An optimal polynomial approximation of Brownian motion

Journal article

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.

Authors: Foster, J; Lyons, T; Oberhauser, H

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Numerical Analysis
• Volume: 58
• Issue: 3
• Pages: 1393–1421
• Publication date: 2020-05-04
• Acceptance date: 2020-02-14
• DOI: 10.1137/19M1261912
• EISSN: 1095-7170
• ISSN: 0036-1429
• Language: English
• Keywords: numerical methods for SDEs; FFR; polynomial approximation; Brownian motion