Generalized Ornstein-Uhlenbeck Processes and Extensions

Abstract

The generalized Ornstein-Uhlenbeck process $V_t$ fulfills the stochastic differential equation $dV_t = V_{t-} dU_t + dL_t$ for a bivariate Levy process $(U_t,L_t)_{t\geq 0}$. In this thesis, for a given bivariate Levy process $(U,L)$, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation $dV_t = V_{t-} \, dU_t + dL_t$ are obtained. Noncausal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for $U$ and $L$ to remain semimartingales with respect to the corresponding expanded filtration is given. Distributional properties of the stationary solutions are analysed. In particular the expectation and autocorrelation function are obtained in terms of the process $(U,L)$ and in several cases of interest the tail behaviour is described. In the case where $U$ has jumps of size $-1$, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given. Finally, a multivariate generalized Ornstein-Uhlenbeck process driven by a Levy process $(X_t,Y_t)_{t\geq 0}$, with $(X_t,Y_t)\in \RR^{d\times d}\times \RR^d,\, d\geq 1,$ is defined. It is shown that this process $(V_t)_{t\geq 0}$ solves the stochastic differential equation $dV_t = dU_t V_{t-} + dL_t$ for another Levy process $(U_t,L_t)_{t\geq 0}$ in $\RR^{d\times d}\times \RR^d$, which is given in terms of $(X,Y)$. Under some extra conditions on the limit behaviour of $\cE(X)$, necessary and sufficient conditions for the existence of strictly stationary solutions are deduced.