Fluctuations of Rare Event Simulation with Monte Carlo Splitting in the Small Noise Asymptotics

TL;DR

This paper rigorously analyzes the fluctuations of the AMS Monte Carlo method for rare event simulation in small noise diffusion processes, deriving large deviations results and efficiency conditions.

Contribution

It provides a large deviations analysis of the AMS estimator in small noise regimes and identifies geometric conditions for its asymptotic efficiency.

Findings

Large deviations logarithmic equivalent derived

Explicit maximization problem in terms of quasi-potential

Conditions for vanishing of the variance established

Abstract

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.