The Annals of Probability

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, and Pierre Vallois

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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.

Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 1772-1820.

Dates
First available in Project Euclid: 12 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1068646366

Digital Object Identifier
doi:10.1214/aop/1068646366

Mathematical Reviews number (MathSciNet)
MR2016600

Zentralblatt MATH identifier
1059.60067

Subjects
Primary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations
Secondary: 60G15: Gaussian processes 60G48: Generalizations of martingales

Keywords
Fractional Brownian motion fourth variation Itô's formula local time

Citation

Gradinaru, Mihai; Russo, Francesco; Vallois, Pierre. Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index ${H \ge \frac{1}{4}}$. Ann. Probab. 31 (2003), no. 4, 1772--1820. doi:10.1214/aop/1068646366. https://projecteuclid.org/euclid.aop/1068646366


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