We study the fluctuations of the power variation of fractional Brownian motion in Brownian time.

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time Z t := X Yt , t 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.

Let X be a (two-sided) fractional Brownian motion of Hurst parameter H ∈ (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 in [17], is by definition the process Z = X • 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 : R → R is smooth and H ∈ [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.

We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H < 1/2. As applications, we study the asymptotic behavior of the trapezoidal weighted odd-power variations of the fractional Brownian motion and the fractional Brownian motion in Brownian time Z t := X Yt , t 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.

No abstract