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.
Publié le : 1995-03-14
Classification:
Dirichlet processes,
Itô's formula,
local time,
quadratic covariation,
Stratonovich integral
@article{1186078365,
author = {F\"ollmer, Hans and Protter, Philip and Shiryayev, Albert N.},
title = {Quadratic covariation and an extension of It\^o's formula},
journal = {Bernoulli},
volume = {1},
number = {3},
year = {1995},
pages = { 149-169},
language = {en},
url = {http://dml.mathdoc.fr/item/1186078365}
}
Föllmer, Hans; Protter, Philip; Shiryayev, Albert N. Quadratic covariation and an extension of Itô's formula. Bernoulli, Tome 1 (1995) no. 3, pp. 149-169. http://gdmltest.u-ga.fr/item/1186078365/