Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence

Abstract

We consider a continuous-time Markov additive process (J_t,S_t) with (J_t) an irreducible Markov chain on E = {1,...,s}; it is known that (S_t/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ^* and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (J_t,S_t): the averaged parameters model (J_t,S_t^(A)) and the fluid model (J_t,S_t^(F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (J_t,S_t^(A)) and (J_t,S_t^(F)) are faster than the corresponding convergences for (J_t,S_t).