Fractional inhomogeneous multi-state models in life insurance

In this paper, we demonstrate through the use of matrix calculus a transparent analysis of fractional inhomogeneous Markov models for life insurance where transition matrices commute. The resulting formulae are intuitive matrix generalizations of their single-state counterparts, and the absorption times are matrix versions of well-known scalar distributions. A further advantage of this approach is that it allows extending the analysis to the non-Markovian case where sojourns are Mittag-Leffler distributed, and where the absorption times are fractional phase-type distributed. Considering deterministic time transforms gives rise to fractional inhomogeneous phase-type distributions as absorption times. The latter underlying processes are an example of a regime where not only the present but also the history of a policyholder influences its future evolution. The sub-exponential nature of stable distributions translates into the multi-state insurance model as a random longevity risk at any given state of the chain.