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# 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

#### 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

Received 4 May 2020

Accepted 1 February 2021

Published 2 August 2021