A bivariate markov process with diffusion and discrete components

Abstract

Let (X (t)Y (t)) be a homogeneous Markov process assuming values in R x S, where S is a finite or countable set, identified by the positive integers. The transition probabilities are described by means of a matrix-valued function P (t; x A), whose entry of index (i j) represents P (X (t)E A Y (t) = j| X (0) =x Y (0) =i). The infinitesimal generator is determined. Under general conditions it is shown that the process has a stochastic representation in terms of a family of diffusionsX _i (t), each having a finite but random lifetime, such that the component X( t) behaves like the diffusion X _i( t) during the stay of Y (t) in state i; and the duration of Y in i coincides with the lifetime of the associated process X _i. The distribution of the time of the first passage of the Y-process out of a given set of states is determined. Explicit results are obtained in the special but important case where the coefficient functions of the generator are piecewise constant. The model is shown to be applicable to describing the progression of HIV in an infected individual.