Maximum Principle for Optimal Control of Infinite Horizon Stochastic Difference Equations Driven by Fractional Noises

In this paper, infinite horizon stochastic difference equations and backward stochastic difference equations with fractional noise are studied. The main difficulty comes from the self-dependence of fractional noise on an infinite horizon. By introducing the infinite horizon stochastic difference equations and backward stochastic difference equations with fractional noise, the stochastic maximum principle for the discrete-time control problem driven by fractional noise on an infinite horizon is proved. As an application, an optimal consumption and investment problem is solved.