Quadratic Covariation and an Extension of Itô's Formula

Abstract

Let [math] be a standard Brownian motion. We show that for any locally square integrable function [math] the quadratic covariation [math] exists as the usual limit of sums converging in probability. For an absolutely continuous function [math] with derivative [math] , Itô's formula takes the form [math] . This is extended to the time-dependent case. As an example, we introduce the local time of Brownian motion at a continuous curve.

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