Approximations of semi-Markov processes and insurance policy valuation

Abstract

Inspired by a duration-dependent life insurance model, we consider continuous-time semi-Markov jump processes, initially assumed to have a finite state-space. We develop approximations using jump processes that are time-homogeneous Markov, conditioned on a high-intensity Poissonian grid (grid-conditional). Our results are based on a recent adaptation of the uniformization principle, which yields a strongly pathwise convergent sequence of jump processes. In contrast to traditional methods that use classical approximations to integro-differential equation solutions to compute value functions, our approximations result in easily implementable expressions, making them valuable in situations where evaluating pathwise distributional functionals for the original semi-Markov process is challenging. Our homogeneous approximation, initially grid-conditional, evolves into an unconditional version that remains effective under reasonable regularity assumptions. We then relax the finite state-space assumption and show how our results can be extended to a general measurable state-space. We illustrate the practicality of our approach with a disability insurance model, using realistic underlying semi-Markov process parameters.