  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.
