Strongly convergent homogeneous approximations to inhomogeneous Markov jump processes and applications

The study of 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 - even in simple settings - 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 homogeneous Markov jump process to meet those of its inhomogeneous counterpart on an increasingly fine Poisson grid - a procedure broadly known as uniformization. Strong convergence of the processes in terms of the Skorokhod $J_1$ topology is established, and convergence rates are provided. 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 ruin probability calculation, univariate hazard-rate density estimation, and multivariate phase-type density evaluation. Finally, extensions to semi-Markovian models are presented.