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

  • Reiß, M.
  • Riedle, M.
  • van Gaans, O.

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.

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Article provided by Elsevier in its journal Stochastic Processes and their Applications.

Volume (Year): 116 (2006)
Issue (Month): 10 (October)
Pages: 1409-1432

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Handle: RePEc:eee:spapps:v:116:y:2006:i:10:p:1409-1432
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  1. Gushchin, Alexander A. & Küchler, Uwe, 2000. "On stationary solutions of delay differential equations driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 195-211, August.
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