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 {(Xε(t)) : ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X(t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψε(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 Ψε(u).