Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis

TL;DR

This paper introduces a large deviation-based framework for particle splitting methods to efficiently estimate rare event probabilities in Markov processes, providing conditions for algorithm growth and variance control.

Contribution

It defines a subsolution concept for the calculus of variations problem and links it to the growth and variance properties of splitting algorithms, advancing rare event simulation techniques.

Findings

Particle growth is subexponential if a scaled importance function is a subsolution.

Variance of the splitting algorithm is characterized asymptotically by the subsolution.

Numerical examples demonstrate the effectiveness of the proposed approach.

Abstract

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.