Random recurrence equations and ruin in a Markov-dependent stochastic economic environment

We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the $\operatorname {ARCH}(1)$ and $\operatorname {GARCH}(1,1)$ time series models. Our results build upon work of Goldie [Ann. Appl. Probab. 1 (1991) 126--166], who has developed tail asymptotics applicable for independent sequences of random variables subject to a random recurrence equation. In contrast, we adopt a general approach based on the theory of Harris recurrent Markov chains and the associated theory of nonnegative operators, and meanwhile develop certain recurrence properties for these operators under a nonstandard "G\"artner--Ellis" assumption on the driving process.