We develop a Fourier method to solve backward stochastic differential equations (BSDEs). General theta-discretization of the time-integrands leads to an induction scheme with conditional expectations. These are approximated by using Fourier-cosine series expansions, relying on the availability of a characteristic function. The method is applied to BSDEs with jumps. Numerical experiments demonstrate the applicability of BSDEs in financial and economic problems and show fast convergence of our efficient probabilistic numerical method.

We develop a Fourier method to solve quite general backward stochastic differential equations (BSDEs) with second-order accuracy. The underlying forward stochastic differential equation (FSDE) is approximated by different Taylor schemes, such as the Euler, Milstein, and Order 2.0 weak Taylor schemes, or by exact simulation. A θ -time-discretization of the time-integrands leads to an induction scheme with conditional expectations. The computation of the conditional expectations appearing relies on the availability of the characteristic function for these schemes. We will use the characteristic function of the discrete forward process. The expected values are approximated by Fourier cosine series expansions. Numerical experiments show rapid convergence of our efficient probabilistic numerical method. Second-order accuracy is observed and also proved. We apply the method to, among others, option pricing problems under the Constant Elasticity of Variance and Cox-Ingersoll-Ross processes.

MSC:We develop three numerical methods to solve coupled forward-backward stochastic differential equations. We propose three different discretization techniques for the forward stochastic differential equation. A theta-discretization of the time-integrands is used to arrive at schemes with conditional expectations. These conditional expectations are approximated by using the COS method, which relies on the availability of the conditional characteristic function of the discrete forward process. The numerical methods are applied to different problems, including a financial problem. Richardson extrapolation is used to obtain more accurate results, resulting in the observation of second-order convergence in the number of time steps. Advantages and disadvantages of each method are compared against each other.

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