Delay differential equations driven by Levy processes: stationarity and Feller properties

Markus Reiß , Markus Riedle , Onno van Gaans

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

60H20 Stochastic integral equations

60H10 Stochastic ordinary differential equations

34K50 Stochastic delay equations

60G48 Generalizations of martingales

93E15 Stochastic stability

Abstract
We consider a stochastic delay differential equation driven by a general Levy process.Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established.