Multilevel branching and splitting algorithm for estimating rare event probabilities

We analyze the splitting algorithm performance in the estimation of rare event probabilities in a discrete multidimensional framework. For this we assume that each threshold is partitioned into disjoint subsets and the probability for a particle to reach the next threshold will depend on the starting subset. A straightforward estimator of the rare event probability is given by the proportion of simulated particles for which the rare event occurs. The variance of this estimator is the sum of two parts: with one part resuming the variability due to each threshold, and the second part resuming the variability due to the number of thresholds. This decomposition is analogous to that of the continuous case. The optimal algorithm is then derived by cancelling the first term leading to optimal thresholds. Then we compare this variance with that of the algorithm in which one of the threshold has been deleted. Finally, we investigate the sensitivity of the variance of the estimator with respect to a shape deformation of an optimal threshold. As an example, we consider a two-dimensional Ornstein-Uhlenbeck process with conformal maps for shape deformation.