High-Efficiency Explicit Multistep Schemes for Coupled Second-Order FBSDEs

In this work, by introducing a new family of recursively defined processes, we propose new explicit multistep schemes for coupled second-order forward backward stochastic differential equations. The explicit schemes avoid calculating the conditional mathematical expectations of the generator $f$ and calculate the required values of $f$ explicitly and accurately. By combining the Sinc quadrature rule for approximating the conditional expectations, we further propose the $k$th order ($1\le k\le 6$) fully discrete explicit multistep schemes. Numerical tests are presented to demonstrate the strong stability, high accuracy, and high efficiency of the explicit schemes.