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.

中文翻译：

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布朗运动的最优多项式逼近

在本文中，我们将通过一类正交多项式呈现标准布朗运动的强（或路径）近似。在这个多项式基础上从布朗运动展开得到的系数是独立的高斯随机变量。因此，它是实用的（需要 $N$ 独立的高斯系数）来生成布朗运动的近似样本路径，该路径尊重次数小于 $N$ 的多项式的积分。此外，由于这些正交多项式自然地表现为由布朗桥协方差函数定义的积分算子的特征函数，因此所提出的近似在某个加权 $L^{2}(\mathbb{P})$ 意义上是最佳的。此外，与标准分段线性方法相比，将布朗路径离散为分段抛物线为随机微分方程 (SDE) 提供了局部高阶数值方法。我们将通过模拟非均匀几何布朗运动 (IGBM) 来证明这些想法。此数值示例还将说明分段抛物线近似与渐近高效 log-ODE（或 Castell-Gaines）方法的新版本相比的缺陷。