Lq(Lp)-theory of stochastic differential equations

Pengcheng Xia; Longjie Xie; Xicheng Zhang; Guohuan Zhao

doi:10.1016/j.spa.2020.03.004

L^q(L^p)-theory of stochastic differential equations

Pengcheng Xia, Longjie Xie, Xicheng Zhang^*, Guohuan Zhao

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

48 Citations (Scopus)

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 R^d: dX_t=b(t,X_t)dt+σ(t,X_t)dW_t,X₀=x, where σ is bounded, uniformly continuous and nondegenerate, b∈L˜_q₁^p₁ and ∇σ∈L˜_q₂^p₂ for some p_i,q_i∈[2,∞) with [Formula presented], i=1,2, where L˜_{q_i}^p_i,i=1,2 are some localized spaces of L^q_i(R₊;L^p_i(R^d)). Moreover, in the endpoint case b∈L˜_∞^d;uni⊂L˜_∞^d, we also show the weak well-posedness.

Original language	English
Pages (from-to)	5188-5211
Number of pages	24
Journal	Stochastic Processes and their Applications
Volume	130
Issue number	8
DOIs	https://doi.org/10.1016/j.spa.2020.03.004
Publication status	Published - Aug 2020
Externally published	Yes

Keywords

Krylov's estimate, L(L)-estimates
Zvonkin's transformation
duality

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10.1016/j.spa.2020.03.004

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Xia, P., Xie, L., Zhang, X., & Zhao, G. (2020). L^q(L^p)-theory of stochastic differential equations. Stochastic Processes and their Applications, 130(8), 5188-5211. https://doi.org/10.1016/j.spa.2020.03.004

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title = "Lq(Lp)-theory of stochastic differential equations",

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 [Formula presented], 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.",

keywords = "Krylov's estimate, L(L)-estimates, Zvonkin's transformation, duality",

author = "Pengcheng Xia and Longjie Xie and Xicheng Zhang and Guohuan Zhao",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier B.V.",

year = "2020",

month = aug,

doi = "10.1016/j.spa.2020.03.004",

language = "English",

volume = "130",

pages = "5188--5211",

journal = "Stochastic Processes and their Applications",

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T1 - Lq(Lp)-theory of stochastic differential equations

AU - Xia, Pengcheng

AU - Xie, Longjie

AU - Zhang, Xicheng

AU - Zhao, Guohuan

PY - 2020/8

Y1 - 2020/8

N2 - 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 [Formula presented], 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.

AB - 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 [Formula presented], 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.

KW - Krylov's estimate, L(L)-estimates

KW - Zvonkin's transformation

KW - duality

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U2 - 10.1016/j.spa.2020.03.004

DO - 10.1016/j.spa.2020.03.004

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SN - 0304-4149

VL - 130

SP - 5188

EP - 5211

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 8

ER -

L^q(L^p)-theory of stochastic differential equations

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Keywords

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