  The Crank-Nicolson (short for C-N) scheme for solving {\it backward
stochastic differential equation} (BSDE), driven by Brownian motions, was first
developed by the authors W. Zhao, L. Chen and S. Peng [SIAM J. Sci. Comput., 28
(2006), 1563--1581], and numerical experiments showed that the accuracy of this
C-N scheme was of second order for solving BSDE. This C-N scheme was extended
to solve decoupled {\it forward-backward stochastic differential equations}
(FBSDEs) by W. Zhao, Y. Li and Y. Fu [Sci. China. Math., 57 (2014), 665--686],
and it was numerically shown that the accuracy of the extended C-N scheme was
also of second order.
  To our best knowledge, among all one-step (two-time level) numerical schemes
with second-order accuracy for solving BSDE or FBSDEs, such as the ones in the
above two papers and the one developed by the authors D. Crisan and K.
Manolarakis [Ann. Appl. Probab., 24, 2 (2014), 652--678], the C-N scheme is the
simplest one in applications. The theoretical proofs of second-order error
estimates reported in the literature for these schemes for solving decoupled
FBSDEs did not include the C-N scheme.
  The purpose of this work is to theoretically analyze the error estimate of
the C-N scheme for solving decoupled FBSDEs. Based on the Taylor and
It\^o-Taylor expansions, the Malliavin calculus theory (e.g., the multiple
Malliavin integration-by-parts formula), and our new truncation error
cancelation techniques, we rigorously prove that the strong convergence rate of
the C-N scheme is of second order for solving decoupled FBSDEs, which fills the
gap between the second-order numerical and theoretical analysis of the C-N
scheme.
