We study the new stochastic integral introduced by Ayed and Kuo in 2008. Our main results are two Itô formulas that extend the one presented by Ayed and Kuo. We generalize the notion of the Itô process onto the class of instantly independent stochastic processes and use it in the formulation of the two Itô formulas we derive.

Keywords: Brownian motion, Itô integral, Itô formula, adapted stochastic processes, instantly independent stochastic processes, anticipating stochastic processes, stochastic integral, anticipating integral