Highly Accurate and Efficient Explicit Multistep Schemes for Forward Backward Stochastic Differential Equations and their Error Analyses

Source abstract

In this paper, by introducing a novel family of recursively defined approximations, we propose new explicit multistep schemes for solving forward backward stochastic differential equations. In contrast to the existing explicit multistep schemes, the proposed schemes avoid computing the associated conditional mathematical expectations of the generator function $f$ and instead use direct explicit evaluations of the $f.$ We provide a rigorous error analysis of the proposed schemes, establishing the optimal $k$-th order convergence rates for $1≤k≤6.$ Numerical tests are conducted to validate the theoretical results and demonstrate the strong stability, high accuracy, and high efficiency of the new schemes.