An extension of the Itô integral: Toward a general theory of stochastic integration

We introduce the class of instantly independent stochastic processes, which serves as the counterpart of the Itô theory of stochastic integration. This class provides a new approach to anticipating stochastic integration. The evaluation points for an adapted stochastic process and an instantly independent stochastic process are taken to be the left endpoint and the right endpoint, respectively. We present some new results on Itô's formula and stochastic differential equations.