Rare event simulation and splitting for discontinuous random variables^∗

¹ CEA, DAM, DIF, 91297 Arpajon, France
walter@math.univ-paris-diderot.fr
² Laboratoire de Probabilités et Modèles Aléatoires, Université Paris Diderot, 75205 Paris cedex 05, France

Received: 3 July 2015
Revised: 2 November 2015

Abstract

Multilevel Splitting methods, also called Sequential Monte−Carlo or Subset Simulation, are widely used methods for estimating extreme probabilities of the form P[S(U) >q] where S is a deterministic real-valued function and U can be a random finite- or infinite-dimensional vector. Very often, X: = S(U) is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in S and/or U is discrete or mixed discrete/continuous this assumption does not hold any more and the estimator is not consistent. In this paper, we study the impact of discontinuities in the cdf of X and present three unbiased corrected estimators to handle them. These estimators do not require to know in advance if X is actually discontinuous or not and become all equal if X is continuous. Especially, one of them has the same statistical properties in any case. Efficiency is shown on a 2-D diffusive process as well as on the Boolean SATisfiability problem (SAT).