Martingale structure of Skorohod integral processes

Let the process Y(t) be a Skorohod integral process with respect to Brownian motion. We use a recent result by Tudor (2004), to prove that Y(t) can be represented as the limit of linear combinations of processes that are products of forward and backward Brownian martingales. Such a result is a further step towards the connection between the theory of continuous-time (semi)martingales, and that of anticipating stochastic integration. We establish an explicit link between our results and the classic characterization, due to Duc and Nualart (1990), of the chaotic decomposition of Skorohod integral processes. We also explore the case of Skorohod integral processes that are time-reversed Brownian martingales, and provide an "anticipating" counterpart to the classic Optional Sampling Theorem for It\^{o} stochastic integrals.