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May 2018 Strong solutions to stochastic differential equations with rough coefficients
Nicolas Champagnat, Pierre-Emmanuel Jabin
Ann. Probab. 46(3): 1498-1541 (May 2018). DOI: 10.1214/17-AOP1208

Abstract

We study strong existence and pathwise uniqueness for stochastic differential equations in $\mathbb{R}^{d}$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^{p}$ bounds for the solution of the corresponding Fokker–Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, including the uniformly elliptic case in any dimension, the one-dimensional case and the Langevin (kinetic) case.

Citation

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Nicolas Champagnat. Pierre-Emmanuel Jabin. "Strong solutions to stochastic differential equations with rough coefficients." Ann. Probab. 46 (3) 1498 - 1541, May 2018. https://doi.org/10.1214/17-AOP1208

Information

Received: 1 July 2013; Revised: 1 July 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894780
MathSciNet: MR3785594
Digital Object Identifier: 10.1214/17-AOP1208

Subjects:
Primary: 35K10, 60H10, 60J60

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 3 • May 2018
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