A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling
Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X-epsilon(t)) : 8 > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let psi(epsilon)(u),(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X-epsilon(t)) as epsilon -> 0. As a consequence, we give exact asymptotics for log psi(epsilon)(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of psi(epsilon)(u).
risk processes large deviations importance sampling ruin probabilities poisson shot noise