发表机构
University of Padua; Wrocław University of Science and Technology（帕多瓦大学; 弗罗茨瓦夫科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有限状态马尔可夫调制加性增加乘性减少过程的一侧、两侧及反射退出问题，通过首次跳跃分解、拉普拉斯变换和递归方法推导退出矩阵，并构造相关杜布鞅。
AI 中文摘要
我们研究有限状态马尔可夫调制加性增加与乘性减少过程的一侧、两侧及反射退出问题。对于向上穿越，首次跳跃分解与空间拉普拉斯变换导出一个由有限相位词索引的递归及一个收敛的格林级数。对于向下穿越，首次跳跃算子为严格压缩，且来自 $[p_i b,b]$ 的已知贡献在同一空间递归中成为显式边界强迫项。这些结果允许推导首达矩阵的相应格林表示及其有限维相容性条件。两侧退出恒等式的推导基于马尔可夫性质，在此背景下等价于拉普拉斯变换的余循环性质，并结合上层递归方程。对于在运行下确界处的反射，一种有限的 $\ar p$-几何步长方法实现某个皮卡映射，从而求得退出矩阵。由此，我们构造一个与反射马尔可夫调制AIMD首达时间相关的杜布鞅。
英文摘要
We study one-sided, two-sided and reflected exit problems for a finite-state Markov-modulated additive-increase and multiplicative-decrease process. For upward passage, a first-jump decomposition and a spatial Laplace transform lead to a recursion indexed by finite words of phases and to a convergent Green series. For downward passage, the first-jump operator is a strict contraction and the known contribution from $[p_i b,b]$ becomes an explicit boundary forcing term in the same spatial recursion. These allow to derive the corresponding Green representation of the first passage matrix and its finite-dimensional compatibility condition. The derivation of the two-sided exit identity is based on the Markov property, which in this context is equivalent to the cocycle property of the Laplace transform, together with the recursive equation at the upper level. For reflection at the running infimum, a finite $\bar p$-geometric method of steps realizes a certain Picard map that allows to find the exit matrix. As a result, we construct a Doob martingale related to the first passage time of the reflected Markov-modulated AIMD.
Comments15 page, 4 figures
