A maximum principle for fully coupled controlled forward–backward stochastic difference systems of mean-field type

Abstract

Abstract In this paper, we consider the optimal control problem for fully coupled forward–backward stochastic difference equations of mean-field type under weak convexity assumption. By virtue of employing a suitable product rule and formulating a mean-field backward stochastic difference equation, we establish the stochastic maximum principle and also derive, under additional assumptions, that the stochastic maximum principle is also a sufficient condition. As an application, a Stackelberg game of mean-field backward stochastic difference equation is presented to demonstrate our results.

Keywords

• Forward–backward stochastic difference equations
• Backward stochastic difference equations
• Mean-field theory
• Stochastic maximum principle
• Adjoint difference equation