On asymptotically efficient simulation of ruin probabilities in a Markovian environment

Let ξ1,ξ2,... be random variables which arise as the additive component of a Markov additive process and let Sn =ξ1 + ξ2 + … + ξ n ,n⩾1. Fix M > 0 and let TM be the first index n so that Sn > M (TM = ∞ if Sn ⩽ M for all n). We consider the estimation of the probability ℙ(TM < ∞) by using Monte Carlo simulation and especially importance sampling techniques. Allowing a wide class of possible simulation kernels and using a large deviations criterion for asymptotic efficiency we prove a theorem which exposes a unique asymptotically optimal (M →∞) simulation kernel. The result is applied to a ruin problem. Key words: Ruin probabilityMarkov additive processlarge deviations theory