非Lipschitz条件下高维McKean-Vlasov随机微分方程解的存在唯一性  

Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions

作　　者：马丽[1,2] 孙芳芳 MA Li;SUN Fangfang(School of Mathematics and Statistics,Hainan Normal University,Haikou 571158,P.R.China;Key Laboratory of Data Science and Smart Education,Ministry of Education,Hainan Normal University,Haikou 571158,P.R.China)

机构地区：[1]海南师范大学数学与统计学院,海口571158 [2]海南师范大学数据科学与智慧教育教育部重点实验室,海口571158

出　　处：《应用数学和力学》2023年第10期1272-1290,共19页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(地区科学基金)项目(11861029);海南省高层次人才项目(120RC589)。

摘　　要：研究了一类漂移系数不连续的高维McKean-Vlasov随机微分方程及相应的粒子系统解的存在唯一性.在漂移系数关于空间变量逐段Lipschitz连续的条件下,首先利用Zvonkin变换将方程转换为漂移系数为Lipschitz连续的McKean-Vlasov随机微分方程,变换后的方程存在唯一解.然后由变换函数的性质可得逆函数的存在性和Lipschitz连续性.最后由Ito公式及逆函数的性质可得原来的McKean-Vlasov随机微分方程及相应的粒子系统解的存在唯一性.The existence and uniqueness of the solutions to high-dimensional McKean-Vlasov stochastic differential equations with discontinuous drift coefficients and corresponding particle systems,were investigated.With the drift coefficient being piecewise Lipschitz continuous about the space variable,through Zvonkin’s transformation,the original equation was converted into a new McKean-Vlasov stochastic differential equation with Lipschitz continuous coefficients.Therefore,the new equation has a unique solution.Moreover,the existence and Lipschitz continuity of the inverse function were proven according to the transformation function characteristics.Finally,based on the It’s formula and the inverse function characteristics,the existence and uniqueness of the solutions to the McKean-Vlasov stochastic differential equation and the corresponding particle system were obtained.

关 键 词：高维McKean-Vlasov随机微分方程 粒子系统 逐段Lipschitz连续 Zvonkin变换 解的存在唯一性 

分 类 号：O211.63[理学—概率论与数理统计]