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Derivative formulas and gradient estimates for SDEs driven by α-stable processes

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  • Zhang, Xicheng

Abstract

In this paper we prove a derivative formula of Bismut–Elworthy–Li’s type as well as a gradient estimate for stochastic differential equations driven by α-stable noises, where α∈(0,2). As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:4:p:1213-1228
    DOI: 10.1016/j.spa.2012.11.012
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    References listed on IDEAS

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    1. Zhang, Xicheng, 2010. "Stochastic flows and Bismut formulas for stochastic Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1929-1949, September.
    2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
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    Citations

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    Cited by:

    1. Wang, Feng-Yu & Wang, Jian, 2013. "Coupling and strong Feller for jump processes on Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1588-1615.
    2. Liu, Xianming, 2022. "Limits of invariant measures of stochastic Burgers equations driven by two kinds of α-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 1-21.
    3. 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.
    4. Xie, Yingchao & Zhang, Qi & Zhang, Xicheng, 2014. "Probabilistic approach for semi-linear stochastic fractal equations," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 3948-3964.
    5. Wang, Ran & Xu, Lihu, 2018. "Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1772-1796.
    6. Zhao, Huiyan & Xu, Siyan, 2020. "A stochastic Fubini theorem for α-stable process," Statistics & Probability Letters, Elsevier, vol. 160(C).
    7. Yan, Litan & Yin, Xiuwei, 2018. "Bismut formula for a stochastic heat equation with fractional noise," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 165-172.
    8. Karipova, Gulnur & Magdziarz, Marcin, 2017. "Pricing of basket options in subdiffusive fractional Black–Scholes model," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 245-253.
    9. Li, Zhi & Yan, Litan, 2018. "Harnack inequalities for SDEs driven by subordinator fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 45-53.
    10. Sun, Xiaobin & Xie, Longjie & Xie, Yingchao, 2020. "Derivative formula for the Feynman–Kac semigroup of SDEs driven by rotationally invariant α-stable process," Statistics & Probability Letters, Elsevier, vol. 158(C).
    11. Zhang, Hua, 2021. "Strong Feller property for one-dimensional Lévy processes driven stochastic differential equations with Hölder continuous coefficients," Statistics & Probability Letters, Elsevier, vol. 169(C).
    12. Wang, Linlin & Xie, Longjie & Zhang, Xicheng, 2015. "Derivative formulae for SDEs driven by multiplicative α-stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 867-885.

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