Multivariate GeneralizedOrnstein-Uhlenbeck Processes
Anita BehmeTU Munchen
Alexander LindnerTU Braunschweig
7th International Conference on Levy Processes:Theory and Applications
Wroclaw, July 15–19, 2013
The Ornstein-Uhlenbeck process : Origin
The Ornstein-Uhlenbeck process: Origin
Einstein (1905) models the movement of a free particle in fluid byBrownian Motion.
Alexander Lindner, 2
The Ornstein-Uhlenbeck process : Origin
The Ornstein-Uhlenbeck process: Origin
Einstein (1905) models the movement of a free particle in fluid byBrownian Motion.
Ornstein and Uhlenbeck (1930) add the concept of friction toEinsteins model.
Alexander Lindner, 2
The Ornstein-Uhlenbeck process : Origin
The Ornstein-Uhlenbeck process: Origin
Einstein (1905) models the movement of a free particle in fluid byBrownian Motion.
Ornstein and Uhlenbeck (1930) add the concept of friction toEinsteins model:A particle moving from left to right gets hit by more particles fromthe right than from the left side which results in a slowdown.The velocity v(t) of the particle is given by
mdv(t) = −λv(t)dt + dB(t)
i.e.
v(t) = e−λt/mv(0) + e−λt/m∫
(0,t]eλs/mdB(s).
Alexander Lindner, 2
The Ornstein-Uhlenbeck process : Origin
The Ornstein-Uhlenbeck process: Origin
Einstein (1905) models the movement of a free particle in fluid byBrownian Motion.
Ornstein and Uhlenbeck (1930) add the concept of friction toEinsteins model:A particle moving from left to right gets hit by more particles fromthe right than from the left side which results in a slowdown.The velocity v(t) of the particle is given by
mdv(t) = −λv(t)dt + dB(t)
i.e.
v(t) = e−λt/mv(0) + e−λt/m∫
(0,t]eλs/mdB(s).
This solution is called an Ornstein-Uhlenbeck (OU) process.Setting λ = 0 yields the original formula by Einstein.
Alexander Lindner, 2
The Ornstein-Uhlenbeck process : From OU to GOU processes
OU processes as AR(1) time series
For every h > 0 the Ornstein-Uhlenbeck process
Vt = e−λtV0 + e−λt∫
(0,t]eλsdBs , t ≥ 0,
fulfills the random recurrence equation
Vnh = e−λhV(n−1)h + e−λnh∫
((n−1)h,nh]eλsdBs , n ∈ N.
Hence it can be seen as a natural generalization in continuous timeof the AR(1) time series
Xn = e−λXn−1 + Zn, n ∈ N,
with i.i.d. noise (Zn)n∈N such that L(Z1) = L(∫
(0,1] e−λ(1−s)dBs).
Alexander Lindner, 3
The Ornstein-Uhlenbeck process : From OU to GOU processes
A more general AR(1) time series
By embedding the more general random sequence
Yn = AnYn−1 + Bn, n ∈ N,
with (An,Bn)n∈N i.i.d., A1 > 0 a.s., into a continuous time settingin 1989 De Haan and Karandikar introduced the generalizedOrnstein-Uhlenbeck process
Vt = e−ξt
(V0 +
∫(0,t]
eξs−dηs
), t ≥ 0.
driven by a bivariate Levy process (ξt , ηt)t≥0 with starting randomvariable V0.
Alexander Lindner, 4
The Ornstein-Uhlenbeck process : From OU to GOU processes
Definition: Levy processesA Levy process in Rd on a probability space (Ω,F ,P) is astochastic process X = (Xt)t≥0, Xt : Ω→ Rd satisfying thefollowing properties:
I X0 = 0 a.s.
I X has independent increments, i.e. for all0 ≤ t0 ≤ t1 ≤ . . . ≤ tn the random variablesXt0 ,Xt1 − Xt0 , . . . ,Xtn − Xtn−1 are independent.
I X has stationary increments, i.e. for all s, t ≥ 0 it holds
Xs+t − Xsd= Xt .
I X has a.s. cadlag paths, i.e. for P-a.e. ω ∈ Ω the patht 7→ Xt(ω) is right-continuous in t ≥ 0 and has left limits int > 0.
Elementary examples of Levy processes include lineardeterministic processes, Brownian motions as well as compoundPoisson processes.
Alexander Lindner, 5
The Ornstein-Uhlenbeck process : From OU to GOU processes
Definition: Levy processesA Levy process in Rd on a probability space (Ω,F ,P) is astochastic process X = (Xt)t≥0, Xt : Ω→ Rd satisfying thefollowing properties:
I X0 = 0 a.s.
I X has independent increments, i.e. for all0 ≤ t0 ≤ t1 ≤ . . . ≤ tn the random variablesXt0 ,Xt1 − Xt0 , . . . ,Xtn − Xtn−1 are independent.
