t2H−1 eνΛt (ϕε (Ytε ) + ψε (Ytε )) + ν
Note that,
s2H−1 ∇ϕε (y)f (s, η, y, z) ≤ s2H−1 |∇ϕε (y)| (L(|η| + |y| + |z|) + |f (s, 0, 0, 0)|)
1
≤ s2H−1 |∇ϕε (y)|2 + 4L2 s2H−1 (|η|2 + |y|2 + |z|2 )
4
+ 4s2H−1 |f (s, 0, 0, 0)|2
-A
s2H−1 ∇ψε (y)f (s, η, y, z) ≤ s2H−1 (∇ϕε (y) · f (s, η, y, z))+
S
s2H−1 ∇ψε (y)g(s, η, y) ≤ s2H−1 |∇ψε (y)|(L|y| + |g(s, η, 0)|)
1
≤ s2H−1 |∇ψε (y)|2
4
+ 2s2H−1 (L2 |y|2 + |g(s, η, 0)|2 )
B
JP
s2H−1 ∇ϕε (y)g(s, η, y) ≤ s2H−1 (∇ψε (y) · g(s, η, y))+ .
Moreover using the fact that ∇ϕε (y)·∇ψε (y) ≥ 0, ϕε (ξ) ≤ ϕ(ξ) and ψε (ξ) ≤
15
GBSVI driven by fBm
ψ(ξ) we get,
e
1
+ E
2
+E
Z
Z
T
t
T
t
(ϕε (Ytε ) +
ψε (Ytε ))
Z
+ νE
T
s2H−1 eνΛs (ϕε (Ysε ) + ψε (Ysε )) dΛs
t
s2H−1 eνΛs |∇ϕε (Ysε )|2 ds + |∇ψε (Ysε )|2 dΛs
s2H−1 eνΛs ∇ϕε (Ysε )∇ψε (Ysε ) (ds + dΛs )
≤ ET 2H−1 eνΛT (ϕ(ξ) + ψ(ξ)) + 8L2 E
Z
t
T
s
2H−1 νΛs
e
2
|f (s, 0, 0, 0)| ds
t
+ 4E
Z
t
T
s2H−1 eνΛs |ηs |2 +|Ysε |2 +|Zsε |2 ds
an
us
+ 8E
Z
T
cr
ip
t
Et
2H−1 νΛt
s2H−1 eνΛs L2 |Ysε |2 + |g(s, ηs , 0|2 dΛs = CΘ2 (t, T ).
cc
ep
te
d
M
From the above inequality (a) is clear. Conditions (b) and (d) follow additionally from inequalities ϕ(Jε (y)) ≤ ϕε (y) and ψ(J˜ε (y)) ≤ ψε (y). From
|y − Jε (y)|2 ≤ 2εϕε (y) and |y − J˜ε (y)|2 ≤ 2εψε (y) follows (c). Finally (e) we
get from y − Jε (y) = ε∇ϕε (y) and y − J˜ε (y) = ε∇ψε (y).
Proposition 5.3. Let assumptions (H1 ) − (H4 ) be satisfied. Then (Y ε , Z ε )
is a Cauchy sequence, i.e. for ε, δ > 0
E eνΛt t2H−1 |Ytε − Ytδ |2 +
-A
+
Z
T
T
Z
t
eνΛs s2H−2 |Ysε − Ysδ |2 (ds + dΛs )
νΛs 2(2H−1)
e
s
t
|Zsε
−
Zsδ |2 ds
≤ C · (ε + δ) · Θ2 (t, T ).
B
JP
S
Proof. Put Yˇ = Y ε − Y δ and Zˇ = Z ε − Z δ . We have
t2H−1 eνΛt Yˇt2 +
Z
t
T
(2H − 1)s2H−2 eνΛs Yˇs2 ds + ν
= T 2H−1 eνΛT YˇT2 − 2
Z
t
T
Z
T
s2H−1 eνΛs Yˇs2 dΛs
t
s2H−1 eνΛs Yˇs dYˇs − 2
Z
t
T
s2H−1 eνΛs
σ
ˆ (s) ˇ 2
Z ds.
