Stochastic maximum principle in the mean-field controls

In Buckdahn, Djehiche, Li, and Peng (2009), the authors obtained mean-field Backward Stochastic Differential Equations (BSDEs) in a natural way as a limit of some highly dimensional system of forward and backward SDEs, corresponding to a great number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying their stochastic maximum principle. This paper studies the stochastic maximum principle (SMP) for mean-field controls, which is different from the classical ones. This paper deduces an SMP in integral form, and it also gets, under additional assumptions, necessary conditions as well as sufficient conditions for the optimality of a control. As an application, this paper studies a linear quadratic stochastic control problem of mean-field type.