The paper is focused on the simulation approach based on the Monte Carlo method. It is used to estimate the probabilities of rare events. Fast simulation based on a splitting method is applied to minimize the variance of the estimator. With this technique (called REpetitive Simulation Trails After Reaching Thresholds – RESTART), the space state is partitioned into a series of nested subsets, and the rare events are considered in this context as a nested sequence of events. When a given subset is entered by a sample trajectory, random retrials are generated from the initial state corresponding to the state of the system at the entry point. Thus, the system trajectory is split into a number of new subtrajectories.
A simple model of splitting is build to introduce a new estimator and to analyze the behavior of the rare event probability. An optimization of the proposed algorithm is accomplished, and a precise confidence interval of the estimator is obtained, using a branching process. Finally, a discussion of the merits of the proposed approach and further research direction are discussed and overall conclusions are presented.