# Numerical methods for stochastic differential equations based on Gaussian mixture

## TL;DR

This paper introduces a novel numerical method for stochastic differential equations that uses Gaussian mixtures to achieve weak second order accuracy, offering an efficient alternative to traditional schemes.

## Contribution

The paper proposes a new Gaussian mixture-based scheme for SDEs that simplifies computation and achieves higher weak order accuracy compared to conventional methods.

## Key findings

- Achieves weak second order accuracy for SDEs.
- Complexity of the method scales linearly with dimension.
- Provides an efficient alternative to Itô-Taylor based schemes.

## Abstract

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