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Derivative formulae for SDEs driven by multiplicative α-stable-like processes

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  • Wang, Linlin
  • Xie, Longjie
  • Zhang, Xicheng

Abstract

By using Bismut’s approach to the Malliavin calculus with jumps, we establish a derivative formula of Bismut–Elworthy–Li’s type for SDEs driven by multiplicative Lévy noises, whose Lévy measure satisfies some order conditions. In particular, α-stable-like noises are allowed. Moreover, we also obtain the sharp gradient estimate in short time for the corresponding transition semigroup provided α∈(1,2).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:3:p:867-885
    DOI: 10.1016/j.spa.2014.10.011
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    References listed on IDEAS

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    1. Chen, Zhen-Qing & Kumagai, Takashi, 2003. "Heat kernel estimates for stable-like processes on d-sets," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 27-62, November.
    2. 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.
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    Cited by:

    1. 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).
    2. Xia, Pengcheng & Xie, Longjie & Zhang, Xicheng & Zhao, Guohuan, 2020. "Lq(Lp)-theory of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5188-5211.
    3. 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.
    4. 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).

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