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 ARCH(1) and GARCH(1,1) time series models.  Our results build upon work of Goldie, 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 nonstandart "Gartner-Ellis" assumption on the driving process.