Fully Coupled Nonlinear Forward-Backward Stochastic Difference Equations: Spectral Contraction and Infinite-Horizon Maximum Principle

Abstract

This paper develops an explicit spectral-contraction approach to studying fully coupled nonlinear forward--backward stochastic difference equations on infinite horizon and their applications to stochastic control. The estimates for this fully coupled system are assembled into an explicit two-dimensional nonnegative matrix. Its spectral radius yields an explicit sufficient discount threshold, and a corresponding equivalent weighted product norm is constructed to establish the contraction property. Moreover, for an infinite-horizon control problem with convex control constraints and an accumulated discounted running cost, we derive a Pontryagin-type stochastic maximum principle, its equivalent pointwise normal-cone formulation, and a verification theorem. Finally, a recursive risk-adjusted portfolio example is given to demonstrate the applications of our theoretical results, and a projection-type sufficient optimality condition for this example is derived.