Random sampling for continuous time Markov additive processes

Let us consider a continuous time Markov additive process with cadlag paths and a sequence of random variables which forms a random walk with positive increments. Then, if they are independent, we prove that the continuous time Markov additive process computed at the epochs of the random walk is a discrete time Markov additive process and we provide an expression for the corresponding MA transition function. Moreover we concentrate our attention on the finite environment's state space case and, under further quite general hypotheses, we present some large deviations results.