An It? type formula for the fractional Brownian motion in Brownian time

摘要

Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $Hin (0,1)$ 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 $Z=Xcirc Y$. 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 It?'s type formula for $f(Z_t)$, when $f:mathbb{R}omathbb{R}$ is smooth and $Hin [1/6,1)$. 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 $H<1/6$.