Ergodic properties of generalized Ornstein-Uhlenbeck processes

We investigate ergodic properties of the solution of the SDE dVt=Vt−dUt+dLt, where (U,L) is a bivariate Lévy process. This class of processes includes the generalized Ornstein-Uhlenbeck processes. We provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster-Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.