Multilevel branching splitting algorithm for estimating rare event probabilities

TL;DR

This paper analyzes a multilevel splitting algorithm for estimating rare event probabilities in a discrete multidimensional setting, deriving optimal thresholds, and examining the impact of threshold shape deformations on estimator variance.

Contribution

It introduces a multilevel splitting algorithm with optimal thresholds for rare event probability estimation in multidimensional discrete frameworks.

Findings

Variance decomposition into threshold variability and number variability

Optimal thresholds derived by cancelling the first variance component

Sensitivity analysis of estimator variance with respect to threshold shape deformation

Abstract

We analyse the splitting algorithm performance in the estimation of rare event probabilities and this 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 we get is the sum of two parts: one part resuming the variability due to each threshold and a second part resuming the variability due to the thresholds number. 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.