Linear Quadratic Stochastic Differential Games with Poisson Jumps and Their Application to Robust Control

In this paper, we investigate a class of linear quadratic stochastic differential games with a Poisson jumps diffusion process, including the Nash equilibrium strategies of a nonzero sum game and the saddle point equilibrium strategies of a zero sum game. Utilizing the maximum principle for differential games, we determine that the existence conditions of the Nash equilibrium strategies are equivalent to the solution for two cross-coupled matrix Riccati equations, and that the existence conditions of the saddle point equilibrium strategies are equivalent to the solution for a matrix Riccati equation. We also provide explicit expressions for the equilibrium strategy and the optimal performance functional value. Finally, we apply the obtained results to problems dealing with stochastic H₂/H_∞ control and stochastic H_∞ control in the fields of modern robust control theory, and obtain the existence conditions of robust control strategies and their explicit expressions. Moreover, we verify the performance of these results in a financial market portfolio optimisation problem.