We propose a method based on the Fourier transform for numerically solving backward stochastic differential equations. Time discretization is applied to the forward equation of the state variable as well as the backward equation to yield a recursive system with terminal conditions. By assuming the integrability of the functions in the terminal conditions and applying truncation, the solutions of the system are shown to be integrable and we derive recursions in the Fourier space. The fractional FFT algorithm is applied to compute the Fourier and inverse Fourier transforms. We showcase the efficiency of our method through various numerical examples.

中文翻译：

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求解倒向随机微分方程的傅立叶变换方法

我们提出了一种基于傅立叶变换的方法，用于数值求解后向随机微分方程。将时间离散化应用于状态变量的前向方程式和后向方程式，以产生具有最终条件的递归系统。通过假设函数在终端条件下的可积性并应用截断，可以证明系统的解是可积的，并且我们在傅立叶空间中获得了递归。分数FFT算法应用于计算傅立叶和傅立叶逆变换。我们通过各种数值示例来展示我们方法的效率。