Communications in Mathematical Sciences

Volume 19 (2021)

Number 6

Numerical methods for stochastic differential equations based on Gaussian mixture

Pages: 1549 – 1577

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n6.a5

Authors

Lei Li (School of Mathematical Sciences, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

Jianfeng Lu (Departments of Mathematics, Chemistry and Physics, Duke University, Durham, North Carolina, U.S.A.)

Jonathan C. Mattingly (Departments of Mathematics, Chemistry and Physics, Duke University, Durham, North Carolina, U.S.A.)

Lihan Wang (Departments of Mathematics, Chemistry and Physics, Duke University, Durham, North Carolina, U.S.A.)

Abstract

We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second-order accuracy based on Gaussian mixture. Unlike conventional higher order schemes for SDEs based on Itô–Taylor expansion and iterated Itô integrals, the scheme we propose approximates the probability measure $\mu (X^{n+1} \vert X^n = x_n)$ using a mixture of Gaussians. The solution at the next time step $X^{n+1}$ is drawn from the Gaussian mixture with complexity linear in dimension $d$. This provides a new strategy to construct efficient high weak order numerical schemes for SDEs.

Keywords

Gaussian mixture, stochastic differential equation, second-order scheme, weak convergence

2010 Mathematics Subject Classification

60H35, 65C30, 65L20

Full Text (PDF format)

Received 4 May 2020

Accepted 1 February 2021

Published 2 August 2021