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

This paper investigates the optimal control problem for a class of nonlinear fully coupled forward-backward stochastic difference equations (FBS$\mathrm{\Delta }$Es). Under the convexity assumption of the control domain, we establish a variational formula for the cost functional involving the Hamiltonian and adjoint system. Both necessary and sufficient conditions for optimal control are derived using the Pontryagin maximum principle. As an application, we present a linear quadratic optimal control problem to illustrate our theoretical results.