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Stochastic differential game of functional forward-backward stochastic system and related path-dependent HJBI equation

This paper is devoted to a stochastic differential game of functional forward-backward stochastic differential equation (FBSDE, for short). The associated upper and lower value functions of the stochastic differential game are defined by controlled functional backward stochastic differential equations (BSDEs, for short). Applying the Girsanov transformation method introduced by Buckdahn and Li [1], the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP, for short), the upper and the lower value functions are shown to be the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.