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Regularity of semigroups generated by Lévy type operators via coupling

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  • Wang, Jian

Abstract

By adopting the coupling method, we obtain new verifiable sufficient conditions about the -Feller continuity, the Lipschitz continuity and the strong Feller continuity of the semigroups associated with Lévy type operators. These results easily apply to jump-diffusion processes and stochastic differential equations driven by Lévy processes. Our results also yield the criterion for the e-property (namely the characterization about the equi-continuity of semigroups acting on bounded Lipschitz functions) of Lévy type operators, and show that both genuine Lévy processes and the Ornstein-Uhlenbeck type processes are e-processes.

Suggested Citation

  • Wang, Jian, 2010. "Regularity of semigroups generated by Lévy type operators via coupling," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1680-1700, August.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:9:p:1680-1700
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    References listed on IDEAS

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    1. Kulik, Alexey M., 2009. "Exponential ergodicity of the solutions to SDE's with a jump noise," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 602-632, February.
    2. Wang, Jian, 2008. "Criteria for ergodicity of Lévy type operators in dimension one," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1909-1928, October.
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    Cited by:

    1. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    2. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    3. Huijie Qiao, 2014. "Exponential Ergodicity for SDEs with Jumps and Non-Lipschitz Coefficients," Journal of Theoretical Probability, Springer, vol. 27(1), pages 137-152, March.
    4. Th'eo Durandard, 2023. "Dynamic delegation in promotion contests," Papers 2308.05668, arXiv.org.
    5. Jian Wang, 2014. "On the Existence and Explicit Estimates for the Coupling Property of Lévy Processes with Drift," Journal of Theoretical Probability, Springer, vol. 27(3), pages 1021-1044, September.
    6. Palczewski, Jan & Stettner, Łukasz, 2014. "Infinite horizon stopping problems with (nearly) total reward criteria," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 3887-3920.
    7. Xi, Fubao & Zhu, Chao, 2018. "On the martingale problem and Feller and strong Feller properties for weakly coupled Lévy type operators," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4277-4308.

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