Multivariate Generalized Ornstein-Uhlenbeck Processes

Multivariate Generalized Ornstein-Uhlenbeck Processes Anita Behme TU München Alexander Lindner TU Braunschweig 7th International Conference on Lévy 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 by Brownian 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 by Brownian Motion. Ornstein and Uhlenbeck (1930) add the concept of friction to Einsteins model: A particle moving from left to right gets hit by more particles from the right than from the left side which results in a slowdown. The velocity v(t) of the particle is given by m dv(t) = λv(t)dt + db(t) i.e. v(t) = e λt/m v(0) + e λt/m (0,t] e λs/m db(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 by Brownian Motion. Ornstein and Uhlenbeck (1930) add the concept of friction to Einsteins model: A particle moving from left to right gets hit by more particles from the right than from the left side which results in a slowdown. The velocity v(t) of the particle is given by i.e. m dv(t) = λv(t)dt + db(t) v(t) = e λt/m v(0) + e λt/m (0,t] e λs/m db(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 V t = e λt V 0 + e λt e λs db s, t 0, (0,t] fulfills the random recurrence equation V nh = e λh V (n 1)h + e λnh ((n 1)h,nh] e λs db s, n N. Hence it can be seen as a natural generalization in continuous time of the AR(1) time series X n = e λ X n 1 + Z n, n N, with i.i.d. noise (Z n ) n N such that L(Z 1 ) = L( (0,1] e λ(1 s) db s ). 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 Y n = A n Y n 1 + B n, n N, with (A n, B n ) n N i.i.d., A 1 > 0 a.s., into a continuous time setting in 1989 De Haan and Karandikar introduced the generalized Ornstein-Uhlenbeck process ) V t = e (V ξt 0 + e ξ s dη s, t 0. (0,t] driven by a bivariate Lévy process (ξ t, η t ) t 0 with starting random variable V 0. Alexander Lindner, 4

Definition: Lévy processes The Ornstein-Uhlenbeck process : From OU to GOU processes A Lévy process in R d on a probability space (Ω, F, P) is a stochastic process X = (X t ) t 0, X t : Ω R d satisfying the following properties: X 0 = 0 a.s. X has independent increments, i.e. for all 0 t 0 t 1... t n the random variables X t0, X t1 X t0,..., X tn X tn 1 are independent. X has stationary increments, i.e. for all s, t 0 it holds X s+t X s d = Xt. X has a.s. càdlàg paths, i.e. for P-a.e. ω Ω the path t X t (ω) is right-continuous in t 0 and has left limits in t > 0. Alexander Lindner, 5

The Ornstein-Uhlenbeck process : The GOU process Definition: Generalized OU processes The generalized Ornstein-Uhlenbeck (GOU) process (V t ) t 0 driven by the bivariate Lévy process (ξ t, η t ) t 0 is given by ) V t = e (V ξt 0 + e ξ s dη s, t 0, (0,t] where V 0 is a finite random variable, usually chosen independent of (ξ, η). In the case that (ξ t, η t ) = (λt, η t ) with a Lévy process (η t ) t 0 and a constant λ 0 the process (V t ) t 0 is called Lévy-driven Ornstein-Uhlenbeck process or Ornstein-Uhlenbeck type process. Obviously, if additionally (η t ) t 0 is a Brownian motion, we get the classical Ornstein-Uhlenbeck process. Alexander Lindner, 6

The Ornstein-Uhlenbeck process : The GOU process The generalized Ornstein-Uhlenbeck process: Applications Example 1: ξ t = t deterministic ) V t = e (V t 0 + e s dη s (0,t] Lévy driven Ornstein-Uhlenbeck process, classical for η t = B t applications in storage theory stochastic volatility model of Barndorff-Nielsen and Shephard (2001): η t subordinator, V t squared volatility, price process G t defined by dg t = (µ + bv t )dt + V t db t for constants µ, b. Alexander Lindner, 7

The corresponding SDE The Ornstein-Uhlenbeck process : The corresponding SDE The generalized Ornstein-Uhlenbeck process driven by (ξ, η) ) V t = e (V ξt 0 + e ξ s dη s, t 0 is the unique solution of the SDE (0,t] dv t = V t du t + dl t, t 0 where (U t, L t ) t 0 is a bivariate Lévy process completely determined by (ξ, η). In particular we have ξ t = log(e(u) t ) Definition: The ( Doléans-Dade Exponential E(U) t = exp U t 1 ) 2 tσ2 U ((1 + U s ) exp( U s )) is the unique solution of the SDE dz t = Z t du t, Z 0 = 1 a.s. s t Alexander Lindner, 8

