Asymptotics of the ruin probability with claims modeled by α-stable aggregated AR(1) process

We study the asymptotics of the ruin probability in a discrete time risk insurance model with stationary claims following the aggregated heavy-tailed AR(1) process discussed in Puplinskaite and Surgailis (2010). The present work is based on the general characterization of the ruin probability with claims modeled by stationary alpha-stable process in Mikosch and Samorodnitsky (2000). We prove that for the aggregated AR(1) claims process, the ruin probability decays with exponent alpha(1-H), where H in [1/alpha, 1) is the asymptotic self-similarity index of the claim process, 1 alpha 2. This result agrees with the decay rate of the ruin probability with claims modeled by increments of linear fractional motion in Mikosch and Samorodnitsky (2000) and also with other characterizations of long memory of the aggregated AR(1) process with infinite variance in Puplinskaite and Surgailis (2010).