Kolmogorov’s forward PIDE and forward transition rates in life insurance

Abstract

We consider a doubly stochastic Markov chain, where the transition intensities are modelled as diffusion processes. Here we present a forward partial integro-differential equation (PIDE) for the transition probabilities. This is a generalisation of Kolmogorov’s forward differential equation. In this set-up, we define forward transition rates, generalising the concept of forward rates, e.g. the forward mortality rate. These models are applicable in e.g. life insurance mathematics, which is treated in the paper. The results presented follow from the general forward PIDE for stochastic processes, of which the Fokker–Planck differential equation and Kolmogorov’s forward differential equation are the two most known special cases. We end the paper by considering the semi-Markov case, which can also be considered a special case of a general forward partial integro-differential equation.