An extension of Itô's formula for anticipating processes

Abstract

In this paper we introduce a class of square integrable processes, denoted by LF, defined in the canonical probability space of the Brownian motion, which contains both the adapted processes and the processes in the Sobolev space L2,2. The processes in the class LF satisfy that for any time t, they are twice weakly differentiable in the sense of the stochastic calculus of variations in points (r, s) such that r ∨ s ≥ t. On the other hand, processes belonging to the class LF are Skorohod integrable, and the indefinite Skorohod integral has properties similar to those of the Ito integral. In particular we prove a change-of-variable formula that extends the classical Itô formula. Those results are generalization of similar properties proved by Nualart and Pardoux(7) for processes in L2,2.