An optimal polynomial approximation of Brownian motion


Foster, J
Oberhauser, H
Lyons, T


SIAM Journal on Numerical Analysis

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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
(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 an integral operator defined by 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 standard
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.

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Journal Article