Bernoulli

  • 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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 2 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186078365

Mathematical Reviews number (MathSciNet)
MR1354459

Zentralblatt MATH identifier
0851.60048

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

Citation

Föllmer, Hans; Protter, Philip; Shiryayev, Albert N. Quadratic covariation and an extension of Itô's formula. Bernoulli 1 (1995), no. 1-2, 149--169. https://projecteuclid.org/euclid.bj/1186078365


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