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.

本文考虑了一类在随机动态博弈中产生的时变Riccati方程。研究了一些全局定义的解，即有界和稳定解的存在性和唯一性问题。作为获得的存在结果的一种应用，我们在第二步骤中解决了随机离散时间动力系统的无限水平零和两个参与者线性二次（LQ）线性博弈（LQ）动态博弈的问题，该动态博弈的系数和乘法噪声。我们表明，在这种最优控制问题的解决方案中，考虑的一类广义Riccati方程的唯一有界和稳定解在其中扮演着至关重要的角色。