Future Expectations Modeling, Random Coefficient Forward–Backward Stochastic Differential Equations, and Stochastic Viscosity Solutions

In this paper we study a class of infinite horizon fully coupled forward–backward stochastic differential equations (FBSDEs) with random coefficients that are stimulated by various continuous time future expectations models. Under standard Lipschitz and monotonicity conditions and by means of the contraction mapping principle, we establish existence and uniqueness of an adapted solution, and we obtain results regarding the dependence of this solution on the data of the problem. Making further the connection with finite horizon quasilinear backward stochastic partial differential equations via a generalization of the well known four-step-scheme, we are led to the notion of stochastic viscosity solutions. As an application of this framework, we also provide a stochastic maximum principle for the optimal control problem of such FBSDEs, which in the linear-quadratic Markovian case boils down to the solvability of an infinite horizon fully coupled system of forward-backward Ricatti differential equations.