The corresponding SDE The Ornstein-Uhlenbeck process : The corresponding SDE The generalized Ornstein-Uhlenbeck process driven by (ξ, η) ) V t = e (V ξt 0 + e ξ s dη s, t 0 is the unique solution of the SDE (0,t] dv t = V t du t + dl t, t 0 where (U t, L t ) t 0 is a bivariate Lévy process completely determined by (ξ, η). In particular we have ξ t = log(e(u) t ) and η t = L t + 0<s t U s L s 1 + U s tcov (B U1, B L1 ). Alexander Lindner, 8

Multivariate generalized Ornstein-Uhlenbeck processes : Multivariate Generalized Ornstein-Uhlenbeck Processes Alexander Lindner, 9

Multivariate generalized Ornstein-Uhlenbeck processes : Construction Recall: An AR(1) time series The generalized Ornstein-Uhlenbeck process ) V t = e (V ξt 0 + e ξ s dη s, t 0. (0,t] driven by a bivariate Lévy process (ξ t, η t ) t 0 with starting random variable V 0 had been derived by embedding the AR(1) time series V n = A n V n 1 + B n, n N, with (A n, B n ) n N i.i.d., A 1 > 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 V n = A n V n 1 + B n, n N, with (A n, B n ) n N i.i.d., (A n, B n ) R m m R m, A 1 a.s. non-singular, into a continuous time setting. More precisely, we want to find all stochastic processes (V t ) t 0 such that V t = A s,t V s + B s,t, 0 s t and is i.i.d. for each h > 0. (A (n 1)h,nh, B (n 1)h,nh ) n N Alexander Lindner, 11

Multivariate generalized Ornstein-Uhlenbeck processes : Construction Constructing a multivariate GOU We aim to embed the random sequence V n = A n V n 1 + B n, n N, with (A n, B n ) n N i.i.d., (A n, B n ) R m m R m, A 1 a.s. non-singular, into a continuous time setting. More precisely, we want to find all stochastic processes (V t ) t 0 such that V t = A s,t V s + B s,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 Assumptions For each 0 s t let (A s,t, B s,t ) GL(R, m) R m s.t.: Assumption (a) For all 0 u s t almost surely A u,t = A s,t A u,s and B u,t = A s,t B u,s + B s,t. Assumption (b) The families of random matrices {(A s,t, B s,t ), a s t b} and {(A s,t, B s,t ), c s t d} are independent for 0 a b c d. Assumption (c) For all 0 s t it holds (A s,t, B s,t ) d = (A 0,t s, B 0,t s ). Alexander Lindner, 12

Multivariate generalized Ornstein-Uhlenbeck processes : Construction Assumptions For each 0 s t let (A s,t, B s,t ) GL(R, m) R m s.t.: Assumption (a) For all 0 u s t almost surely A u,t = A s,t A u,s and B u,t = A s,t B u,s + B s,t. Assumption (b) The families of random matrices {(A s,t, B s,t ), a s t b} and {(A s,t, B s,t ), c s t d} are independent for 0 a b c d. Assumption (c) For all 0 s t it holds Assumption (d) It holds (A s,t, B s,t ) d = (A 0,t s, B 0,t s ). P lim t 0 A 0,t = A 0,0 = I and P lim t 0 B 0,t = B 0,0 = 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 A t := A 0,t Lemma: Every stochastic process A t := A 0,t in the autoregressive model above which fulfills Assumptions (a) to (d) has a version which is a multiplicative right Lévy process in the general linear group GL(R, m) of order m. That means, (A t ) t 0 is a stochastic process with values in GL(R, m) with the following properties: A 0 = I a.s. it has independent left increments, i.e. for all 0 t 1... t n, the random variables A 0, A t1 A 1 0,..., A t n A 1 t n 1 are independent. it has stationary left increments, i.e. for all s, t 0 it holds A s+t A 1 d s = A t. it has a.s. càdlàg paths, i.e. for P-a.e. ω Ω the path t A t (ω) is right-continuous in t 0 and has left limits in t > 0. Alexander Lindner, 13

