STRONG SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH ROUGH COEFFICIENTS

• 成果类型: Article
• 署名作者: Champagnat, Nicolas; Jabin, Pierre-Emmanuel
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine; University System of Maryland; University of Maryland College Park; University System of Maryland; University of Maryland College Park
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1208
• 发表日期: 2018
• 页码: 1498-1541
• 关键词: continuous local martingales; explicit formulas; well-posedness; sdes; uniqueness; degenerate; EXISTENCE; FLOWS; time

摘要

We study strong existence and pathwise uniqueness for stochastic differential equations in 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.