Fractional Inhomogeneous Multi-state Models in Life Insurance

TL;DR

This paper develops a matrix calculus framework for fractional inhomogeneous Markov models in life insurance, extending classical models to include non-Markovian features like Mittag-Leffler distributions, capturing complex regime-dependent and memory effects.

Contribution

It introduces a transparent matrix-based analysis of fractional inhomogeneous Markov models, extending to non-Markovian cases with Mittag-Leffler distributed sojourns and fractional phase-type absorption times.

Findings

Matrix generalizations of scalar distributions for absorption times.

Extension to non-Markovian models with Mittag-Leffler sojourns.

Inclusion of regime-dependent and history-influenced processes.

Abstract

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.