In this paper we propose a numerical
scheme for a class of backward stochastic differential equations
(BSDEs) with possible path-dependent terminal values. We prove that
our scheme converges in the strong $L^2$ sense and derive its
rate of convergence. As an intermediate step we prove an
$L^2$-type
regularity of the solution to such BSDEs. Such a notion of regularity,
which can be thought of as the modulus of continuity of the paths in
an $L^2$ sense, is new.
Some other features of our scheme include the following: (i) both components of the
solution are approximated by step processes (i.e., piecewise constant
processes); (ii) the regularity requirements on the coefficients are
practically "minimum"; (iii) the dimension of the integrals
involved in the approximation is independent of the
partition size.