Approximating the signature of Brownian motion for high order SDE simulation

TL;DR

This paper introduces new and improved methods for approximating the signature of Brownian motion, which are crucial for high-order SDE simulations, addressing the challenge of simulating complex nonlinear integrals.

Contribution

It presents novel and recent approximation techniques for Brownian motion signatures, enhancing the accuracy and efficiency of high-order stochastic differential equation simulations.

Findings

New approximation methods for Brownian motion signatures

Improved simulation accuracy for high-order SDEs

Applications demonstrated in stochastic numerical methods

Abstract

The signature is a collection of iterated integrals describing the "shape" of a path. It appears naturally in the Taylor expansions of controlled differential equations and, as a consequence, is arguably the central object within rough path theory. In this paper, we will consider the signature of Brownian motion with time, and present both new and recently developed approximations for some of its integrals. Since these integrals (or equivalent L\'{e}vy areas) are nonlinear functions of the Brownian path, they are not Gaussian and known to be challenging to simulate. To conclude the paper, we will present some applications of these approximations to the high order numerical simulation of stochastic differential equations (SDEs).