This paper focuses on estimating reach probability of a closed unsafe set by a stochastic process. A well-developed approach is to make use of multi-level MC simulation, which consists of encapsulating the unsafe set by a sequence of increasing closed sets and conducting a sequence of MC simulations to estimate the reach probability of each inner set from the previous set. An essential step is to copy (split) particles that have reached the next level (inner set) prior to conducting a MC simulation to the next level. The aim of this paper is to prove that the variance of the multi-level MC estimated reach probability under fixed assignment splitting is smaller or equal than under random assignment splitting methods. The approaches are illustrated for a geometric Brownian motion example.

本文着重于通过随机过程估计封闭不安全集的到达概率。一个成熟的方法是利用多级 MC 模拟，它包括通过一系列增加的封闭集来封装不安全集，并进行一系列 MC 模拟以估计每个内部集与前一个集合的到达概率. 一个重要的步骤是在进行 MC 模拟到下一个级别之前复制（拆分）已经到达下一个级别（内部集）的粒子。本文的目的是证明在固定分配拆分方法下多级 MC 估计到达概率的方差小于或等于随机分配拆分方法下的方差。针对几何布朗运动示例说明了这些方法。