Ruin probability with claims modeled by a stationary ergodic stable process

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 i n this situation and evaluate the asymptotic behavior of the ruin probabili ty for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significan tly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit long-range dependence. The proofs exploit lar ge deviation techniques for sums of dependent stable random variables and t he series representation of a stable process as a function of a Poisson pro cess.