Approximating the signature of Brownian motion for high order SDE simulation

James Foster
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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).
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近似布朗运动特征,用于高阶 SDE 仿真
签名是描述路径 "形状 "的迭代积分集合。它很自然地出现在受控微分方程的泰勒展开中,因此可以说是粗糙路径理论的核心对象。在本文中,我们将考虑布朗运动随时间变化的特征,并对其某些积分提出新的和最近发展的近似值。由于这些积分(或等价L'{e}vy区域)是布朗路径的非线性函数,因此它们不是高斯函数,已知模拟起来具有挑战性。最后,我们将介绍这些近似值在随机微分方程(SDEs)高阶数值模拟中的一些应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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