σ(s) s
16
D. Borkowski, K. Ja´
nczak-Borkowska
Therefore
+ν
Z
T
Z
T
t
− 2E
Z
T
t
− 2E
t
T
s
(2H − 1)s2H−2 eνΛs Yˇs2 ds +
2H−1 νΛs
e
t
Yˇs2 dΛs
Z
t
Z
T
t
s2(2H−1) eνΛs Zˇs2 ds
T
s2H−1 eνΛs Yˇs g(s, ηs , Ysε ) − g(s, ηs , Ysδ ) dΛs
s2H−1 eνΛs Yˇs ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ) ds
s2H−1 eνΛs Yˇs ∇ψε (Ysε ) − ∇ψδ (Ysδ ) dΛs .
ε
ε
δ
δ
M
Note that
2
M
s2H−1 eνΛs Yˇs f (s, ηs , Ysε , Zsε ) − f (s, ηs , Ysδ , Zsδ ) ds
t
+ 2E
T
an
us
≤ 2E
Z
Z
cr
ip
t
E t2H−1 eνΛt Yˇt2 +
cc
ep
te
d
2ˇ
y · f (s, η, y , z ) − f (s, η, y , z ) ≤
L2 M
2L + 2H−1
s
!
yˇ2 +
1 2H−1 2
s
zˇ ,
M
2ˇ
y · g(s, η, y ε ) − g(s, η, y δ ) ≤ 2Lˇ
y2 .
Moreover, by the definition of ϕε we get
0 ≤ ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ) · Jε (Ysε ) − Jδ (Ysδ )
= ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ) · Ysε − Ysδ − ε∇ϕε (Ysε ) + δ∇ϕδ (Ysδ )
-A
= ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ) · Ysε − Ysδ − ε|∇ϕε (Ysε )|2 − δ|∇ϕδ (Ysδ )|2
+ (ε + δ)∇ϕε (Ysε ) · ∇ϕδ (Ysδ )
S
and then
B
JP
∇ϕε (Ysε ) − ∇ϕδ (Ysδ ) · Ysε − Ysδ ≥ −(ε + δ)∇ϕε (Ysε ) · ∇ϕδ (Ysδ ).
Similarly,
∇ψε (Ysε ) − ∇ψδ (Ysδ ) · Ysε − Ysδ ≥ −(ε + δ)∇ψε (Ysε ) · ∇ψδ (Ysδ ).
17
GBSVI driven by fBm
Therefore, since ν ≥ 2L + 2,
≤E
Z
Z
T
s
T
t
(2H − 1)s2H−2 eνΛs Yˇs2 ds
Zˇs2 ds +
2(2H−1) νΛs
e
t
T
t
L2 M
2L + 2H−1
s
+ 2(ε + δ)E
Z
T
t
+ 2(ε + δ)E
Z
t
T
!
e
Z
T
t
e
t
Yˇs2 dΛs
s2H−1 eνΛs ∇ϕε (Ysε ) ∇ϕδ (Ysδ )ds
s2H−1 eνΛs ∇ψε (Ysε ) ∇ψδ (Ysδ )dΛs .
Yˇt2 ≤ C(ε + δ)E
+ C(ε + δ)E
s
2H−1 νΛs
Z
T
s2H−1 eνΛs ∇ϕε (Ysε ) ∇ϕδ (Ysδ )ds
t
M
Et
T
s2H−1 eνΛs Yˇs2 ds
By the Gronwall lemma
2H−1 νΛt
Z
cr
ip
t
1
+
M
Z
an
us
E t2H−1 eνΛt Yˇt2 +
s2H−1 eνΛs ∇ψε (Ysε ) ∇ψδ (Ysδ )dΛs .
cc
ep
te
d
By the simple inequality ab ≤ a2 /2 + b2 /2 and by Proposition 5.2 a) we get
the result.
Now we can give a proof of Theorem 3.3.
