Maximum Principle for Stochastic Control System with Elephant Memory and Jump Diffusion

Motivated by a duopoly game problem, the authors study an optimal control problem where the system is driven by Brownian motion and Poisson point process and has elephant memory for the control variable and the state variable. Firstly, the authors establish the unique solvability of an anticipated backward stochastic differential equation, derive a stochastic maximum principle, and prove a verification theorem for the aforementioned optimal control problem. Furthermore, the authors generalize these results to nonzero-sum stochastic differential game problems. Finally, the authors apply the theoretical results to the duopoly game problem and obtain the corresponding Nash equilibrium solution.