1. No category

Asymptotic Results for Sample ACF and Extremes
of Generalized Ornstein-Uhlenbeck Processes
Vicky Fasen
[email protected]
Graduate Program ”Applied Algorithmic Mathematics”
Munich University of Technology
www-m4.ma.tum.de/pers/fasen/
Overview
• Generalized Ornstein-Uhlenbeck (genOU) process
• Tail behavior of a genOU process
• Extremal behavior of a genOU process
• Convergence of the sample ACF of a genOU
process
• Asymptotic behavior of a log-price process
GenOU process
A genOU (generalized Ornstein-Uhlenbeck) process is
defined by
Z t
Vt = e−ξt
eξs− dηs + V0
for t > 0
0
where
• (ξt , ηt )t≥0 is a bivariate Lévy process
• independent of the starting random variable V0
Examples for Vt = e−ξt
R
t ξs−
e
0
dηs + V0
• Ornstein-Uhlenbeck process
Z t
Vt = e−λt
eλs dηs + V0
0
Examples for Vt = e−ξt
R
t ξs−
e
0
dηs + V0
• Ornstein-Uhlenbeck process
Z t
Vt = e−λt
eλs dηs + V0
0
• Volatility process of the COGARCH(1,1) process
Z t
Vt = e−ξt β
eξs− ds + V0
0
where
ξt = ct −
X
log(1 + λec (∆Ls )2 )
0<s≤t
c > 0, λ ≥ 0, β > 0 and (Lt )t≥0 is a Lévy process
0
50
L
100
150
Volatility process of the COGARCH(1,1) model
2000
4000
6000
8000
10000
0
2000
4000
6000
8000
10000
10
6
8
sigma
12
14
0
L(t) =
P N (t)
k=1
Zk where Z1 ∼ N (0, 1), EN (1) = 1 and corresponding sample path of the volatility
process of the COGARCH process with parameters β = 1, λ = 0.04 and c = 0.062 (bottom)
Assumptions for the genOU process
• (ξ, η) is a bivariate Lévy process of finite variation
• η is a subordinator
• The drift of ξ is non-zero, or there is no r > 0 such
that the support of Πξ is concentrated on rZ
• (Vt )t≥0 is not degenerate to a constant process
Assumptions for the genOU process
• (ξ, η) is a bivariate Lévy process of finite variation
• η is a subordinator
• The drift of ξ is non-zero, or there is no r > 0 such
that the support of Πξ is concentrated on rZ
• (Vt )t≥0 is not degenerate to a constant process
• For some α > 0:
Ee−αξ1 = 1
Assumptions for the genOU process
• (ξ, η) is a bivariate Lévy process of finite variation
• η is a subordinator
• The drift of ξ is non-zero, or there is no r > 0 such
that the support of Πξ is concentrated on rZ
• (Vt )t≥0 is not degenerate to a constant process
• For some α > 0:
Ee−αξ1 = 1
• For some p, q > 1 with 1/p + 1/q = 1 and d > α:
E|η1 |q max{1,d} < ∞,
Ee−p max{1,d}ξ1 < ∞
Tail behavior of V0
V0 denotes the stationary distribution of V
Reference: Lindner and Maller (2005)
Tail behavior of V0
V0 denotes the stationary distribution of V
lim xα P(V0 > x) = C
x→∞
where C > 0.
Reference: Lindner and Maller (2005)
Tail behavior of V0
V0 denotes the stationary distribution of V
lim xα P(V0 > x) = C
x→∞
where C > 0. In the following
an = C 1/α n1/α
such that
lim nP(V0 > an ) = 1
n→∞
Reference: Lindner and Maller (2005)
Extremal behavior
Discrete time skeleton
Let h > 0 and
Hk =
sup
(k−1)h≤s≤kh
Vs
for k ∈ N
Discrete time skeleton
Let h > 0 and
Hk =
sup
Vs
(k−1)h≤s≤kh
z
z
0
sup Vt = max Hk
k=1,2,3
0≤t≤3h
}|
H1
H2
H3
}| {z }| {z }|
h
2h
{
{
3h
for k ∈ N
Discrete time skeleton
Let h > 0 and
Hk =
sup
Vs
for k ∈ N
(k−1)h≤s≤kh
z
z
0
H1
}|
{z
h
sup Vt = max Hk
k=1,...,l
0≤t≤lh
}|
H2
H3
}| {z }| {
2h
3h
{
Discrete time skeleton
Let h > 0 and
Hk =
sup
for k ∈ N
Vs
(k−1)h≤s≤kh
Then
lim nP(H1 > an ) = E
n→∞
z
z
0
H1
}|
{z
h
sup e−αξt
0≤t≤h
sup Vt = max Hk
k=1,...,l
0≤t≤lh
}|
H2
H3
}| {z }| {
2h
3h
{
Discrete time skeleton
Let h > 0 and
Hk =
sup
for k ∈ N
Vs
(k−1)h≤s≤kh
Then
lim nP(H1 > an ) = E
n→∞
sup e−αξt
0≤t≤h
sup e−ξs V0 ≤ H1 ≤ sup e−ξs V0 + sup
0≤s≤h
0≤s≤h
0≤s≤h
Z
0
s
e−(ξt −ξs− ) dηs
Discrete time skeleton
Let
H(l) = (H1 , . . . , Hl ) =
sup Vs , . . . ,
0≤s≤h
sup
(l−1)h≤s≤lh
Vs
!
Multivariate regular variation of H(l)
Let
H(l) = (H1 , . . . , Hl ) =
sup Vs , . . . ,
0≤s≤h
sup
(l−1)h≤s≤lh
Vs
!
Then
H(l)
(l)
P(|H | > ux, |H(l) | ∈ ·)
n→∞
=⇒ x
P(|H(l) | > u)

