Author
Foster, J
Lyons, T
Oberhauser, H
Journal title
SIAM Journal on Numerical Analysis
DOI
10.1137/19M1261912
Issue
3
Volume
58
Last updated
2024-03-24T16:47:33.58+00:00
Page
1393-1421
Abstract
In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore, it is practical (i.e., requires $N$ independent Gaussian coefficients) to generate an approximate sample path of Brownian motion that respects integration of polynomials with degree less than $N$. Moreover, since these orthogonal polynomials appear naturally as eigenfunctions of the Brownian bridge covariance function, the proposed approximation is optimal in a certain weighted $L^{2}(\mathbb{P})$ sense. In addition, discretizing Brownian paths as piecewise parabolas gives a locally higher order numerical method for stochastic differential equations (SDEs) when compared to the piecewise linear approach. We shall demonstrate these ideas by simulating inhomogeneous geometric Brownian motion (IGBM). This numerical example will also illustrate the deficiencies of the piecewise parabola approximation when compared to a new version of the asymptotically efficient log-ODE (or Castell--Gaines) method.
Symplectic ID
995012
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Publication type
Journal Article
Publication date
04 May 2020
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