WEAK SOLUTIONS OF MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATION TO ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES

This work is devoted to the study of stochastic differential equations (SDEs) whose diffusion coefficient sigma(s, X. (boolean AND s)) is Lipschitz continuous with respect to the path of the solution process X, while its drift coefficient b(s, X. (boolean AND s), QX(s)) is only measurable with respect to X and depends continuously (in the sense of the 1-Wasserstein metric) on the law of the solution process. Embedded in a mean-field game, the weak existence for such SDEs with mean-field term was recently studied by Lacker [Stochastic Process. Appl., 125 (2015), pp. 2856-2894] and Carmona and Lacker [Ann. Appl. Probab., 25 (2015), pp. 1189-1231] under only sequential continuity of b(s, X. (boolean AND s), QX(s)) in Q(X) with respect to a weak topology, but for uniqueness, Carmona and Lacker supposed that b is independent of the mean-field term. We prove the uniqueness in law for b depending on the mean-field, and the proof of the existence of a weak solution, relatively short in comparison with Carmona and Lacker's work, is extended in section 5 of this paper to the study of 2-person zero-sum stochastic differential games described by doubly controlled coupled mean-field forward-backward SDEs with dynamics whose drift coefficient is only measurable with respect to the state process.