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Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

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  • Luo, Dejun
  • Wang, Jian

Abstract

We establish the exponential convergence with respect to the L1-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation dXt=dZt+b(Xt)dt,where (Zt)t≥0 is a pure jump Lévy process whose Lévy measure ν fulfills infx∈Rd,|x|≤κ0[ν∧(δx∗ν)](Rd)>0for some constant κ0>0, and the drift term b satisfies that for any x,y∈Rd, 〈b(x)−b(y),x−y〉≤Φ1(|x−y|)|x−y|,|x−y|

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  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3129-3173
    DOI: 10.1016/j.spa.2018.09.003
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    References listed on IDEAS

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    1. Böttcher, Björn & Schilling, René L. & Wang, Jian, 2011. "Corrigendum to âConstructions of coupling processes for Lévy processesâ," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2711-2714, November.
    2. 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.
    3. Majka, Mateusz B., 2017. "Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4083-4125.
    4. Mátyás Barczy & Zenghu Li & Gyula Pap, 2015. "Yamada-Watanabe Results for Stochastic Differential Equations with Jumps," International Journal of Stochastic Analysis, Hindawi, vol. 2015, pages 1-23, January.
    5. Zhang, Xicheng, 2013. "Derivative formulas and gradient estimates for SDEs driven by α-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1213-1228.
    6. Böttcher, Björn & Schilling, René L. & Wang, Jian, 2011. "Constructions of coupling processes for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1201-1216, June.
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    Cited by:

    1. Franziska Kühn, 2021. "Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1506-1578, September.
    2. Jianhai Bao & Jian Wang, 2023. "Coupling methods and exponential ergodicity for two‐factor affine processes," Mathematische Nachrichten, Wiley Blackwell, vol. 296(5), pages 1716-1736, May.
    3. Shukai Chen & Rongjuan Fang & Xiangqi Zheng, 2023. "Wasserstein-Type Distances of Two-Type Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1572-1590, September.
    4. Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    5. Wang, Tao, 2022. "Ergodic convergence rates for time-changed symmetric Lévy processes in dimension one," Statistics & Probability Letters, Elsevier, vol. 183(C).
    6. Huang, Lu-Jing & Majka, Mateusz B. & Wang, Jian, 2022. "Strict Kantorovich contractions for Markov chains and Euler schemes with general noise," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 307-341.
    7. Liang, Mingjie & Wang, Jian, 2020. "Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3053-3094.

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