I X has stationary increments, i.e. for all s, t ≥ 0 it holds
Xs+t − Xsd= Xt .
I X has a.s. cadlag paths, i.e. for P-a.e. ω ∈ Ω the patht 7→ Xt(ω) is right-continuous in t ≥ 0 and has left limits int > 0.
Elementary examples of Levy processes include lineardeterministic processes, Brownian motions as well as compoundPoisson processes.
Alexander Lindner, 5
The Ornstein-Uhlenbeck process : The GOU process
Definition: Generalized OU processes
The generalized Ornstein-Uhlenbeck (GOU) process (Vt)t≥0
driven by the bivariate Levy process (ξt , ηt)t≥0 is given by
Vt = e−ξt
(V0 +
∫(0,t]
eξs−dηs
), t ≥ 0,
where V0 is a finite random variable, usually chosen independent of(ξ, η).
In the case that (ξt , ηt) = (λt, ηt) with a Levy process (ηt)t≥0 anda constant λ 6= 0 the process (Vt)t≥0 is called Levy-drivenOrnstein-Uhlenbeck process or Ornstein-Uhlenbeck typeprocess.Obviously, if additionally (ηt)t≥0 is a Brownian motion, we get theclassical Ornstein-Uhlenbeck process.
Alexander Lindner, 6
The Ornstein-Uhlenbeck process : The GOU process
The generalized Ornstein-Uhlenbeck process: Applications
Example 1: ξt = t deterministic
⇒ Vt = e−t
(V0 +
∫(0,t]
esdηs
)
Levy driven Ornstein-Uhlenbeck process, classical for ηt = Bt
I applications in storage theory
I stochastic volatility model of Barndorff-Nielsen and Shephard(2001): ηt subordinator, Vt squared volatility, price process Gt
defined by dGt = (µ+ bVt)dt +√VtdBt for constants µ, b.
Example 2: ηt = t deterministic.Applications for Asian options, or COGARCH(1,1) model ofKluppelberg, L., Maller (2004).
Alexander Lindner, 7
The Ornstein-Uhlenbeck process : The GOU process
The generalized Ornstein-Uhlenbeck process: Applications
Example 1: ξt = t deterministic
⇒ Vt = e−t
(V0 +
∫(0,t]
esdηs
)
Levy driven Ornstein-Uhlenbeck process, classical for ηt = Bt
I applications in storage theory
I stochastic volatility model of Barndorff-Nielsen and Shephard(2001): ηt subordinator, Vt squared volatility, price process Gt
defined by dGt = (µ+ bVt)dt +√VtdBt for constants µ, b.
Example 2: ηt = t deterministic.Applications for Asian options, or COGARCH(1,1) model ofKluppelberg, L., Maller (2004).
Alexander Lindner, 7
The Ornstein-Uhlenbeck process : The corresponding SDE
The corresponding SDEThe generalized Ornstein-Uhlenbeck process driven by (ξ, η)
Vt = e−ξt
(V0 +
∫(0,t]
eξs−dηs
), t ≥ 0
is the unique solution of the SDE
dVt = Vt−dUt + dLt , t ≥ 0
where (Ut , Lt)t≥0 is a bivariate Levy process completelydetermined by (ξ, η).
In particular we haveξt = − log(E(U)t)
Alexander Lindner, 8
The Ornstein-Uhlenbeck process : The corresponding SDE
The corresponding SDEThe generalized Ornstein-Uhlenbeck process driven by (ξ, η)
Vt = e−ξt
(V0 +
∫(0,t]
eξs−dηs
), t ≥ 0
is the unique solution of the SDE
dVt = Vt−dUt + dLt , t ≥ 0
where (Ut , Lt)t≥0 is a bivariate Levy process completelydetermined by (ξ, η).In particular we have
ξt = − log(E(U)t)
Definition: The Doleans-Dade Exponential
E(U)t = exp
(Ut −
1
2tσ2
U
)∏s≤t
((1 + ∆Us) exp(−∆Us))
is the unique solution of the SDE dZt = Zt−dUt , Z0 = 1 a.s.
Alexander Lindner, 8
The Ornstein-Uhlenbeck process : The corresponding SDE
The corresponding SDEThe generalized Ornstein-Uhlenbeck process driven by (ξ, η)
Vt = e−ξt
(V0 +
∫(0,t]
eξs−dηs
), t ≥ 0
is the unique solution of the SDE
dVt = Vt−dUt + dLt , t ≥ 0
where (Ut , Lt)t≥0 is a bivariate Levy process completelydetermined by (ξ, η).In particular we have
ξt = − log(E(U)t)
and
ηt = Lt +∑
0<s≤t
∆Us∆Ls1 + ∆Us
− tCov (BU1 ,BL1).
