The Itô Integral

Abstract

The Itô integral allows us to integrate stochastic processes with respect to the increments of a Brownian motion or a somewhat more general stochastic process. We develop the Itô integral first for Brownian motion and then for generalized diffusion processes (so-called Itô processes). In the third section, we derive the celebrated Itô formula. This is the chain rule for the Itô integral that enables us to do explicit calculations with the Itô integral. In the fourth section, we use the Itô formula to obtain a stochastic solution of the classical Dirichlet problem. This in turn is used in the fifth section in order to show that like symmetric simple random walk, Brownian motion is recurrent in low dimensions and transient in high dimensions.