# An optimal polynomial approximation of Brownian motion

## TL;DR

This paper introduces an optimal polynomial-based method for approximating Brownian motion, leveraging orthogonal polynomials with Gaussian coefficients, leading to improved numerical solutions for stochastic differential equations.

## Contribution

It presents a new strong approximation of Brownian motion using orthogonal polynomials with independent Gaussian coefficients, optimizing in a weighted $L^{2}$ sense and enhancing SDE discretization.

## Key findings

- Orthogonal polynomial expansion provides an optimal pathwise approximation.
- Piecewise parabola discretization yields higher order numerical methods.
- Demonstrated improved simulation of Inhomogeneous Geometric Brownian Motion.

## 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 (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.