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On asymptotically efficient simulation of ruin probabilities in a Markovian environment. (English) Zbl 0755.62080

Summary: Let \(\xi_ 1,\xi_ 2,\dots\) be random variables which arise as the additive component of a Markov additive process and let \(S_ n=\xi_ 1+\xi_ 2+\dots+\xi_ n\), \(n\geq 1\). Fix \(M > 0\) and let \(T_ M\) be the first index \(n\) so that \(S_ n > M\) (\(T_ M = \infty\) if \(S_ n \leq M\) for all \(n\)). We consider the estimation of the probability \(\mathbb{P}(T_ M < \infty)\) 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 \((M \to \infty)\) simulation kernel. The result is applied to a ruin problem.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
65C05 Monte Carlo methods
60F10 Large deviations
60J27 Continuous-time Markov processes on discrete state spaces
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