Multilevel splitting for estimating rare event probabilities

We analyze the performance of a splitting technique for the estimation of rare event probabilities by simulation. A straightforward estimator of the probability of an event evaluates the proportion of simulated paths on which the event occurs. If the event is rare, even a large number of paths may produce little information about its probability using this approach. The method we study reinforces promising paths at intermediate thresholds by splitting them into subpaths which then evolve independently. If implemented appropriately, this has the effect of dedicating a greater fraction of the computational effort to informative runs. We analyze the method for a class of models in which, roughly speaking, the number of states through which each threshold can be crossed is bounded. Under additional assumptions, we identify the optimal degree of splitting at each threshold as the rarity of the event increases: It should be set so that the expected number of subpaths reaching each threshold remains roughly constant. Thus implemented, the method is provably effective in a sense appopriate to rare event simulations. These results follow from a branching-process analysis of the method. We illustrate our theoretical results with some numerical examples for queueing models.