2019/04/30 by James Foster, Terry Lyons, Harald Oberhauser · 1 citation
Mathematics · Computer Science · #math.NA #cs.NA #math.PR #msc:41A10 #msc:60J65 #msc:60L90 #msc:65C30
paper · pdf · doi:10.1137/19m1261912
published as SIAM Journal on Numerical Analysis, vol. 58, no. 3, pp. 1393-1421, 2020 · 27 pages, 8 figures
arxiv created 2020/05/20 · arxiv updated 2020/05/21
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 L2(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.