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

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

2009-08-31

arxiv.org/abs/0908.4479v1

Annals of Applied Probability 2009, Vol. 19, No. 4, 1404-1458

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP584 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP584

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.

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