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Quadratic Covariation And An Extension Of Itô's Formula

H. Föllmer, P. Protter, A. Shiryayev
Published 1995 · Mathematics

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Let X be a standard Brownian motion. We show that for any locally square integrable functionfthe quadratic covariation [f(X), X] exists as the usual limit of sums converging in probability. For an absolutely continuous function F with derivativef, It6's formula takes the form F(Xt) = F(Xo) + Jj f(X,) dXs + ? [f(X), X]t. 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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