Stackelberg Game Approach to Mixed Stochastic H_2/H_∞ Control for Mean-Field Jump-Diffusions Systems

In this paper, we address the mixed stochastic H_2/H_∞ control problem for a mean-field jump-diffusion system, where the state equation is influenced by both a standard Brownian motion and a Poisson random martingale measure. The control input is treated as a leader, while the disturbance is considered a follower, employing the Stackelberg game approach. Our paper makes the following key contributions: (i) Initially, we treat the H_∞ problem as the follower problem. Utilizing the stochastic maximum principle, we obtain the optimal open-loop solution for the follower and derive an explicit feedback representation of the optimal control, associated with the follower problem. This feedback representation is obtained through coupled Riccati differential equations (CRDEs) using the Four-Step Scheme. (ii) Subsequently, we address the H_2 problem as the leader problem. By applying the convex variational method, we establish the mean-field stochastic maximum principle. However, due to technical complexities related to the jump process, we consider two distinct situations. In both cases, we introduce new state variables and effectively employ the Four-Step Scheme to obtain the explicit feedback representation of the optimal control for the leader problem. (iii) We demonstrate that the open-loop Stackelberg equilibrium point can be characterized by a feedback representation, which incorporates both the state and its expected value. (iv) To enhance the practicality of our model, we present an example of a PA problem involving moral hazard, illustrating the corresponding optimal contract and optimal strategy. Therefore with our proposed methodologies and solutions, we contribute valuable insights into the mixed stochastic H_2/H_∞ control problem for mean-field jump-diffusion systems. And our findings offer significant implications for understanding and addressing such complex control problems in various real-world scenarios.