The last zero crossing of an iterated Brownian motion with drift

In this paper we consider the iterated Brownian motion ${}_{{\mu }_2}^{{\mu }_1}\text{\hspace{0.17em}⁣}I(t)=B_1^{{\mu }_1}(\mathrm{\mid }{B}_2^{{\mu }_2}(t)\mathrm{\mid })$ 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 ${}_{{\mu }_2}^{{\mu }_1}\text{\hspace{0.17em}⁣}I(t)$ 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 ${^I_\mu T_0} = \sup { s < \max_{0\leq z\leq t} |B_2(z)| : B_1^\mu (s) = 0 }$\mathrm{.}Alsotheiteratedzero-crossing$ {^{\mu_1} T}{0, {^{\mu_2} T}{0,t}} $ is analyzed and extended to the case where the level of nesting is arbitrary.