Efficient numerical Fourier methods for coupled forward–backward SDEs

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.