No abstract

In this paper we study zero-sum two-player stochastic differential games with the help of theory of Backward Stochastic Differential Equations (BSDEs). At the one hand we generalize the results of the pioneer work of Fleming and Souganidis [8] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game (which implies that the cost functionals become random variables), on the other hand the application of BSDE methods, in particular that of the notion of stochastic "backward semigroups" introduced by Peng [14] allows to prove a dynamic programming principle for the upper and the lower value functions of the game in a straight-forward way, without passing by additional approximations. The upper and the lower value functions are proved to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method (Peng [14]) is translated from the framework of stochastic control theory into that of stochastic differential games.

In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Ann. Probab. (2007) (in press). Available online: http://www.imstat.org/aop/future papers. htm] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of "particles" (or "agents"). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we study a mean-field BSDE in a Markovian framework, associated with a McKean-Vlasov forward equation. By combining classical BSDE methods, in particular that of "backward semigroups" introduced by Peng ) (in Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions.