Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index ${H \ge \frac{1}{4}}$
Gradinaru, Mihai ; Russo, Francesco ; Vallois, Pierre
Ann. Probab., Tome 31 (2003) no. 1, p. 1772-1820 / Harvested from Project Euclid
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.
Publié le : 2003-10-14
Classification:  Fractional Brownian motion,  fourth variation,  Itô's formula,  local time,  60H05,  60H10,  60H20,  60G15,  60G48
@article{1068646366,
     author = {Gradinaru, Mihai and Russo, Francesco and Vallois, Pierre},
     title = {Generalized covariations, local time and Stratonovich It\^o's formula for fractional Brownian motion with Hurst index ${H \ge \frac{1}{4}}$},
     journal = {Ann. Probab.},
     volume = {31},
     number = {1},
     year = {2003},
     pages = { 1772-1820},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1068646366}
}
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., Tome 31 (2003) no. 1, pp.  1772-1820. http://gdmltest.u-ga.fr/item/1068646366/