High order recombination and an application to cubature on Wiener space

TL;DR

This paper introduces a dynamic recombination technique for particle methods that maintains high order accuracy and prevents particle explosion, with an application to cubature on Wiener space.

Contribution

It proposes a novel support simplification algorithm enabling high order particle methods with dynamic recombination, improving efficiency and accuracy.

Findings

Supports high order accuracy in particle methods

Prevents particle explosion during iteration

Effective application to cubature on Wiener space

Abstract

Particle methods are widely used because they can provide accurate descriptions of evolving measures. Recently it has become clear that by stepping outside the Monte Carlo paradigm these methods can be of higher order with effective and transparent error bounds. A weakness of particle methods (particularly in the higher order case) is the tendency for the number of particles to explode if the process is iterated and accuracy preserved. In this paper we identify a new approach that allows dynamic recombination in such methods and retains the high order accuracy by simplifying the support of the intermediate measures used in the iteration. We describe an algorithm that can be used to simplify the support of a discrete measure and give an application to the cubature on Wiener space method developed by Lyons and Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169-198].