• Bernoulli
  • Volume 1, Number 1-2 (1995), 149-169.

Quadratic covariation and an extension of Itô's formula

Hans Föllmer, Philip Protter, and Albert N. Shiryayev

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Let X be a standard Brownian motion. We show that for any locally square integrable function f the quadratic covariation [ f(X),X] exists as the usual limit of sums converging in probability. For an absolutely continuous function F with derivative f , Itô's formula takes the form F (X t)=F(X 0)+ 0 tf(X s)dX s+1 2 [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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Bernoulli, Volume 1, Number 1-2 (1995), 149-169.

First available in Project Euclid: 2 August 2007

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Dirichlet processes Itô's formula local time quadratic covariation Stratonovich integral


Föllmer, Hans; Protter, Philip; Shiryayev, Albert N. Quadratic covariation and an extension of Itô's formula. Bernoulli 1 (1995), no. 1-2, 149--169.

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