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.
翻译:我们在这项工作中开发了一种基于高斯混合物的二阶精确度弱的随机差分方程式(SDEs)的数字方法。与基于 Itçóo-Taylor扩展和迭代 It ⁇ o 集成的常规SDEs较高顺序方案不同,拟议办法以高斯人混合体的近似概率计量$mu(X ⁇ n+1 ⁇ X ⁇ n=x_n)美元。下一步骤的解决方案$X ⁇ n+1}美元则取自具有维度线性复杂线性的高斯混合体。这为SDEs构建高效高低序数字方案提供了新的总体战略。