标题： 罕见事件仿真与分裂法应用于间断随机变量 显示英文标题

标题： Rare Event Simulation and Splitting for Discontinuous Random Variables

摘要： 分裂方法（也称为顺序蒙特卡洛或\emph{子集模拟}）是广泛用于估计形如$P[S(\mathbf{U}) > q]$的极端概率的方法，其中$S$是一个确定的实值函数，而$\mathbf{U}$可以是随机的有限维或无限维向量。 通常情况下，$X := S(\mathbf{U})$被假定为连续随机变量，并且许多关于估计器统计行为的理论结果都是在这种假设下得出的。 然而，一旦$S$中出现阈值效应和/或$\mathbf{U}$是离散或混合离散/连续时，该假设不再成立，估计器也不再具有一致性。 本文研究了\emph{累积分布函数}中$X$不连续性的影响，并提出了三种无偏的\emph{校正的}估计器来处理这些问题。这些估计器无需提前知道$X$是否真的存在不连续性，在$X$连续的情况下，所有估计器都相等。特别是，其中一种在任何情况下都具有相同的统计性质。效率通过二维扩散过程以及\emph{布尔可满足性问题}（SAT）得到了验证。 显示英文摘要

摘要： Multilevel Splitting methods, also called Sequential Monte-Carlo or \emph{Subset Simulation}, are widely used methods for estimating extreme probabilities of the form $P[S(\mathbf{U}) > q]$ where $S$ is a deterministic real-valued function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Very often, $X := S(\mathbf{U})$ 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 $\mathbf{U}$ 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 \emph{cdf} of $X$ and present three unbiased \emph{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 \emph{Boolean SATisfiability problem} (SAT).