We introduce the class of instantly independent stochastic processes,
which serve as the anticipating counterpart of adapted stochastic
processes in the Ito theory. This leads to a new stochastic integral
with integrands being sums of product of adapted and instantly
independent stochastic processes. The crucial idea, in forming
Riemann-like sums, is to use the left endpoints of subintervals to
evaluate the adapted factors and to use the right endpoints to
evaluate the instantly independent factors. The new stochastic
integral has many advantages than the white noise methods, e.g.,
it has a probabilistic interpretation and is connected to the
underlying filtration. Moreover, by using this new stochastic
integral, we can treat a multiple Wiener-Ito integral as a
true iterated stochastic integral. We also obtain extensions of
Ito's formula and study stochastic differential equation with an
anticipating initial condition or integrands being anticipating.