An It\^o's type formula for the fractional Brownian motion in Brownian time

Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\in (0,1)$andlet$Y$beastandardBrownianmotionindependentof$X$. Fractional Brownian motion in Brownian motion time (of index $H$), recently studied in \cite{13}, is by definition the process $Z=X\circ Y$. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index $H\mathrm{/}2$. The main result of the present paper is an It^{o}'s type formula for $f(Z_t)$, when $f:\mathbb{R}\to \mathbb{R}$ is smooth and $H\in [1\mathrm{/}6,1)$. When $H>1\mathrm{/}6$, the change-of-variable formula we obtain is similar to that of the classical calculus. In the critical case $H=1\mathrm{/}6$, 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 $H<1\mathrm{/}6$.