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2019

An optimal polynomial approximation of Brownian motion

Foster, James, Lyons, Terry, Oberhauser, Harald

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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.

An optimal polynomial approximation of Brownian motion · Around

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