A process with chain dependent growth rate

Abstract

Additive processes on finite Markov chains have been investigated by Miller ([8], [9]), Keilson and Wishart ([2], [3], [4]) and by Fukushima and Hitsuda [1]. These papers study a two-dimensional Markov Process {X(t), R(t)} whose state space is R₁ × {1, 2, ···, R} characterized by the following properties: (i)R(t) is an irreducible Markov chain on states 1,2, …,R governed by atransition probability matrix B_o = {b_rs}. (ii)X(t) is a sum of random increments dependent on the chain, i.e., if the ith transition takes the chain from state r to state s, then the increment has the distribution (iii)N_t, is t in discrete time while in the continuous time case N_t, might be an independent Poisson process.