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Approximating the Signature of Brownian Motion for High Order SDE Simulation

In: Stochastic Analysis and Applications 2025

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  • James Foster

    (University of Bath, Department of Mathematical Sciences)

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 Lévy 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 discuss some applications of these approximations to the high-order numerical simulation of stochastic differential equations (SDEs).

Suggested Citation

  • James Foster, 2026. "Approximating the Signature of Brownian Motion for High Order SDE Simulation," Springer Books, in: Dan Crisan & Ilya Chevyrev & Thomas Cass & James Foster & Christian Litterer & Cristopher Salvi (ed.), Stochastic Analysis and Applications 2025, pages 1-49, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-03914-9_1
    DOI: 10.1007/978-3-032-03914-9_1
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