On fluid model and averaged parameters model for Markov additive processes with finite environment’s state space

We deal with a continuous time Markov additive process where the environment is a Markov chain with finite state space. Moreover we always assume that the environment is irreducible. We consider two further Markov additive processes derived in a suitable way from the original one: the fluid model and the averaged parameters model. We prove inequalities between rate functions which can be related to suitable inequalities between Perron Frobenius eigenvalues. We also present some convergence results for rate functions and Lundberg parameters for both the fluid model and the averaged parameters model. A final section is devoted to the discrete time case