On finiteness and tails of perpetuities under a Lamperti-Kiu MAP

Consider a Lamperti-Kiu Markov additive process $(J_t,{\xi }_t:t\ge 0)$ on $\{+,-\}\times \mathbb{R}\cup \mathrm{\infty }$ where $J$ is the modulating Markov chain component. First, we study the finiteness of the exponential functional and then consider its moments and tail asymptotics under Cramer’s condition. In the strong subexponential case we determine the subexponential tails of the exponential functional under some further assumptions.