Weak solutions of mean-field stochastic differential equations. Issue 3 (4th May 2017)

• Record Type: Journal Article
• Authors: Li, Juan; Min, Hui
• Is Part Of: Stochastic analysis and applications. Volume 35:Issue 3(2017)
• Page Start: 542
• Page End: 568
• Publication Date: 2017-05-04
• Subjects: Weak solution -- uniqueness in law -- mean-field stochastic differential equations -- local martingale problem; 65C30; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2017.1278706
• Languages: English
• ISSNs: 0736-2994

Abstract

ABSTRACT: In this article, we study weak solutions of mean-field stochastic differential equations (SDEs), also known as McKean–Vlasov equations, whose drift, and diffusion coefficient depend not only on the state process Xs but also on its law. We suppose that b and σ are bounded and continuous in the state as well as the probability law; the continuity with respect to the probability law is understood in the sense of the 2-Wasserstein metric. Using the approach through a local martingale problem, we prove the existence and the uniqueness in law of the weak solution of mean-field SDEs. The uniqueness in law is obtained if the associated Cauchy problem possesses for all initial condition a classical solution. However, unlike the classical case, the Cauchy problem is a mean-field PDE as recently studied by Buckdahn et al. [arXiv:1407.1215, 2014]. In our approach, we also extend the Itô formula associated with mean-field problems given by Buckdahn et al. to a more general case of coefficients.