Diffusion processes with small noise conditioned to reach a target set are considered. The AMS algorithm is a Monte Carlo method that is used to sample such rare events by iteratively simulating clones of the process and selecting trajectories that have reached the highest value of a so-called importance function. In this paper, the large sample size relative variance of the AMS small probability estimator is considered. The main result is a large deviations logarithmic equivalent of the latter in the small noise asymptotics, which is rigorously derived. It is given as a maximisation problem explicit in terms of the quasi-potential cost function associated with the underlying small noise large deviations. Necessary and sufficient geometric conditions ensuring the vanishing of the obtained quantity ('weak' asymptotic efficiency) are provided. Interpretations and practical consequences are discussed.
翻译:AMS算法是一种蒙特卡洛方法,用来通过反复模拟过程的克隆人和选择达到所谓重要功能最高值的轨迹来抽样这种稀有事件。在本文件中,考虑了AMS小概率估测器的较大样本大小相对差异。主要结果是在微小噪音杂音中与后者大偏差对数等值,这种偏差是严格推算出来的。它被作为与潜在的小噪声大偏差相关的准潜在成本函数的一个最大问题。提供了必要和充分的几何条件,确保获得的数量消失(“微弱”微量效率)。讨论了解释和实际后果。