Delay differential equations driven by Lévy processes: Stationarity and Feller properties

TY - JOUR

T1 - Delay differential equations driven by Lévy processes: Stationarity and Feller properties

AU - Reiß, Markus

AU - Riedle, Markus

AU - van Neerven, Onno

PY - 2006/10

Y1 - 2006/10

N2 - We consider a stochastic delay differential equation driven by a general Lévy 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.

AB - We consider a stochastic delay differential equation driven by a general Lévy 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.

KW - Invariant measure

KW - Lévy process

KW - Semimartingale characteristic

KW - Stationary solution

KW - Stochastic equation with delay

KW - Stochastic functional differential equation

U2 - 10.1016/j.spa.2006.03.002

DO - 10.1016/j.spa.2006.03.002

M3 - Article

SN - 0304-4149

VL - 116

SP - 1409

EP - 1432

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

IS - 10

ER -