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

Reiß, M. and Riedle, M. and Van Gaans, O. (2006) Delay differential equations driven by Lévy processes: Stationarity and Feller properties. Stochastic Processes and their Applications, 116 (10). pp. 1409-1432. ISSN 0304-4149

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Abstract

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.

Item Type:	Article
Uncontrolled Keywords:	Feller process; Invariant measure; Lévy process; Semimartingale characteristic; Stationary solution; Stochastic equation with delay; Stochastic functional differential equation
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User:	Ms Lucy van Russelt
Date Deposited:	19 Nov 2007
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/906

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