  This article concerns the tail probabilities of a light-tailed
Markov-modulated L\'evy process stopped at a state-dependent Poisson rate. The
tails are shown to decay exponentially at rates given by the unique positive
and negative roots of the spectral abscissa of a certain matrix-valued
function. We illustrate the use of our results with an application to the
stationary distribution of wealth in a simple economic model in which agents
with constant absolute risk aversion are subject to random mortality and income
fluctuation.
