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

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 $\mathrm{ARCH}(1)$ and $\mathrm{GARCH}(1,1)$ 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.

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

References in corpus (2)

• A survey of max-type recursive distributional equations
• Efficient importance sampling for Monte Carlo evaluation of exceedance probabilities

Cited by in corpus (12)

• Ruin models with investment income
• Interplay of insurance and financial risks in a discrete-time model with strongly regular variation
• Determination of Pareto exponents in economic models driven by Markov multiplicative processes
• Rare event simulation for processes generated via stochastic fixed point equations
• Stability of perpetuities in Markovian environment
• Asymptotic behaviour of ruin probabilities in a general discrete risk model using moment indices
• Pemantle's min-plus binary tree
• Large deviation estimates for exceedance times of perpetuity sequences and their dual processes
• Relative growth of the partial sums of certain random Fibonacci-like sequences
• Efficient Rare-event Simulation for Perpetuities
• Tail estimates for stochastic fixed point equations via nonlinear renewal theory
• Random linear recursions with dependent coefficients