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October 2003 Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index ${H \ge \frac{1}{4}}$
Mihai Gradinaru, Francesco Russo, Pierre Vallois
Ann. Probab. 31(4): 1772-1820 (October 2003). DOI: 10.1214/aop/1068646366

Abstract

Given a locally bounded real function g, we examine the existence of a 4-covariation $[g(B^H), B^H, B^H, B^H]$, where $B^H$ is a fractional Brownian motion with a Hurst index $H \ge \tfrac{1}{4}$. We provide two essential applications. First, we relate the 4-covariation to one expression involving the derivative of local time, in the case $H = \tfrac{1}{4}$, generalizing an identity of Bouleau--Yor type, well known for the classical Brownian motion. A second application is an Itô formula of Stratonovich type for $f(B^H)$. The main difficulty comes from the fact $B^H$ has only a finite 4-variation.

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Mihai Gradinaru. Francesco Russo. Pierre Vallois. "Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index ${H \ge \frac{1}{4}}$." Ann. Probab. 31 (4) 1772 - 1820, October 2003. https://doi.org/10.1214/aop/1068646366

Information

Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1059.60067
MathSciNet: MR2016600
Digital Object Identifier: 10.1214/aop/1068646366

Subjects:
Primary: 60H05, 60H10, 60H20
Secondary: 60G15, 60G48

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 4 • October 2003
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