Pathwise and distributional approximations of semi-Markov processes

Abstract: Continuous-time semi-Markov finite state-space jump processes are considered, inspired by a duration-dependent life insurance model. New approximations using grid-conditional homogeneous Markov jump-processes are developed, based on a recent adaptation of the uniformization principle which results in a strong pathwise convergent sequence of jump processes. Unlike traditional methods that use classical approximations to integro-differential equation solutions to compute their value functions, the proposed grid-conditional homogeneous Markov jump-processes allows for a direct and tractable approximation. In particular, these approximations simplify to easily implementable expressions, making them useful in areas where evaluating pathwise distributional functionals is difficult. Our homogeneous approximation, initially of a grid-conditional kind, is evolved into an unconditional version that holds well under fair regularity assumptions. The practicality of this approach is demonstrated on a disability life insurance model, with realistic underlying semi-Markov process parameters, showcasing its broader applicability in operations research and related fields.