Alexander Lindner, 8
Multivariate generalized Ornstein-Uhlenbeck processes :
Multivariate GeneralizedOrnstein-Uhlenbeck Processes
Alexander Lindner, 9
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Recall: An AR(1) time series
The generalized Ornstein-Uhlenbeck process
Vt = e−ξt
(V0 +
∫(0,t]
eξs−dηs
), t ≥ 0.
driven by a bivariate Levy process (ξt , ηt)t≥0 with starting randomvariable V0 had been derived by embedding the AR(1) time series
Vn = AnVn−1 + Bn, n ∈ N,
with (An,Bn)n∈N i.i.d., A1 > 0 a.s., into a continuous time setting.
Alexander Lindner, 10
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Constructing a multivariate GOU
We aim to embed the random sequence
Vn = AnVn−1 + Bn, n ∈ N,
with (An,Bn)n∈N i.i.d., (An,Bn) ∈ Rm×m × Rm, A1 a.s.non-singular, into a continuous time setting.
More precisely, we want to find all stochastic processes (Vt)t≥0
such thatVt = As,tVs + Bs,t , ∀ 0 ≤ s ≤ t
and(A(n−1)h,nh,B(n−1)h,nh)n∈N
is i.i.d. for each h > 0.Assuming slightly more leads to the following requirements:
Alexander Lindner, 11
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Constructing a multivariate GOU
We aim to embed the random sequence
Vn = AnVn−1 + Bn, n ∈ N,
with (An,Bn)n∈N i.i.d., (An,Bn) ∈ Rm×m × Rm, A1 a.s.non-singular, into a continuous time setting.More precisely, we want to find all stochastic processes (Vt)t≥0
such thatVt = As,tVs + Bs,t , ∀ 0 ≤ s ≤ t
and(A(n−1)h,nh,B(n−1)h,nh)n∈N
is i.i.d. for each h > 0.
Assuming slightly more leads to the following requirements:
Alexander Lindner, 11
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Constructing a multivariate GOU
We aim to embed the random sequence
Vn = AnVn−1 + Bn, n ∈ N,
with (An,Bn)n∈N i.i.d., (An,Bn) ∈ Rm×m × Rm, A1 a.s.non-singular, into a continuous time setting.More precisely, we want to find all stochastic processes (Vt)t≥0
such thatVt = As,tVs + Bs,t , ∀ 0 ≤ s ≤ t
and(A(n−1)h,nh,B(n−1)h,nh)n∈N
is i.i.d. for each h > 0.Assuming slightly more leads to the following requirements:
Alexander Lindner, 11
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
AssumptionsFor each 0 ≤ s ≤ t let (As,t ,Bs,t) ∈ GL(R,m)× Rm s.t.:Assumption (a) For all 0 ≤ u ≤ s ≤ t almost surely
Au,t = As,tAu,s and Bu,t = As,tBu,s + Bs,t .
Assumption (b) The families of random matrices(As,t ,Bs,t), a ≤ s ≤ t ≤ b and (As,t ,Bs,t), c ≤ s ≤ t ≤ d areindependent for 0 ≤ a ≤ b ≤ c ≤ d .
Assumption (c) For all 0 ≤ s ≤ t it holds
(As,t ,Bs,t)d= (A0,t−s ,B0,t−s).
Assumption (d) It holds
P − limt↓0
A0,t = A0,0 = I and P − limt↓0
B0,t = B0,0 = 0,
where I denotes the identity matrix and 0 the vector (or matrix)only having zero entries.
Alexander Lindner, 12
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
AssumptionsFor each 0 ≤ s ≤ t let (As,t ,Bs,t) ∈ GL(R,m)× Rm s.t.:Assumption (a) For all 0 ≤ u ≤ s ≤ t almost surely
Au,t = As,tAu,s and Bu,t = As,tBu,s + Bs,t .
Assumption (b) The families of random matrices(As,t ,Bs,t), a ≤ s ≤ t ≤ b and (As,t ,Bs,t), c ≤ s ≤ t ≤ d areindependent for 0 ≤ a ≤ b ≤ c ≤ d .
Assumption (c) For all 0 ≤ s ≤ t it holds
(As,t ,Bs,t)d= (A0,t−s ,B0,t−s).
Assumption (d) It holds
P − limt↓0
A0,t = A0,0 = I and P − limt↓0
B0,t = B0,0 = 0,
where I denotes the identity matrix and 0 the vector (or matrix)only having zero entries.
Alexander Lindner, 12
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
AssumptionsFor each 0 ≤ s ≤ t let (As,t ,Bs,t) ∈ GL(R,m)× Rm s.t.:Assumption (a) For all 0 ≤ u ≤ s ≤ t almost surely
Au,t = As,tAu,s and Bu,t = As,tBu,s + Bs,t .
