Discretizing Malliavin calculus

Suppose B is a Brownian motion and Bn is an approximating sequence of rescaled random walks on the same probability space converging to B pointwise in probability. We provide necessary and sufficient conditions for weak and strong L2-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark-Ocone derivative to their continuous counterparts. Moreover, given a sequence (Xn) of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to Bn, we derive necessary and sufficient conditions for strong L2-convergence to a σ(B)-measurable random variable X via convergence of the discrete chaos coefficients of Xn to the continuous chaos coefficients.