Multivariate Generalized Ornstein-Uhlenbeck Processes

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




Ornstein and Uhlenbeck (1930) add the concept of friction toEinsteins model.





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).





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.


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).



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.



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.


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.


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).


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)




Vt = e−ξt

(V0 +

∫(0,t]

eξs−dηs

), 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.




Vt = e−ξt

(V0 +

∫(0,t]

eξs−dηs

), 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).


Multivariate generalized Ornstein-Uhlenbeck processes :

Multivariate GeneralizedOrnstein-Uhlenbeck Processes


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.



Constructing a multivariate GOU

We aim to embed the random sequence


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:






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



is i.i.d. for each h > 0.

Assuming slightly more leads to the following requirements:






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



is i.i.d. for each h > 0.Assuming slightly more leads to the following requirements:



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.



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.



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 .



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.



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.



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.



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.



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.


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.


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.

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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 →

∞.



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.

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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.



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 .



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


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 .







with As,t in GL(R,m) and Bs,t ∈ Rm×m positive semidefinite.

This is equivalent to








with As,t in GL(R,m) and Bs,t ∈ Rm×m positive semidefinite.This is equivalent to




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).



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).



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.


:

Thank you for your attention!


:

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.


Multivariate Generalized Ornstein-Uhlenbeck Processes

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