# The Itô Integral and Itô's Lemma

The Itô Integral and Itô's Lemma

This Demonstration illustrates (a discrete version of) the most fundamental concept in stochastic analysis—the Itô integral and its most fundamental property—Itô's lemma. Choose the integrand from the dropdown menu. The graph displays four curves (two of which coincide in the case of the first integrand) that show approximations of a path of Brownian motion (the integrator), the chosen integrand, and the left- and right-hand sides in Itô's formula (see the details). As you decrease the size of the time step the latter two curves come closer together, showing that they coincide in the limit (i.e., actual Brownian motion).

Mouse over a curve to see the stochastic concept for which the curve is an approximation.