A process with chain dependent growth rate

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 isR1× {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 matrixBo= {brs}. (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) Nt, is t in discrete time while in the continuous time case Nt, might be an independent Poisson process.