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2014 An Itô-type formula for the fractional Brownian motion in Brownian time
Ivan Nourdin, Raghid Zeineddine
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Electron. J. Probab. 19: 1-15 (2014). DOI: 10.1214/EJP.v19-3184


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 $H<1/6$.


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Ivan Nourdin. Raghid Zeineddine. "An Itô-type formula for the fractional Brownian motion in Brownian time." Electron. J. Probab. 19 1 - 15, 2014.


Accepted: 24 October 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1307.60041
MathSciNet: MR3305825
Digital Object Identifier: 10.1214/EJP.v19-3184

Primary: 60F05
Secondary: 60G15 , 60H05 , 60H07

Keywords: change-of-variable formula in law , Fractional Brownian motion in Brownian time , Malliavin calculus


Vol.19 • 2014
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