Assumption (b) The families of random matrices(As,t ,Bs,t), a ≤ s ≤ t ≤ b and (As,t ,Bs,t), c ≤ s ≤ t ≤ d areindependent for 0 ≤ a ≤ b ≤ c ≤ d .
Assumption (c) For all 0 ≤ s ≤ t it holds
(As,t ,Bs,t)d= (A0,t−s ,B0,t−s).
Assumption (d) It holds
P − limt↓0
A0,t = A0,0 = I and P − limt↓0
B0,t = B0,0 = 0,
where I denotes the identity matrix and 0 the vector (or matrix)only having zero entries.
Alexander Lindner, 12
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
AssumptionsFor each 0 ≤ s ≤ t let (As,t ,Bs,t) ∈ GL(R,m)× Rm s.t.:Assumption (a) For all 0 ≤ u ≤ s ≤ t almost surely
Au,t = As,tAu,s and Bu,t = As,tBu,s + Bs,t .
Assumption (b) The families of random matrices(As,t ,Bs,t), a ≤ s ≤ t ≤ b and (As,t ,Bs,t), c ≤ s ≤ t ≤ d areindependent for 0 ≤ a ≤ b ≤ c ≤ d .
Assumption (c) For all 0 ≤ s ≤ t it holds
(As,t ,Bs,t)d= (A0,t−s ,B0,t−s).
Assumption (d) It holds
P − limt↓0
A0,t = A0,0 = I and P − limt↓0
B0,t = B0,0 = 0,
where I denotes the identity matrix and 0 the vector (or matrix)only having zero entries.
Alexander Lindner, 12
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
The Process At := A0,t
Lemma: Every stochastic process At := A0,t in the autoregressivemodel above which fulfills Assumptions (a) to (d) has a versionwhich is a multiplicative right Levy process in the general lineargroup GL(R,m) of order m.
That means, (At)t≥0 is a stochastic process with values inGL(R,m) with the following properties:
I A0 = I a.s.
I it has independent left increments, i.e. for all0 ≤ t1 ≤ . . . ≤ tn, the random variablesA0,At1A
−10 , . . . ,AtnA
−1tn−1
are independent.
I it has stationary left increments, i.e. for all s, t ≥ 0 it holds
As+tA−1s
d= At .
I it has a.s. cadlag paths, i.e. for P-a.e. ω ∈ Ω the patht 7→ At(ω) is right-continuous in t ≥ 0 and has left limits int > 0.
Alexander Lindner, 13
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
The Process At := A0,t
Lemma: Every stochastic process At := A0,t in the autoregressivemodel above which fulfills Assumptions (a) to (d) has a versionwhich is a multiplicative right Levy process in the general lineargroup GL(R,m) of order m.
That means, (At)t≥0 is a stochastic process with values inGL(R,m) with the following properties:
I A0 = I a.s.
I it has independent left increments, i.e. for all0 ≤ t1 ≤ . . . ≤ tn, the random variablesA0,At1A
−10 , . . . ,AtnA
−1tn−1
are independent.
I it has stationary left increments, i.e. for all s, t ≥ 0 it holds
As+tA−1s
d= At .
I it has a.s. cadlag paths, i.e. for P-a.e. ω ∈ Ω the patht 7→ At(ω) is right-continuous in t ≥ 0 and has left limits int > 0.
Alexander Lindner, 13
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
The Multivariate Stochastic Exponential ILemma: By an observation due to Skorokhod every right Levyprocess in (GL(R,m), ·) is the right stochastic exponential of aLevy process in (Rm×m,+).
Definition: Let (Xt)t≥0 be a semimartingale in (Rm×m,+). Then
its left stochastic exponential←E (X )t is defined as the unique
Rm×m-valued, adapted, cadlag solution of the SDE
Zt = I +
∫(0,t]
Zs−dXs , t ≥ 0,
while the unique adapted, cadlag solution of the SDE
Zt = I +
∫(0,t]
dXs Zs−, t ≥ 0,
will be called right stochastic exponential and denoted by→E (X )t .
Alexander Lindner, 14
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
The Multivariate Stochastic Exponential ILemma: By an observation due to Skorokhod every right Levyprocess in (GL(R,m), ·) is the right stochastic exponential of aLevy process in (Rm×m,+).
Definition: Let (Xt)t≥0 be a semimartingale in (Rm×m,+). Then
its left stochastic exponential←E (X )t is defined as the unique
Rm×m-valued, adapted, cadlag solution of the SDE
Zt = I +
∫(0,t]
Zs−dXs , t ≥ 0,
while the unique adapted, cadlag solution of the SDE
Zt = I +
∫(0,t]
dXs Zs−, t ≥ 0,
will be called right stochastic exponential and denoted by→E (X )t .
