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

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

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H_2/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.

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.