Proof of Theorem 3.3. First, we show the uniqueness. From the proof of
Proposition 4.2 it follows that for (Y, Z, U, V ) and (Y ′ , Z ′ , U ′ , V ′ ) being two
solutions of (3.2), we have
νΛt
-A
E e
|Yt − Yt′ |2 +
Z
t
T
νΛs 2H−1
e
s
|Zs − Zs′ |2 ds +
Z
t
T
νΛs
e
|Ys − Ys′ |2 dΛs
= 0,
and
Z
T
S
E
t
eνΛs (Ys − Ys′ )(Us − Us′ )ds +
Z
t
T
eνΛs (Ys − Ys′ )(Vs − Vs′ )dΛs
≤ 0,
B
JP
which means that the solution is unique.
Now we will show that the limit of (Y ε , Z ε , ∇ϕε (Y ε ), ∇ψε (Y ε )) converges
to a solution of (3.2).
Since by Proposition 5.3 (Y ε , Z ε ) is a Cauchy sequence, there exists its
1/2
1/2,Λ
H
limit, i.e. there exists a pair of processes (Y, Z) ∈ V˜[0,T ] ∩ V˜[0,T ] × V˜[0,T
] such
18
D. Borkowski, K. Ja´
nczak-Borkowska
that
εց0
T
t
eνΛs s2H−1 |Ysε − Ys |2 (ds + dΛs )
+
Z
T
t
eνΛs s2(2H−1) |Zsε − Zs |2 ds = 0.
cr
ip
t
lim E
Z
From Proposition 5.2 c),
νΛt 2H−1
lim Ee
t
εց0
|Ytε
−
Jε (Ytε )|2
ε
ε 2
˜
+ Yt − Jε (Yt ) = 0,
T
E
0
an
us
1/2
1/2,Λ
and we have limεց0 Jε (Y ε ) = Y in V˜[0,T ] and limεց0 J˜ε (Y ε ) = Y in V˜[0,T ] .
Denoting U ε = ∇ϕε (Y ε ) and V ε = ∇ψε (Y ε ) from Proposition 5.2 a) we
obtain
Z
eνΛs s2H−1 |Usε |2 ds + |Vsε |2 dΛs ≤ C.
M
Hence there exist a subsequence εn ց 0 and processes U , V such that
1/2
U εn → U weakly in V˜[0,T ]
1/2,Λ
V εn → V weakly in V˜[0,T ]
cc
ep
te
d
and
and from the Fatou Lemma
E
Z
0
T
eνΛs s2H−1 |Us |2 ds + |Vs |2 dΛs ≤ C.
Passing now with ε to 0 in (5.6) we obtain (3.2).
1/2
Moreover since Utε ∈ ∂ϕ(Jε (Ytε )) and Vtε ∈ ∂ψ(J˜ε (Ytε )), for all u ∈ V˜[0,T ]
1/2,Λ
and v ∈ V˜
we have
[0,T ]
-A
Utε · (ut − Jε (Ytε )) + ϕ(Jε (Ytε )) ≤ ϕ(ut )
and
S
Vtε · vt − J˜ε (Ytε ) + ψ(J˜ε (Ytε )) ≤ ψ(vt ).
B
JP
Therefore we can deduce (passing to limes infimum) that
Ut · (ut − Yt ) + ϕ(Yt ) ≤ ϕ(ut ) and
Vt · (vt − Yt ) + ψ(Yt ) ≤ ψ(vt ),
which mean that (Yt , Ut ) ∈ ∂ϕ and (Yt , Vt ) ∈ ∂ψ, t ∈ [0, T ]. That completes
the proof.
19
GBSVI driven by fBm
Acknowledgement
This research was supported by the National Science Centre (Poland) under
decision number DEC-2012/07/D/ST6/02534.
cr
ip
t
References
B
JP
S
-A
cc
ep
te
d
M
an
us
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Department of Mathematics
and Computer Science
Nicolaus Copernicus University
Toru´
n, Poland
E-mail: [email protected]
Institute of Mathematics and Physics
University of Technology
and Life Sciences
Bydgoszcz, Poland
E-mail: [email protected]