The Doob-Meyer decomposition for anticipating processes

Abstract

In [13], Skorohod introduced a stochastic integral of non-adapted random processes with respect to a Gaussian measure with orthogonal increments. The Skorohod integral is an extension of the classical Ito integral and coincides with the adjoint of the derivative operator on the Wiener space (see [5]). The relation between the Skorohod integral and the Malliavin calculus has been analyzed by Nualart and Zakai in [8]. More recently, a generalized or anticipating stochastic calculus based on the Skorohod integral has been developed by Nualart and Pardoux [9] (see also [12, 14, 15]). We also refer to [10] for an exposition of the basic ideas of this theory...