Alexander Lindner, 14
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
The Multivariate Stochastic Exponential II
Let (Xt)t≥0 be a semimartingale in (Rm×m,+). Then we observe:
I A stochastic exponential of X is invertible for all t ≥ 0 if andonly if
det(I + ∆Xt) 6= 0 for all t ≥ 0. (∗)
I Suppose (Xt)t≥0 fulfills (∗). Then for (Ut)t≥0 given by
Ut := −Xt +[X ,X ]ct +∑
0<s≤t
((I + ∆Xs)−1 − I + ∆Xs
), t ≥ 0
it holds
[←E (X )t ]
−1 =→E (U)t , t ≥ 0.
Alexander Lindner, 15
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
The Multivariate Stochastic Exponential II
Let (Xt)t≥0 be a semimartingale in (Rm×m,+). Then we observe:
I A stochastic exponential of X is invertible for all t ≥ 0 if andonly if
det(I + ∆Xt) 6= 0 for all t ≥ 0. (∗)I Suppose (Xt)t≥0 fulfills (∗). Then for (Ut)t≥0 given by
Ut := −Xt +[X ,X ]ct +∑
0<s≤t
((I + ∆Xs)−1 − I + ∆Xs
), t ≥ 0
it holds
[←E (X )t ]
−1 =→E (U)t , t ≥ 0.
Alexander Lindner, 15
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Choice of As,t
We have
I Every stochastic process At := A0,t in the autoregressivemodel Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, which fulfillsAssumptions (a) to (d) is a multiplicative right Levy processin GL(R,m).
I Every right Levy process in (GL(R,m), ·) is the rightstochastic exponential of a Levy process in (Rm×m,+), i.e. we
have At =→E (U)t .
I There exists another Levy process X in (Rm×m,+) such that
At =←E (X )−1, t ≥ 0,
and the increments As,t = AtA−1s of At take the form
As,t =←E (X )−1
t
←E (X )s , 0 ≤ s ≤ t.
Alexander Lindner, 16
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Choice of As,t
We have
I Every stochastic process At := A0,t in the autoregressivemodel Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, which fulfillsAssumptions (a) to (d) is a multiplicative right Levy processin GL(R,m).
I Every right Levy process in (GL(R,m), ·) is the rightstochastic exponential of a Levy process in (Rm×m,+), i.e. we
have At =→E (U)t .
I There exists another Levy process X in (Rm×m,+) such that
At =←E (X )−1, t ≥ 0,
and the increments As,t = AtA−1s of At take the form
As,t =←E (X )−1
t
←E (X )s , 0 ≤ s ≤ t.
Alexander Lindner, 16
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Choice of As,t
We have
I Every stochastic process At := A0,t in the autoregressivemodel Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, which fulfillsAssumptions (a) to (d) is a multiplicative right Levy processin GL(R,m).
I Every right Levy process in (GL(R,m), ·) is the rightstochastic exponential of a Levy process in (Rm×m,+), i.e. we
have At =→E (U)t .
I There exists another Levy process X in (Rm×m,+) such that
At =←E (X )−1, t ≥ 0,
and the increments As,t = AtA−1s of At take the form
As,t =←E (X )−1
t
←E (X )s , 0 ≤ s ≤ t.
Alexander Lindner, 16
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Choice of (As,t ,Bs,t)0≤s≤tTheorem: (i) Suppose (Xt ,Yt)t≥0 to be a Levy process in
(Rm×m × Rm,+) such that←E (X ) is non-singular. For 0 ≤ s ≤ t
define (As,t
Bs,t
):=
←E (X )−1
t
←E (X )s
←E (X )−1
t
∫(s,t]
←E (X )u−dYu
.
Then (As,t ,Bs,t)0≤s≤t satisfies Assumptions (a) to (d) above andfor any starting random variable V0 the process
Vt :=←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs
)satisfies Vt = As,tVs + Bs,t , 0 ≤ s ≤ t.
(ii) All processes satisfying Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, with(As,t ,Bs,t)0≤s≤t satisfying Assumptions (a) to (d), can beobtained in this way.
Alexander Lindner, 17
Multivariate generalized Ornstein-Uhlenbeck processes : Construction
Choice of (As,t ,Bs,t)0≤s≤tTheorem: (i) Suppose (Xt ,Yt)t≥0 to be a Levy process in
(Rm×m × Rm,+) such that←E (X ) is non-singular. For 0 ≤ s ≤ t
define (As,t
Bs,t
):=
←E (X )−1
t
←E (X )s
←E (X )−1
t
∫(s,t]
←E (X )u−dYu
.
Then (As,t ,Bs,t)0≤s≤t satisfies Assumptions (a) to (d) above andfor any starting random variable V0 the process
Vt :=←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs
)satisfies Vt = As,tVs + Bs,t , 0 ≤ s ≤ t.(ii) All processes satisfying Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, with(As,t ,Bs,t)0≤s≤t satisfying Assumptions (a) to (d), can beobtained in this way.
