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Lq(Lp)-theory of stochastic differential equations

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  • Xia, Pengcheng
  • Xie, Longjie
  • Zhang, Xicheng
  • Zhao, Guohuan

Abstract

In this paper we show the weak differentiability of the unique strong solution with respect to the starting point x as well as Bismut–Elworthy–Li’s derivative formula for the following stochastic differential equation in Rd: dXt=b(t,Xt)dt+σ(t,Xt)dWt,X0=x, where σ is bounded, uniformly continuous and nondegenerate, b∈L˜q1p1 and ∇σ∈L˜q2p2 for some pi,qi∈[2,∞) with dpi+2qi<1, i=1,2, where L˜qipi,i=1,2 are some localized spaces of Lqi(R+;Lpi(Rd)). Moreover, in the endpoint case b∈L˜∞d;uni⊂L˜∞d, we also show the weak well-posedness.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:5188-5211
    DOI: 10.1016/j.spa.2020.03.004
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    References listed on IDEAS

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    1. 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.
    2. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
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    Cited by:

    1. Wu, Mingyan & Hao, Zimo, 2023. "Well-posedness of density dependent SDE driven by α-stable process with Hölder drifts," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 416-442.
    2. Ren, Panpan, 2023. "Singular McKean–Vlasov SDEs: Well-posedness, regularities and Wang’s Harnack inequality," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 291-311.
    3. Wang, Feng-Yu, 2023. "Exponential ergodicity for singular reflecting McKean–Vlasov SDEs," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 265-293.
    4. Xie, Longjie & Yang, Li, 2022. "The Smoluchowski–Kramers limits of stochastic differential equations with irregular coefficients," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 91-115.
    5. Suo, Yongqiang & Yuan, Chenggui & Zhang, Shao-Qin, 2022. "Transportation cost inequalities for SDEs with irregular drifts," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 288-311.

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