Markov Additive Processes II. Large Deviations

Abstract

Let $\{(X_n, S_n); n = 0,1,\ldots\}$ be a Markov additive process, $\{X_n\}$ taking values in a general state space $\mathbb{E}$, while $\{S_n\} \subset \mathbb{R}^d$. The large deviation principle is shown to hold for $P_x\{(X_n, S_n) \in A \times n\Gamma\}, A \subset \mathbb{E}, \Gamma \subset \mathbb{R}^d$, the upper bound holding for closed sets $\Gamma$, the lower bound for open sets. The only hypothesis for the lower bound is irreducibility of $\{X_n\}$, and nonsingularity of $\{S_n\}$. The rate function is characterized in terms of the transform kernel of $P_x$.