@article{EJP3184,
author = {Ivan Nourdin and Raghid Zeineddine},
title = {An Itô type formula for the fractional Brownian motion in Brownian time},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {Fractional Brownian motion in Brownian time; change-of-variable formula in law; Malliavin calculus},
abstract = {Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\in (0,1)$ and let $Y$ be a standard Brownian motion independent of $X$. Fractional Brownian motion in Brownian motion time (of index $H$), recently studied, is by definition the process $Z=X\circ Y$. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index $H/2$. The main result of the present paper is an Itô's type formula for $f(Z_t)$, when $f:\mathbb{R}\to\mathbb{R}$ is smooth and $H\in [1/6,1)$. When $H>1/6$, the change-of-variable formula we obtain is similar to that of the classical calculus. In the critical case $H=1/6$, our change-of-variable formula is in law and involves the third derivative of $f$ as well as an extra Brownian motion independent of the pair $(X,Y)$. We also discuss briefly the case $H<1/6$.},
pages = {no. 98, 1-15},
issn = {1083-6489},
doi = {10.1214/EJP.v19-3184},
url = {http://ejp.ejpecp.org/article/view/3184}}