−α
E  sup e−αξt 1
0≤t≤lh


sup
e
!
−ξs
(k−1)h≤s≤kh
sup e−ξs

0≤s≤lh
E sup e−αξs
0≤s≤lh
k=1,...,m


∈· 

Multivariate regular variation of H(l)
Let
H(l) = (H1 , . . . , Hl ) =
sup Vs , . . . ,
0≤s≤h
sup
Vs
(l−1)h≤s≤lh
!
Then
H(l)
(l)
P(|H | > ux, |H(l) | ∈ ·)
n→∞
=⇒ x
P(|H(l) | > u)

−α
E  sup e−αξt 1
0≤t≤lh


sup
e
!
−ξs
(k−1)h≤s≤kh
sup e−ξs

0≤s≤lh
E sup e−αξs
0≤s≤lh
k=1,...,m


∈· 

(Vt )0≤t≤1 is regularly varying (Hult and Lindkøg (2005))
Point process
Nn =
∞
X
ε(k/n,a−1
n Xk )
k=1
30
Nn([s,t)×(x,∞)) =3
25
Xk
20
anx
15
10
5
0
20
40
60
ns
80
100
120
nt
140
160
180
200
Point process behavior
Nn =
∞
X
n→∞
ε(k/n,a−1
=⇒
n Hk )
where
ε(sk ,Qkj Pk )
k=1 j=0
k=1
P∞
∞ X
∞
X
k=1 ε(sk ,Pk )
is PRM(ϑ) with
+
ϑ(dt×dx)=dt×E sup e−αξs −sup e−αξs αx−α−11(0,∞) (x) dx
0≤s≤h
P∞
s≤h
and j=0 εQkj for k ∈ N are i. i. d point processes on
[0, 1] with support in 1
Running maxima
Let M (n) = sup0≤t≤n Vt = maxk=1,...,n Hk and
h = 1. Then
lim P(a−1
n M (n) ≤ x)
T →∞
= lim P(Nn ((0, 1] × (x, ∞)) = 0)
n→∞
= P(N ((0, 1] × (x, ∞)) = 0)
= exp(−θx−α )
for x > 0
where
θ=E
sup e−αξt − sup e−αξt
0≤t≤1
t≥1
+
Asymptotic behavior of the
sample autocovariance function
Stochastic recurrence equation
Let h > 0. Then (Vnh )n∈N is the solution of a stochastic
recurrence equation:
Z nh
Anh = e−(ξnh −ξ(n−1)h ) and Bnh =
e−(ξnh −ξs− ) dηs
(n−1)h
for n ∈ N. Then
Vnh = Anh V(n−1)h + Bnh
for n ∈ N
Point process behavior
∞
X
n→∞
ε(k/n,a−2
=⇒
n Vkh V(k+l)h )
ε(s(l) ,Q(l) P (l) )
k=1 j=0
k=1
where
∞ X
∞
X
P∞
k
kj
k
(l)
ε
) with
(l)
(l) is PRM(ϑ
k=1 (s ,P )
k
k
α −α/2−1
ϑ (dt × dx) = dt × θl x
1(0,∞) (x) dx,
2
P∞
θl > 0 and j=0 εQ(l) for k ∈ N are i. i. d point processes
(l)
kj
on [0, 1] with support in 1
Convergence of the sample ACF
α ∈ (0, 2):
Let h > 0. We define the sample autocovariance
function
γn,V (lh) = n−1
n−l
X
k=1
Vkh V(k+l)h
for l ≥ 0
Convergence of the sample ACF
α ∈ (0, 2):
Let h > 0. We define the sample autocovariance
function
γn,V (lh) = n−1
n−l
X
Vkh V(k+l)h
for l ≥ 0
k=1
Then
n1−2/α γn,V (lh)
n→∞
l=0,...,m
=⇒ (Wl )l=0,...,m
where (W0 , . . . , Wm ) is jointly α/2-stable in Rm+1
Convergence of the sample ACF
α ∈ (2, 4):
n−l
Let h > 0.
X
γn,V (lh) = n−1
Vkh V(k+l)h
for l ≥ 0
k=1
Suppose for l = 0, . . . , m,
!
n−l
X
lim lim sup Var n−2/α
Vkh V(k+l)h 1{|Vkh V(k+l)h |≤n2/α } = 0
↓0
n→∞
k=1
Convergence of the sample ACF
α ∈ (2, 4):
n−l
Let h > 0.
X
γn,V (lh) = n−1
Vkh V(k+l)h
for l ≥ 0
k=1
Suppose for l = 0, . . . , m,
!
n−l
X
lim lim sup Var n−2/α
Vkh V(k+l)h 1{|Vkh V(k+l)h |≤n2/α } = 0
↓0
n→∞
k=1
Then
n1−2/α (γn,V (lh) − γV (lh))
n→∞
