In this paper, we consider infinite horizon linear-quadratic (LQ) Nash games for stochastic differential equations (SDEs) with infinite Markovian jumps and (x,u,v)-dependent noise. An indefinite stochastic LQ result is first derived for the considered system. Then, under the condition of strong detectability, a necessary and sufficient condition for the existence of a Nash equilibrium is put forward in terms of the solvability of a countably infinite set of coupled generalized algebraic Riccati equations (ICGAREs). Moreover, the mixed H₂/H_∞ control is investigated by Nash game approach as an important application. At last, we present an iterative algorithm to solve the ICGAREs, and a numerical simulation is given to illustrate its efficiency.

中文翻译：

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无限跳跃的 SDE 的无限地平线 LQ Nash 游戏

在本文中，我们考虑了具有无限马尔可夫跳跃和 ( x , u , v ) 相关噪声的随机微分方程 (SDE) 的无限水平线线性二次 (LQ) 纳什博弈。首先为所考虑的系统推导出不确定的随机 LQ 结果。然后，在强可检测性的条件下，根据可数无穷大耦合广义代数Riccati方程组（ICGAREs）的可解性，提出了纳什均衡存在的充要条件。此外，混合H ₂ / H _∧控制被纳什博弈方法研究为一个重要的应用。最后，我们提出了求解ICGAREs的迭代算法，并给出了数值模拟来说明其效率。