Let (Sn : n ≥ 0) be a mean zero random walk (rw) with light-tailed increments. One of the most fundamental problems in rare-event simulation involves computing P (Sn nβ) for β 0 when n is large. It is well known that the optimal exponential tilting (OET), although logarithmically efficient, is not strongly efficient (the squared coefficient of variation of the estimator grows at rate n1/2). Our analysis of the zero-variance change-of-measure provides useful insights into why OET is not strongly efficient. In particular, the iid nature of OET induces an overshoot over the boundary nβ that is too big and causes the coefficient of variation to grow as [EQUATION]. We study techniques used to provide a state-dependent change-of-measure that yields a strongly efficient estimator. The application of our state-dependent algorithm to the Gaussian case reveals the fine structure of the zero-variance change-of-measure. We see how (Sn : n ≥ 0) transitions from a rw under OET to a process that looks "mean reverting" aroung βn, indicating that less bias is required as the process approaches the boundary βn.
展开▼
机译：令（ S n ：n≥0）为具有轻尾增量的平均零随机游动（rw）。稀有事件模拟中最基本的问题之一是，当 n 大时，计算β> 0的 P（S n >nβ）。众所周知，最佳指数倾斜（OET）尽管对数有效，但并不是很有效（估计量的平方变异系数以 n 1/2 ）。我们对零方差度量的分析提供了有用的见解，说明了OET为何效率不高。特别是，OET的同性性质会导致边界 n β的过冲，该过大太大并导致变异系数随[方程式]增大。我们研究了用于提供状态相关的量度变化的技术，该方法产生了非常有效的估计量。我们的状态相关算法在高斯情况下的应用揭示了零方差测量值的精细结构。我们看到（ S n ：n ≥0）如何从OET下的rw过渡到一个看起来在β n 周围“平均回复”的过程。 ，表明随着该过程接近边界β n ，所需的偏差较小。
展开▼
