Ito Integral

The Ito integral is a stochastic integral with respect to a Wiener process (Kuo 2006). For a partition and square-integrable random variables determined by information available at time , the integral of the corresponding simple adapted process is

The integral for a general square-integrable adapted process is defined as a mean-square limit of such sums. The use of left-endpoint information is essential because the integrand cannot depend on future increments of .

The map sending an integrand to its Ito integral preserves the corresponding mean-square norm. This isometry, called the Ito isometry, is expressed by

In particular, an Ito integral has expectation value zero under the usual integrability conditions, and its indefinite integral is a martingale. The change-of-variables rule for these integrals is Ito's lemma.