A logarithmic efficient estimator of the probability of ruin with recuperation for spectrally negative Lévy risk processes

This article provides an importance sampling algorithm for computing the probability of ruin with recuperation of a spectrally negative Lévy risk process with light-tailed downwards jumps. Ruin with recuperation corresponds to the following double passage event: for some t∈(0,∞), the risk process starting at level x∈[0,∞) falls below the null level during the period [0,t] and returns above the null level at the end of the period t. The proposed Monte Carlo estimator is logarithmic efficient, as t,x→∞, when y=t/x is constant and below a certain bound.