Abstract A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. Asmussen (Stochastic Models, 1995) proved that such distributions can be obtained by solving a suitable quadratic matrix equation (QME), while, more recently, Ahn and Ramaswami (Stochastic Models, 2017) derived the distributions from the solution of a suitable algebraic Riccati equation (NARE). In this paper we provide an explicit algebraic relation between the QME and the NARE, based on a linearization of a matrix polynomial. Moreover, we discuss the doubling algorithms such as the structure-preserving doubling algorithm (SDA) and alternating-directional doubling algorithm (ADDA), with shifting technique, which are used for finding the sought of the NARE.

中文翻译：

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马尔可夫调制布朗运动中的矩阵方程：理论性质和数值解

摘要 马尔可夫调制布朗运动 (MMBM) 是经典布朗运动的实质推广，它是通过允许布朗参数由基础马尔可夫环境链调制而获得的。与布朗运动一样，一旦确定了级别之间的第一次通过时间的分布，MMBM 的平稳分析就变得容易了。Asmussen（随机模型，1995 年）证明可以通过求解合适的二次矩阵方程 (QME) 来获得此类分布，而最近，Ahn 和 Ramaswami（随机模型，2017 年）从合适的代数 Riccati等式（NARE）。在本文中，我们基于矩阵多项式的线性化，提供了 QME 和 NARE 之间的显式代数关系。而且，