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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. 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.
    2. 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.
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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. 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.
    3. Th'eo Durandard, 2023. "Dynamic delegation in promotion contests," Papers 2308.05668, arXiv.org.
    4. 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.
    5. 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.
    6. 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.
    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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