研究带泊松跳的线性Markov切换系统的随机微分博弈问题, 首先在有限时域内, 借助动态规划原理和配方法, 得到了Nash均衡解存在的条件等价于其相应的微分Riccati方程存在解, 并给出了均衡解及最优性能泛函值函数的显式表达. 然后延伸到无限时域进行分析, 得到了Nash均衡解存在的条件等价于其相应的代数Riccati方程存在解. 最后讨论了金融市场中的投资组合的最优化问题, 假设风险资产的价格服从带Markov切换参数的跳扩散过程, 两个投资者在相互竞争的情形下进行非零和随机微分投资博弈, 利用上述结论得到了最优投资组合策略的解.

This paper investigates the stochastic differential game problem of linear Markov switching systems with Poisson jumps. Firstly, the finite time horizon Nash games are discussed, by using the dynamic programming principle and the completion square method, the existence condition of the Nash equilibrium solution is obtained, which is equivalent to the solvability of the associated differential Riccati equations, and the explicit expression of equilibrium solution and optimal cost function are given. Then, we extend the Nash games to the infinite time horizon. For this case, we conclude that the existence condition of Nash equilibrium solution is equivalent to the solvability of the algebraic Riccati equation. Finally, a portfolio optimization problem is discussed in which the price process of the risky asset is assumed to be governed by a jump-diffusion process with Markov switching parameters. By introducing the definition of relative performance, we transform this problem into a nonzero-sum stochastic differential game with two investors in a competitive situation. By using the obtained results of stochastic differential game, the optimal portfolio strategies of the two investors are obtained.