Martingale structure of Skorohod integral processes

Abstract

Let the process {Yt,t∈[0,1]} have the form Yt=δ(u1[0,t]), where δ stands for a Skorohod integral with respect to Brownian motion and u is a measurable process that verifies some suitable regularity conditions. We use a recent result by Tudor to prove that Yt 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 toward 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 (owing to Duc and Nualart) 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 Ito stochastic integrals.