A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations Based on the Fractional FFT

Abstract

BSDEs are applied in many areas, particularly in finance and economics. In this paper, we extended the convolution method to numerically solve FBSDEs. First, a generalized θ-scheme is applied to discretize the backwards component. Second, the convolution method is used to solve the conditional expectation. Third, the resulting convolution is dealt with numerically by the Fourier transform. Therefore, the fractional FFT algorithm is applied to compute the Fourier and inverse the transforms. Then, we prove some error estimates. Finally, a numerical example is implemented to test the efficiency and stability of the proposed method.

Keywords

• convolution method
• generalized <i>θ</i>-scheme
• backward stochastic differential equations
• the fractional FFT
• Fourier transform