Alexander Lindner, 17
Multivariate generalized Ornstein-Uhlenbeck processes : Definition
The Multivariate Generalized Ornstein-Uhlenbeck Process
Definition: Let (Xt ,Yt)t≥0 be a Levy process in (Rm×m × Rm,+)such that det(I + ∆Xt) 6= 0 for all t ≥ 0 and let V0 be a randomvariable in Rm. Then the process (Vt)t≥0 in Rm given by
Vt :=←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs
)
will be called multivariate generalized Ornstein-Uhlenbeck(MGOU) process driven by (Xt ,Yt)t≥0.
Remark:
I V0 not a priori independent of (X ,Y ).
I←E (X )−1
t may take negative (definite) values.
Alexander Lindner, 18
Multivariate generalized Ornstein-Uhlenbeck processes : Definition
The Multivariate Generalized Ornstein-Uhlenbeck Process
Definition: Let (Xt ,Yt)t≥0 be a Levy process in (Rm×m × Rm,+)such that det(I + ∆Xt) 6= 0 for all t ≥ 0 and let V0 be a randomvariable in Rm. Then the process (Vt)t≥0 in Rm given by
Vt :=←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs
)
will be called multivariate generalized Ornstein-Uhlenbeck(MGOU) process driven by (Xt ,Yt)t≥0.
Remark:
I V0 not a priori independent of (X ,Y ).
I←E (X )−1
t may take negative (definite) values.
Alexander Lindner, 18
Multivariate generalized Ornstein-Uhlenbeck processes : Definition
The Multivariate Generalized Ornstein-Uhlenbeck Process
Definition: Let (Xt ,Yt)t≥0 be a Levy process in (Rm×m × Rm,+)such that det(I + ∆Xt) 6= 0 for all t ≥ 0 and let V0 be a randomvariable in Rm. Then the process (Vt)t≥0 in Rm given by
Vt :=←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs
)
will be called multivariate generalized Ornstein-Uhlenbeck(MGOU) process driven by (Xt ,Yt)t≥0.
Remark:
I V0 not a priori independent of (X ,Y ).
I←E (X )−1
t may take negative (definite) values.
Alexander Lindner, 18
Multivariate generalized Ornstein-Uhlenbeck processes : The SDE
The corresponding SDETheorem: The MGOU process
Vt :=←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs
)
driven by the Levy process (Xt ,Yt)t≥0 in (Rm×m × Rm,+) is theunique solution of the SDE
dVt = dUtVt− + dLt , t ≥ 0,
for the Levy process (Ut , Lt)t≥0 in (Rm×m × Rm,+) given by(Ut
Lt
):=
(−Xt + [X ,X ]ct +
∑0<s≤t
((I + ∆Xs)−1 − I + ∆Xs
)Yt +
∑0<s≤t
((I + ∆Xs)−1 − I
)∆Ys − [X ,Y ]ct
),
for t ≥ 0.
skip subsection
Alexander Lindner, 19
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions - Part 1Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in (Rm×m × Rm,+). Let (Ut , Lt)t≥0 be theLevy process defined as above.
(i) Suppose limt→∞←E (U)t = 0 in probability, then:
A finite random variable V0 can be chosen such that(Vt)t≥0 is strictly stationary
⇔∫(0,t]
←E (U)s−dLs converges in distribution.
In this case, the distribution of the strictly stationary process (Vt)t≥0
is uniquely determined and is obtained by choosing V0 independent
of (Xt ,Yt)t≥0 as the distributional limit of∫
(0,t]
←E (U)s−dLs as t →
∞.
Alexander Lindner, 20
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions - Part 1Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in (Rm×m × Rm,+). Let (Ut , Lt)t≥0 be theLevy process defined as above.
(i) Suppose limt→∞←E (U)t = 0 in probability, then:
A finite random variable V0 can be chosen such that(Vt)t≥0 is strictly stationary
⇔∫(0,t]
←E (U)s−dLs converges in distribution.
In this case, the distribution of the strictly stationary process (Vt)t≥0
is uniquely determined and is obtained by choosing V0 independent
of (Xt ,Yt)t≥0 as the distributional limit of∫
(0,t]
←E (U)s−dLs as t →
∞.
Alexander Lindner, 20
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions - Part 1Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in (Rm×m × Rm,+). Let (Ut , Lt)t≥0 be theLevy process defined as above.
(ii) Suppose limt→∞←E (X )t = 0 in probability, then:
A finite random variable V0 can be chosen such that(Vt)t≥0 is strictly stationary
⇔∫(0,t]
←E (X )s−dYs converges in probability.
