Error Estimates of the Crank-Nicolson Scheme for Solving Backward Stochastic Differential Equations

Abstract

In this paper, we study error estimates of a special $\theta$-scheme — the Crank-Nicolson scheme proposed in [25] for solving the backward stochastic differential equation with a general generator, $-dy_t = f(t, y_t, z_t)dt-z_tdW_t$. We rigorously prove that under some reasonable regularity conditions on $\varphi$ and $f$, this scheme is second-order accurate for solving both $y_t$ and $z_t$ when the errors are measured in the $L^p (p \geq 1)$ norm.