Efficient discretisation of stochastic differential equations

TL;DR

This paper introduces a more efficient discretisation method for stochastic differential equations by using hitting times of spheres, reducing error compared to standard schemes, and providing practical implementation strategies.

Contribution

It develops a novel discretisation approach based on hitting times of spheres that improves accuracy over Euler-Maruyama and is easier to implement.

Findings

Error reduced by a factor of (d+2)/d with sphere hitting times

Hitting times of moving spheres achieve near-optimal error reduction

Method outperforms standard Euler-Maruyama scheme in efficiency

Abstract

The aim of this study is to find a generic method for generating a path of the solution of a given stochastic differential equation which is more efficient than the standard Euler-Maruyama scheme with Gaussian increments. First we characterize the asymptotic distribution of pathwise error in the Euler-Maruyama scheme with a general partition of time interval and then, show that the error is reduced by a factor (d+2)/d when using a partition associated with the hitting times of sphere for the driving d-dimensional Brownian motion. This reduction ratio is the best possible in a symmetric class of partitions. Next we show that a reduction which is close to the best possible is achieved by using the hitting time of a moving sphere which is easier to implement.