In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we propose a new multi-step scheme by adopting the high-order multi-step method in Zhao et al. (SIAM J. Sci. Comput., 36(4): A1731-A1751, 2014) with the combination technique. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by using the finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the temporal direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.

在这项工作中，为了获得解决前向后向随机微分方程的高阶方案，我们采用了赵等人的高阶多步法，提出了一种新的多步法。（SIAM J. Sci.Comput。，36（4）：A1731-A1751，2014）。从后向分量中导出了两个包含条件期望及其导数的参考常微分方程。这些导数通过使用具有多个步骤组合的有限差分方法来近似。所得方案是在时间方向上涉及条件期望的半离散化，可以通过使用高斯正交规则和空间网格上的多项式插值来解决。我们新提出的多步方案允许更高的收敛速度，直到第9阶，而且效率更高。最后，我们提供了所提出方法收敛性的数值说明。