Monte Carlo simulation for estimating rare event probabilities and parameters in Markov process models

TL;DR: This work considers a simulation technique to estimate very small entrance probabilities for Markov processes by splitting sample paths at various level surfaces of the so called importance function before reaching the set of interest, and proposes a new variant, called fixed number of successes.

Abstract: Markov processes models appear in many different fields, for example in queueing theory, epidemic models, stochastic kinetics of chemical reactions or financial engineering. Often, simple algorithms exist to generate sample paths of this processes. This makes analysis based on simulation, so called Monte Carlo methods, very attractive. We study two specific estimation problems in this context. First we consider a simulation technique, called importance splitting, to estimate very small entrance probabilities for Markov processes by splitting sample paths at various level surfaces of the so called importance function before reaching the set of interest. This can be done in many ways, yielding different variants of the method. In this context, we propose a new one, called fixed number of successes. We prove unbiasedness for the new and some known variants, because in many papers, the proof is based on an incorrect argument. Further, we analyze its behavior in a simplified setting, which is appropriate if the importance function is well chosen, in terms of efficiency and asymptotics in comparison to the standard variant. The main difference is that the new variant controls the precision of the estimator rather than the computational effort. Our analysis and simulation examples show that it is rather robust in terms of parameter choice and we present a two-stage procedure which also yields confidence intervals. The choice of the importance function which governs the placement