In this paper, a large class of time-varying Riccati equations arising in stochastic dynamic games is considered. The problem of the existence and uniqueness of some globally defined solution, namely the bounded and stabilizing solution, is investigated. As an application of the obtained existence results, we address in a second step the problem of infinite-horizon zero-sum two players linear quadratic (LQ) dynamic game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such an optimal control problem, a crucial role is played by the unique bounded and stabilizing solution of the considered class of generalized Riccati equations.

中文翻译：

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零和随机差分博弈中广义Riccati方程稳定解的存在性：时变情况

在本文中，考虑了在随机动态博弈中出现的一大类时变 Riccati 方程。研究了一些全局定义的解，即有界解和稳定解的存在唯一性问题。作为所获得的存在结果的应用，我们在第二步中解决了随机离散时间动态系统的无限范围零和双玩家线性二次 (LQ) 动态博弈问题，该系统的系数随机切换和乘法噪声。我们表明，在这种最优控制问题的解决方案中，所考虑的广义 Riccati 方程类的唯一有界和稳定解起着至关重要的作用。