The last zero crossing of an iterated Brownian motion with drift

In this paper we consider the iterated Brownian motion $^{\mu_1}_{\mu_2}\!I(t) = B_1^{\mu_1} ( | B_{2}^{\mu_2} (t)|)$ where $B_j^{\mu_j} , j=1,2$ are two independent Brownian motions with drift $\mu_j$. Here we study the last zero crossing of $^{\mu_1}_{\mu_2}\!I(t)$ and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before $t$ and of the first passage time through the zero level of a Brownian motion with drift $\mu$ after $t$. All these results permit us to derive explicit formulas for ${^I_\mu T_0} = \sup \{ s < \max_{0\leq z\leq t} |B_2(z)| : B_1^\mu (s) = 0 \}$. Also the iterated zero-crossing ${^{\mu_1} T}_{0, {^{\mu_2} T}_{0,t}}$ is analyzed and extended to the case where the level of nesting is arbitrary.