A higher-order numerical method is presented for scalar valued, coupled forward-backward stochastic differential equations. Unlike most classical references, the forward component is not only discretized by an Euler-Maruyama approximation but also by higher-order Taylor schemes. This includes the famous Milstein scheme, providing an improved strong convergence rate of order 1; and the simplified order 2.0 weak Taylor scheme exhibiting weak convergence rate of order 2. In order to have a fully-implementable scheme in case of these higher-order Taylor approximations, which involve the derivatives of the decoupling fields, we use the COS method built on Fourier cosine expansions to approximate the conditional expectations arising from the numerical approximation of the backward component. Even though higher-order numerical approximations for the backward equation are deeply studied in the literature, to the best of our understanding, the present numerical scheme is the first which achieves strong convergence of order 1 for the whole coupled system, including the forward equation, which is often the main interest in applications such as stochastic control. Numerical experiments demonstrate the proclaimed higher-order convergence, both in case of strong and weak convergence rates, for various equations ranging from decoupled to the fully-coupled settings.

翻译：本文针对标量值耦合正倒向随机微分方程提出了一种高阶数值方法。与大多数经典文献不同，正向分量不仅采用欧拉-丸山近似进行离散化，还采用了高阶泰勒格式。这包括著名的米尔斯坦格式（提供阶数为1的改进强收敛速率）以及简化的2.0阶弱泰勒格式（展现阶数为2的弱收敛速率）。为了在涉及解耦场导数的高阶泰勒近似下获得完全可实施的格式，我们采用基于傅里叶余弦展开的COS方法来近似由倒向分量数值逼近产生的条件期望。尽管倒向方程的高阶数值逼近在文献中已有深入研究，但据我们所知，本数值格式首次实现了整个耦合系统（包括正向方程）的阶数为1的强收敛，而正向方程往往是随机控制等应用中的主要关注对象。数值实验验证了所宣称的高阶收敛性（包括强收敛速率与弱收敛速率），测试范围涵盖从解耦到完全耦合的多种方程类型。