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

TL;DR

This paper develops a maximum principle for nonlinear fully coupled forward-backward stochastic difference equations and applies it to linear quadratic control problems, providing new theoretical insights.

Contribution

It introduces a variational formula and necessary and sufficient optimality conditions for nonlinear coupled FBSΔEs, extending control theory in stochastic difference equations.

Findings

Established a variational formula for the cost functional.

Derived Pontryagin maximum principle for coupled FBSΔEs.

Applied results to a linear quadratic control problem.

Abstract

This paper investigates the optimal control problem for a class of nonlinear fully coupled forward-backward stochastic difference equations (FBS$\Delta$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.