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Ruin probability with claims modeled by a stationary ergodic stable process. (English) Zbl 1044.60028
Summary: For a random walk with negative drift we study the exceedance probability (ruin probability) of a high threshold. The steps of this walk (claim sizes) constitute a stationary ergodic stable process. We study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significantly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit long-range dependence. The proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a Poisson process.

MSC:
60G10 Stationary stochastic processes
60G52 Stable stochastic processes
91B30 Risk theory, insurance (MSC2010)
60E07 Infinitely divisible distributions; stable distributions
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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