Change-of-variable formula for the bi-dimensional fractional Brownian motion in Brownian time

Let X^{1}, X^{2} be two independent (two-sided) fractional Brownian motions having the same Hurst parameter H in (0,1), and let Y be a standard (one-sided) Brownian motion independent of (X^{1},X^{2}). In dimension 2, fractional Brownian motion in Brownian motion time (of index H) is, by definition, the process Z_t:= (Z^1_t, Z^2_t)= (X^{1}{Y_t},X^{2}{Y_t}). The main result of the present paper is an Ito’s type formula for f(Z_t), when f:\R^2\to\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 partial derivatives of f as well as an extra Brownian motion independent of (X^1,X^2,Y). We also discuss the case H<1/6.