An extension of the Itô integral

We introduce the concept of instant independence for certain anticipating stochastic processes and take the class of instantly independent stochastic processes as a counterpart of adapted stochastic processes for the Itô theory of stochastic integration. Then we define the stochastic integral of a stochastic process which is a linear combination of the products of instantly independent and adapted stochastic processes. The crucial idea is to use the right endpoints as the evaluation points for the instantly independent factors, while the left endpoints are used for the adapted factors. We prove a special case of Itô’s formula for this new stochastic integral and present some examples of stochastic differential equations.