Multivariate generalized Ornstein-Uhlenbeck processes : Construction The Multivariate Stochastic Exponential I Lemma: By an observation due to Skorokhod every right Lévy process in (GL(R, m), ) is the right stochastic exponential of a Lévy process in (R m m, +). Definition: Let (X t ) t 0 be a semimartingale in (R m m, +). Then its left stochastic exponential E (X ) t is defined as the unique R m m -valued, adapted, càdlàg solution of the SDE Z t = I + Z s dx s, t 0, (0,t] while the unique adapted, càdlàg solution of the SDE Z t = I + dx s Z s, t 0, (0,t] 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 (X t ) t 0 be a semimartingale in (R m m, +). Then we observe: A stochastic exponential of X is invertible for all t 0 if and only if det(i + X t ) 0 for all t 0. ( ) Suppose (X t ) t 0 fulfills ( ). Then for (U t ) t 0 given by U t := X t +[X, X ] c t + it holds 0<s t ( (I + Xs ) 1 I + X s ), t 0 [ E (X ) t ] 1 = E (U) t, t 0. Alexander Lindner, 15

Multivariate generalized Ornstein-Uhlenbeck processes : Construction Choice of A s,t We have Every stochastic process A t := A 0,t in the autoregressive model V t = A s,t V s + B s,t, 0 s t, which fulfills Assumptions (a) to (d) is a multiplicative right Lévy process in GL(R, m). Every right Lévy process in (GL(R, m), ) is the right stochastic exponential of a Lévy process in (R m m, +), i.e. we have A t = E (U) t. Alexander Lindner, 16

Multivariate generalized Ornstein-Uhlenbeck processes : Construction Choice of (A s,t, B s,t ) 0 s t Theorem: (i) Suppose (X t, Y t ) t 0 to be a Lévy process in (R m m R m, +) such that E (X ) is non-singular. For 0 s t define ( ) As,t := E (X ) 1 t E (X )s B. s,t E (X ) 1 t (s,t] E (X )u dy u Then (A s,t, B s,t ) 0 s t satisfies Assumptions (a) to (d) above and for any starting random variable V 0 the process ( ) V t := E (X ) 1 t V 0 + E (X )s dy s (0,t] satisfies V t = A s,t V s + B s,t, 0 s t. Alexander Lindner, 17

Multivariate generalized Ornstein-Uhlenbeck processes : Definition The Multivariate Generalized Ornstein-Uhlenbeck Process Definition: Let (X t, Y t ) t 0 be a Lévy process in (R m m R m, +) such that det(i + X t ) 0 for all t 0 and let V 0 be a random variable in R m. Then the process (V t ) t 0 in R m given by ( ) V t := E (X ) 1 t V 0 + E (X )s dy s (0,t] will be called multivariate generalized Ornstein-Uhlenbeck (MGOU) process driven by (X t, Y t ) t 0. Alexander Lindner, 18

Multivariate generalized Ornstein-Uhlenbeck processes : The SDE The corresponding SDE Theorem: The MGOU process ( V t := E (X ) 1 t V 0 + (0,t] E (X )s dy s ) driven by the Lévy process (X t, Y t ) t 0 in (R m m R m, +) is the unique solution of the SDE dv t = du t V t + dl t, t 0, for the Lévy process (U t, L t ) t 0 in (R m m R m, +) given by ( ) ( Ut Xt + [X, X ] c t + ( 0<s t (I + Xs ) 1 ) I + X s := L t Y t + ( 0<s t (I + Xs ) 1 I ) Y s [X, Y ] c t ), for t 0. skip subsection Alexander Lindner, 19

Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity Stationary Solutions - Part 1 Theorem: Suppose (V t ) t 0 is a MGOU process driven by the Lévy process (X t, Y t ) t 0 in (R m m R m, +). Let (U t, L t ) t 0 be the Lévy process defined as above. (i) Suppose lim t E (U)t = 0 in probability, then: A finite random variable V 0 can be chosen such that (V t ) t 0 is strictly stationary E (U)s dl s converges in distribution. (0,t] In this case, the distribution of the strictly stationary process (V t ) t 0 is uniquely determined and is obtained by choosing V 0 independent of (X t, Y t ) t 0 as the distributional limit of (0,t] E (U)s dl s as t. Alexander Lindner, 20

Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity Stationary Solutions - Part 1 Theorem: Suppose (V t ) t 0 is a MGOU process driven by the Lévy process (X t, Y t ) t 0 in (R m m R m, +). Let (U t, L t ) t 0 be the Lévy process defined as above. (ii) Suppose lim t E (X )t = 0 in probability, then: A finite random variable V 0 can be chosen such that (V t ) t 0 is strictly stationary E (X )s dy s converges in probability. (0,t] In this case the strictly stationary solution is unique and given by V t = E (X ) 1 t E (X )s dy s a.s. for all t 0. (t, ) skip now Alexander Lindner, 20

Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity MGOU processes on affine subspaces Definition: Suppose (X t, Y t ) t 0 is a Lévy process in (R m m R m, +) such that E (X ) is non-singular and define (A s,t, B s,t ) 0 s t by ( ) As,t := E (X ) 1 t E (X )s B. s,t E (X ) 1 t (s,t] E (X )u dy u Then an affine subspace H of R m is called invariant under the autoregressive model V t = A s,t V s + B s,t, 0 s t, if A s,t H + B s,t H almost surely, holds for all 0 s t. If R m is the only invariant affine subspace, the model is called irreducible. Alexander Lindner, 21

Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity MGOU processes on affine subspaces Theorem: The autoregressive model V t = A s,t V s + B s,t, 0 s t, is irreducible if and only if there exists no pair (O, K) of an orthogonal transformation O R m m and a constant K = (k 1,..., k d ) T R d, 1 d m, such that a.s. ( ) ( ) OX t O 1 X 1 = t 0 X 1 and OY t = t K X 2 t X 3 t where X 1 t R d d, t 0. With (U t, L t ) t 0 as defined above this is equivalent to ( ) ( ) OU t O 1 U 1 = t 0 U 1 and OL t = t K a.s. with U 1 t R d d. U 2 t U 3 t Y 2 t L 2 t Alexander Lindner, 22

Multivariate generalized Ornstein-Uhlenbeck processes : Stationarity Stationary Solutions of MGOU processes - Part 2 Theorem: Suppose (V t ) t 0 is a MGOU process driven by the Lévy process (X t, Y t ) t 0 in R m m R m such that the corresponding autoregressive model V t = A s,t V s + B s,t, 0 s t, with (A s,t, B s,t ) 0 s t as defined before is irreducible. Let (U t, L t ) t 0 be defined as above. Then A finite random variable V 0, independent of (X t, Y t ) t 0, can be chosen such that (V t ) t 0 is strictly stationary lim t E (U)t = 0 in probability and (0,t] E (U)s dl s converges in distribution. Alexander Lindner, 23

Extensions : Multivariate volatilities Extensions Behme (2012) obtains various further results, in particular the moment structure of multivariate generalized Ornstein Uhlenbeck processes. She further considers matrix valued positive semidefinite generalized Ornstein Uhlenbeck processes: Often, in the one dimensional case volatilities are modeled as square-root process of a generalized Ornstein-Uhlenbeck process. Hence to construct a multivariate volatility model similarly, we have to ensure our processes to be positive semidefinite. One possibility hereby is to consider processes which fulfill V t = A s,t V s A T s,t + B s,t, 0 s t with A s,t in GL(R, m) and B s,t R m m positive semidefinite. Alexander Lindner, 24

Extensions : Multivariate volatilities A Construction Arguing as above we see that the only process which fulfills the above random recurrence equation is given by ( ) V t = E (X ) 1 t V 0 + E (X )s dy s ( E (X ) s ) T (0,t] for a Lévy process (X, Y ) R m m R m m and that ( vec V t = E (X ) 1 t t E (X ) 1 t vec V 0 + = E (X) 1 t ( vec V 0 + t is a MGOU process driven by the Lévy process (X, Y) R m2 m 2 R m2 with and Y t = vec (Y t ). 0 0 ( E (X ) 1 t ) T, E (X )s E (X ) s dy s ) E (X)s dy s ), t 0 X t = I X t + X t I + [X I, I X ] t, t 0 Alexander Lindner, 25

Extensions : Multivariate volatilities A Condition for Positive Semidefiniteness The process V t = E (X ) 1 t ( V 0 + (0,t] E (X )s dy s ( E (X ) s ) T ) ( E (X ) 1 t ) T, is positive semidefinite for all t 0 and all positive semidefinite starting random variables V 0 if and only if Y is a matrix subordinator. Alexander Lindner, 26

: Thank you for your attention! Alexander Lindner, 27

: Main references: A. Behme and A. Lindner (2012) Multivariate Generalized Ornstein-Uhlenbeck Processes. Stoch. Proc. Appl. 122. A. Behme (2012) Moments of MGOU Processes and Positive Semidefinite Matrix Processes. JMVA 111. P. Bougerol and N. Picard (1992) Strict stationarity of generalized autoregressive processes. Ann. Probab. 20. L. de Haan and R.L. Karandikar (1989) Embedding a stochastic difference equation into a continuous-time process. Stoch. Proc. Appl. 32. Alexander Lindner, 28