  In this paper we study a class of infinite horizon fully coupled
forward-backward stochastic differential equations (FBSDEs), that are
stimulated by various continuous time future expectations models with random
coefficients. Under standard Lipschitz and monotonicity conditions, and by
means of the contraction mapping principle, we establish existence, uniqueness,
a comparison property and dependence on a parameter of adapted solutions.
Making further the connection with infinite horizon quasilinear backward
stochastic partial differential equations (BSPDEs) via a generalization of the
well known four-step-scheme, we are led to the notion of stationary stochastic
viscosity solutions. A stochastic maximum principle for the optimal control
problem of such FBSDEs is also provided as an application to this framework.
