Matrix calculations for inhomogeneous Markov reward processes, with applications to life insurance and point processes

TL;DR

This paper introduces a matrix-based method for analyzing inhomogeneous Markov reward processes, enabling explicit calculation of moments, distributions, and other properties in complex multi-state life insurance models, simplifying classical differential equation approaches.

Contribution

It develops a general matrix-oriented framework for Markov reward processes, providing explicit formulas for moments and distributions, and streamlining classical methods in life insurance modeling.

Findings

Explicit formulas for moments using matrix exponentials

Framework simplifies complex multi-state models

Methods for distribution and quantile calculations

Abstract

A multi--state life insurance model is naturally described in terms of the intensity matrix of an underlying (time--inhomogeneous) Markov process which describes the dynamics for the states of an insured person. Between and at transitions, benefits and premiums are paid, defining a payment process, and the technical reserve is defined as the present value of all future payments of the contract. Classical methods for finding the reserve and higher order moments involve the solution of certain differential equations (Thiele and Hattendorf, respectively). In this paper we present an alternative matrix--oriented approach based on general reward considerations for Markov jump processes. The matrix approach provides a general framework for effortlessly setting up general and even complex multi--state models, where moments of all orders are then expressed explicitly in terms of so--called product integrals (matrix--exponentials) of certain matrices. As Thiele and Hattendorf type of theorems can be retrieved immediately from the matrix formulae, this methods also provides a quick and transparent approach to proving these classical results. Methods for obtaining distributions and related properties of interest (e.g. quantiles or survival functions) of the future payments are presented from both a theoretical and practical point of view (via Laplace transforms and methods involving orthogonal polynomials).