l=0,...,m
=⇒ (Wl )l=0,...,m
where (W0 , . . . , Wm ) is jointly α/2-stable in Rm+1
Convergence of the sample ACF
α > 4:
Let h > 0.
γn,V (lh) = n−1
n−l
X
Vkh V(k+l)h
for l ≥ 0,
k=1
Then
n1/2 (γn,V (lh) − γV (lh))
n→∞
l=0,...,m
=⇒ (Wl )l=0,...,m
where (W1 , . . . , Wm ) is multivariate normal with mean
zero, covariance matrix
!
∞
X
Cov(V0 Vih , Vkh V(k+j)h )
and W0 = EV02
k=−∞
i,j=1,...,m
Asymptotic behavior of a
log-price process
Log-price process
The log-price process G = (Gt )t≥0 is defined as
Z tp
Gt =
Vs− dLs for t ≥ 0
0
where
• (ξ, η, L) is a three-dimensional Lévy process
• L is of finite variation
• (−Lt )t≥0 is not a subordinator
• E|L1 |q max{1,4d} < ∞
Point process behavior
∞
X
n→∞
ε(k/n,a−1/2
=⇒
(G
−G
))
n
k
k−1
ε(sk ,Qkj Pk )
k=1 j=0
k=1
where
∞ X
∞
X
P∞
k=1 ε(sk ,Pk )
is PRM(ϑ) with
−2α−1
e
ϑ(dt × dx)=dt × θ2αx
1(0,∞) (x) dx,
P∞
e
θ > 0 and j=0 εQkj for k ∈ N are i. i. d point processes
on [0, 1] with support in 1
Asymptotic behavior of the price process
0 < α < 1/2:
t
−1/(2α)
where S is 2α-stable
t→∞
Gt =⇒ S
Asymptotic behavior of the price process
0 < α < 1/2:
t
−1/(2α)
t→∞
Gt =⇒ S
where S is 2α-stable
α > 1:
t
−1/2
t→∞
(Gt − EGt ) =⇒ N
where N is normal distributed
Asymptotic behavior of the price process
1/2 ≤ α < 1:
Let
(,n)
e
Gk
=
Z
k
k−1
p
Vs− dLs 1{n−1/(2α) R k √V
k−1
dL
s−
s ≤}
Suppose for > 0, n, k ∈ N, and for all δ > 0:
" n
#
!
X
−1/(2α)
(,n)
(,n) e
e
Gk − nEG1
lim lim sup P n
>δ =0
↓0 n→∞
k=1
Asymptotic behavior of the price process
1/2 ≤ α < 1:
Let
(,n)
e
Gk
=
Z
k
k−1
p
Vs− dLs 1{n−1/(2α) R k √V
k−1
dL
s−
s ≤}
Suppose for > 0, n, k ∈ N, and for all δ > 0:
" n
#
!
X
−1/(2α)
(,n)
(,n) e
e
Gk − nEG1
lim lim sup P n
>δ =0
↓0 n→∞
k=1
Then
t
−1/(2α)
t→∞
(Gt − tEG1 ) =⇒ S
where S is 2α-stable
COGARCH(1,1) process
The COGARCH(1,1) process is defined as
Z tp
Gt =
Vs− dLs for t ≥ 0
0
where
Z t
eξs− ds + V0
Vt = e−ξt β
0
X
ξt = ct −
log(1 + λec (∆Ls )2 )
0<s≤t
COGARCH(1,1) process
The COGARCH(1,1) process is defined as
Z tp
Gt =
Vs− dLs for t ≥ 0
0
where
Z t
eξs− ds + V0
Vt = e−ξt β
0
X
ξt = ct −
log(1 + λec (∆Ls )2 )
0<s≤t
Suppose L is symmetric and α ∈ (0, 1). Then
t
−1/(2α)
where S is 2α-stable
t→∞
Gt =⇒ S
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with applications to ARCH. Ann. Statist. 26 (5), 2049–2080.
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¨
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, C., AND L INDNER , A. (2006). Extremal behavior of stochastic
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¨
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