Delay differential equations driven by Lévy processes: Stationarity and Feller properties
Delay differential equations driven by Lévy processes: Stationarity and Feller properties
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DOI：
10.1016/j.spa.2006.03.002
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发表时间：
2005-05
影响因子：
1.4
通讯作者：
Markus Reiß;M. Riedle;O. Gaans
中科院分区：
文献类型：
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作者：
Markus Reiß;M. Riedle;O. Gaans
考虑一类由一般Lévy过程驱动的随机时滞微分方程。漂移和噪声项都可能依赖于过去，但只有漂移项被假定为线性的。我们证明了Skorokhod空间上的段过程是最终Feller的，但一般不是最终强Feller的。不变测度的存在性证明了使用半鞅特征和Krylov-Bogoliubov方法的片段的紧密性。一个反例表明，在完全一般的情况下，定态解可能不是唯一的，但在更具体的情况下，唯一性成立。
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.