带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈，包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理，得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解，鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论，并给出了均衡策略的显式表达及最优性能泛函值．最后，将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题，得到了鲁棒控制策略的存在条件及显式表达，并验证所得结果在金融市场投资组合优化问题中的应用．

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.