Numerical methods for stochastic differential equations based on Gaussian mixture

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 Ito-Taylor expansion and iterated Ito 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.

在这项工作中，我们基于高斯混合，为随机微分方程（SDEs）开发了一种具有弱二阶精度的数值方法。与基于伊藤 - 泰勒展开和迭代伊藤积分的传统随机微分方程高阶格式不同，所提出的格式通过高斯混合来逼近概率测度\(\mu(X^{n + 1}|X^n = x_n)\)。然后从复杂度与维度\(d\)呈线性关系的高斯混合中抽取下一个时间步\(X^{n + 1}\)的解。这为构建有效的随机微分方程高弱阶数值格式提供了一种新的通用策略。