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

Jeffrey F. Collamore

doi:10.1214/08-aap584

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arXiv · 0908.4479

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

Jeffrey F. Collamore

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Abstract

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 "Gärtner--Ellis" assumption on the driving process.

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Jeffrey F. Collamore. 2009-08-31. Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. https://doi.org/10.1214/08-aap584

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