## Electronic Journal of Probability

### An Itô-type formula for the fractional Brownian motion in Brownian time

#### 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 $H<1/6$.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 99, 15 pp.

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

https://projecteuclid.org/euclid.ejp/1465065741

Digital Object Identifier
doi:10.1214/EJP.v19-3184

Mathematical Reviews number (MathSciNet)
MR3305825

Zentralblatt MATH identifier
1307.60041

Rights

#### Citation

Nourdin, Ivan; Zeineddine, Raghid. An Itô-type formula for the fractional Brownian motion in Brownian time. Electron. J. Probab. 19 (2014), paper no. 99, 15 pp. doi:10.1214/EJP.v19-3184. https://projecteuclid.org/euclid.ejp/1465065741

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