In this case the strictly stationary solution is unique and given by
Vt = −←E (X )−1
t
∫(t,∞)
←E (X )s−dYs a.s. for all t ≥ 0.
skip now
Alexander Lindner, 20
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions - Part 1Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in (Rm×m × Rm,+). Let (Ut , Lt)t≥0 be theLevy process defined as above.
(ii) Suppose limt→∞←E (X )t = 0 in probability, then:
A finite random variable V0 can be chosen such that(Vt)t≥0 is strictly stationary
⇔∫(0,t]
←E (X )s−dYs converges in probability.
In this case the strictly stationary solution is unique and given by
Vt = −←E (X )−1
t
∫(t,∞)
←E (X )s−dYs a.s. for all t ≥ 0.
skip now
Alexander Lindner, 20
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
MGOU processes on affine subspaces
Definition: Suppose (Xt ,Yt)t≥0 is a Levy process in
(Rm×m × Rm,+) such that←E (X ) is non-singular and define
(As,t ,Bs,t)0≤s≤t by
(As,t
Bs,t
):=
←E (X )−1
t
←E (X )s
←E (X )−1
t
∫(s,t]
←E (X )u−dYu
.
Then an affine subspace H of Rm is called invariant under theautoregressive model Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, if
As,tH + Bs,t ⊆ H almost surely,
holds for all 0 ≤ s ≤ t.If Rm is the only invariant affine subspace, the model is calledirreducible.
Alexander Lindner, 21
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
MGOU processes on affine subspaces
Theorem: The autoregressive model Vt = As,tVs + Bs,t ,0 ≤ s ≤ t, is irreducible if and only if there exists no pair (O,K ) ofan orthogonal transformation O ∈ Rm×m and a constantK = (k1, . . . , kd)T ∈ Rd , 1 ≤ d ≤ m, such that a.s.
OXtO−1 =
(X1t 0
X2t X3
t
)and OYt =
(X1tKY2t
)where X1
t ∈ Rd×d , t ≥ 0. With (Ut , Lt)t≥0 as defined above this isequivalent to
OUtO−1 =
(U1t 0
U2t U3
t
)and OLt =
(−U1
tKL2t
)a.s. with U1
t ∈ Rd×d .
Alexander Lindner, 22
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions of MGOU processes - Part 2
Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in Rm×m × Rm such that the correspondingautoregressive model Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, with(As,t ,Bs,t)0≤s≤t as defined before is irreducible. Let (Ut , Lt)t≥0 bedefined as above. Then
A finite random variable V0, independent of (Xt ,Yt)t≥0,can be chosen such that (Vt)t≥0 is strictly stationary
⇔limt→∞
←E (U)t = 0 in probability
and∫
(0,t]
←E (U)s−dLs converges in distribution.
A similar result for strictly noncausal strictly stationary solutions ofMGOU processes can be obtained, too.
skip now
Alexander Lindner, 23
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions of MGOU processes - Part 2
Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in Rm×m × Rm such that the correspondingautoregressive model Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, with(As,t ,Bs,t)0≤s≤t as defined before is irreducible. Let (Ut , Lt)t≥0 bedefined as above. Then
A finite random variable V0, independent of (Xt ,Yt)t≥0,can be chosen such that (Vt)t≥0 is strictly stationary
⇔limt→∞
←E (U)t = 0 in probability
and∫
(0,t]
←E (U)s−dLs converges in distribution.
A similar result for strictly noncausal strictly stationary solutions ofMGOU processes can be obtained, too.
skip now
Alexander Lindner, 23
Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity
Stationary Solutions of MGOU processes - Part 2
Theorem: Suppose (Vt)t≥0 is a MGOU process driven by the Levyprocess (Xt ,Yt)t≥0 in Rm×m × Rm such that the correspondingautoregressive model Vt = As,tVs + Bs,t , 0 ≤ s ≤ t, with(As,t ,Bs,t)0≤s≤t as defined before is irreducible. Let (Ut , Lt)t≥0 bedefined as above. Then
A finite random variable V0, independent of (Xt ,Yt)t≥0,can be chosen such that (Vt)t≥0 is strictly stationary
⇔limt→∞
←E (U)t = 0 in probability
and∫
(0,t]
←E (U)s−dLs converges in distribution.
A similar result for strictly noncausal strictly stationary solutions ofMGOU processes can be obtained, too.
skip now
Alexander Lindner, 23
Extensions : Multivariate volatilities
ExtensionsBehme (2012) obtains various further results, in particular themoment structure of multivariate generalized Ornstein–Uhlenbeckprocesses. She further considers matrix valued positive semidefinitegeneralized Ornstein–Uhlenbeck processes:
Often, in the one dimensional case volatilities are modeled assquare-root process of a generalized Ornstein-Uhlenbeck process.Hence to construct a multivariate volatility model similarly, wehave to ensure our processes to be positive semidefinite.
