{"title":"Approximating the signature of Brownian motion for high order SDE simulation","authors":"James Foster","doi":"arxiv-2409.10118","DOIUrl":null,"url":null,"abstract":"The signature is a collection of iterated integrals describing the \"shape\" of\na path. It appears naturally in the Taylor expansions of controlled\ndifferential equations and, as a consequence, is arguably the central object\nwithin rough path theory. In this paper, we will consider the signature of\nBrownian motion with time, and present both new and recently developed\napproximations for some of its integrals. Since these integrals (or equivalent\nL\\'{e}vy areas) are nonlinear functions of the Brownian path, they are not\nGaussian and known to be challenging to simulate. To conclude the paper, we\nwill present some applications of these approximations to the high order\nnumerical simulation of stochastic differential equations (SDEs).","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"65 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10118","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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).