MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATIONS AND ASSOCIATED PDES

• 成果类型: Article
• 署名作者: Buckdahn, Rainer; Li, Juan; Peng, Shige; Rainer, Catherine
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Bretagne Occidentale; Shandong University; Shandong University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1076
• 发表日期: 2017
• 页码: 824-878
• 关键词: mckean-vlasov; propagation; chaos; DYNAMICS; limit

摘要

In this paper we consider a mean-field stochastic differential equation, also called Mc Kean-Vlasov equation, with initial data (t,x)is an element of[0,T]xR(d), which coefficients depend on both the solution X-s(t,x) but also its law. By considering square integrable random variables xi as initial condition for this equation, we can easily show the flow property of the solution X-s(t,xi) of this new equation. Associating it with a process X-s(t,x,P xi) which coincides with X-s(t,xi), when one substitutes xi for x, but which has the advantage to depend on xi only through its law P xi, we characterise the function V(t,x,P xi)=E[phi(X-T(t,x,P xi),P-XTt,P-xi)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a non local PDE of mean-field type, involving the first and second order derivatives of V with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first and second order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding It\<^>{o} formula. In our approach we use the notion of derivative with respect to a square integrable probability measure introduced in \cite{PL} and we extend it in a direct way to second order derivatives.