Stackelberg Game Approach to Mixed Stochastic H2/H∞ Control for Mean‐Field Jump‐Diffusions Systems With Markovian Switching

ABSTRACT This article focuses on the mixed stochastic control problem for mean‐field jump‐diffusion systems with Markovian switching. We investigate the stationarity condition and state‐feedback representations of the open‐loop optimal solution using the Stackelberg approach, treating the criterion as the follower problem and the criterion as the leader problem. To achieve this, we develop the Markov‐type stochastic maximum principle to obtain the open‐loop optimal solutions for both the follower and leader problems. Furthermore, we extend the Markov‐type Four‐Step Scheme and use coupled Riccati differential equations to derive state feedback representations of the optimal control for the follower and leader. To incorporate the impact of Markovian switching and Poisson jump processes, we introduce the application of conditional mean fields, providing a more comprehensive and accurate solution analysis. Finally, we establish the feedback representation of the open‐loop Stackelberg equilibrium, considering both the solvability of the leader and follower problems, and incorporating the state and its expected value.