Fully Coupled Nonlinear FBS$\mathrm{\Delta }$Es: Solvability and LQ Control Insights

This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBS$\mathrm{\Delta }$Es). Building on insights from linear quadratic optimal control problems, we introduce a more relaxed framework of domination-monotonicity conditions specifically designed for discrete systems. Utilizing these conditions, we apply the method of continuation to demonstrate the unique solvability of the fully coupled FBS$\mathrm{\Delta }$Es and derive a set of solution estimates. Moreover, our results have considerable implications for various related linear quadratic (LQ) problems, particularly where stochastic Hamiltonian systems are aligned with the FBS$\mathrm{\Delta }$Es meeting these introduced domination-monotonicity conditions. As a result, solving the associated stochastic Hamiltonian systems allows us to derive explicit expressions for the unique optimal controls.