The operation
extends
to all Ito processes through the linearity in both
arguments:
Suppose
is a smooth function of its arguments. The Ito formula
states


(Ito formula)

The Ito formula is a direct consequence of the Taylor formula
and the considerations of the previous section. Indeed, we are always
interested in the representation of the difference
in terms of the infinitesimal
increments:
Hence, we apply the Taylor formula to the
and keep the leading terms.
The term
is intended to be used under integral sign. The terms
and
vanish when we take the integral (pass to the dt=0 limit and take a sum) but
the term
does not vanish by the formula (
Ito isometry
).
This test of survival under the limit dt=0 and sum determines the rules
(
Ito calculus
) at the beginning of this section.
For several variables
the Ito formula takes
form


(Ito formula 2)

The following is an important consequence of the Ito
formula:


(Ito derivative of product)

To see this consider a linear combination of
,
and
.