One possibility hereby is to consider processes which fulfill
Vt = As,tVsATs,t + Bs,t , 0 ≤ s ≤ t
with As,t in GL(R,m) and Bs,t ∈ Rm×m positive semidefinite.This is equivalent to
vecVt = (As,t ⊗ As,t)vecVs + vecBs,t .
Alexander Lindner, 24
Extensions : Multivariate volatilities
ExtensionsBehme (2012) obtains various further results, in particular themoment structure of multivariate generalized Ornstein–Uhlenbeckprocesses. She further considers matrix valued positive semidefinitegeneralized Ornstein–Uhlenbeck processes:
Often, in the one dimensional case volatilities are modeled assquare-root process of a generalized Ornstein-Uhlenbeck process.Hence to construct a multivariate volatility model similarly, wehave to ensure our processes to be positive semidefinite.
One possibility hereby is to consider processes which fulfill
Vt = As,tVsATs,t + Bs,t , 0 ≤ s ≤ t
with As,t in GL(R,m) and Bs,t ∈ Rm×m positive semidefinite.
This is equivalent to
vecVt = (As,t ⊗ As,t)vecVs + vecBs,t .
Alexander Lindner, 24
Extensions : Multivariate volatilities
ExtensionsBehme (2012) obtains various further results, in particular themoment structure of multivariate generalized Ornstein–Uhlenbeckprocesses. She further considers matrix valued positive semidefinitegeneralized Ornstein–Uhlenbeck processes:
Often, in the one dimensional case volatilities are modeled assquare-root process of a generalized Ornstein-Uhlenbeck process.Hence to construct a multivariate volatility model similarly, wehave to ensure our processes to be positive semidefinite.
One possibility hereby is to consider processes which fulfill
Vt = As,tVsATs,t + Bs,t , 0 ≤ s ≤ t
with As,t in GL(R,m) and Bs,t ∈ Rm×m positive semidefinite.This is equivalent to
vecVt = (As,t ⊗ As,t)vecVs + vecBs,t .
Alexander Lindner, 24
Extensions : Multivariate volatilities
A ConstructionArguing as above we see that the only process which fulfills theabove random recurrence equation is given by
Vt =←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs(
←E (X )s−)T
)(←E (X )−1
t )T ,
for a Levy process (X ,Y ) ∈ Rm×m × Rm×m
and that
vecVt =←E (X )−1
t ⊗←E (X )−1
t
(vecV0 +
∫ t
0
←E (X )s− ⊗
←E (X )s−dYs
)=
←E (X)−1
t
(vecV0 +
∫ t
0
←E (X)s−dYs
), t ≥ 0
is a MGOU process driven by the Levy process(X,Y) ∈ Rm2×m2 × Rm2
with
Xt = I ⊗ Xt + Xt ⊗ I + [X ⊗ I , I ⊗ X ]t , t ≥ 0
and Yt = vec (Yt).
Alexander Lindner, 25
Extensions : Multivariate volatilities
A ConstructionArguing as above we see that the only process which fulfills theabove random recurrence equation is given by
Vt =←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs(
←E (X )s−)T
)(←E (X )−1
t )T ,
for a Levy process (X ,Y ) ∈ Rm×m × Rm×m and that
vecVt =←E (X )−1
t ⊗←E (X )−1
t
(vecV0 +
∫ t
0
←E (X )s− ⊗
←E (X )s−dYs
)=
←E (X)−1
t
(vecV0 +
∫ t
0
←E (X)s−dYs
), t ≥ 0
is a MGOU process driven by the Levy process(X,Y) ∈ Rm2×m2 × Rm2
with
Xt = I ⊗ Xt + Xt ⊗ I + [X ⊗ I , I ⊗ X ]t , t ≥ 0
and Yt = vec (Yt).
Alexander Lindner, 25
Extensions : Multivariate volatilities
A Condition for Positive Semidefiniteness
The process
Vt =←E (X )−1
t
(V0 +
∫(0,t]
←E (X )s−dYs(
←E (X )s−)T
)(←E (X )−1
t )T ,
is positive semidefinite for all t ≥ 0 and all positive semidefinitestarting random variables V0 if and only if Y is a matrixsubordinator.
Alexander Lindner, 26
:
Thank you for your attention!
Alexander Lindner, 27
:
Main references:
I A. Behme and A. Lindner (2012) Multivariate GeneralizedOrnstein-Uhlenbeck Processes. Stoch. Proc. Appl. 122.
I A. Behme (2012) Moments of MGOU Processes and PositiveSemidefinite Matrix Processes. JMVA 111.
I P. Bougerol and N. Picard (1992) Strict stationarity ofgeneralized autoregressive processes. Ann. Probab. 20.
I L. de Haan and R.L. Karandikar (1989) Embedding astochastic difference equation into a continuous-time process.Stoch. Proc. Appl. 32.
Alexander Lindner, 28