Optimal control problems subject to forward and backward stochastic difference equations

Abstract Backward and forward stochastic difference equations are two distinct types of difference equations. In this paper, we explore optimal control problems based on both forward and backward stochastic difference equations. Initially, we study optimal control problems grounded in backward stochastic difference equations and present backward recursive equations as a means of resolving such problems. Through solving these equations, we delve into a bang‐bang optimal control problem, offering an analytical expression for its optimal solution. Building upon these findings, we extend our investigation to encompass a linear quadratic optimal control problem, involving both forward and backward stochastic difference equations. Leveraging backward recursive equations, we derive an analytical expression for the optimal solution to the linear quadratic optimal control problem. Finally, we substantiate the validity of our conclusions through a numerical example.