The study of time-inhomogeneous Markov jump processes is a traditional topic within probability theory that has recently attracted substantial attention in various applications. However, their flexibility also incurs a substantial mathematical burden which is usually circumvented by using well-known generic distributional approximations or simulations. This article provides a novel approximation method that tailors the dynamics of a time-homogeneous Markov jump process to meet those of its time-inhomogeneous counterpart on an increasingly fine Poisson grid. Strong convergence of the processes in terms of the Skorokhod $J_1$ metric is established, and convergence rates are provided. Under traditional regularity assumptions, distributional convergence is established for unconditional proxies, to the same limit. Special attention is devoted to the case where the target process has one absorbing state and the remaining ones transient, for which the absorption times also converge. Some applications are outlined, such as univariate hazard-rate density estimation, ruin probabilities, and multivariate phase-type density evaluation.

翻译：时间非齐次马尔可夫跳过程的研究是概率论中的一个传统课题，近年来已在多种应用中引发广泛关注。然而，其灵活性也带来了显著的数学负担，通常通过使用通用的分布近似或模拟方法来规避。本文提出一种新颖的近似方法，通过在逐渐精细化的泊松网格上调整时间齐次马尔可夫跳过程的动态特性，使其匹配非齐次对应过程。基于Skorokhod $J_1$ 度量建立了过程的强收敛性，并给出了收敛速率。在传统正则性假设下，证明了无条件代理变量到相同极限的分布收敛性。特别关注目标过程具有一个吸收态而其余为瞬态的情形，此时吸收时间也收敛。文中概述了一些应用，例如单变量风险率密度估计、破产概率以及多元相位型密度评估。