Numerical methods for stochastic differential equations based on Gaussian mixture

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 $\mu(X<sup>{n+1}|X<sup>n=x_n)$ by a mixture of Gaussians. The solution at next time step $X<sup>{n+1}$ 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.