The last zero crossing of an iterated Brownian motion with drift

Abstract

In this paper we consider the iterated Brownian motion ${}_{{\mu }_2}^{{\mu }_1}I(t)=B_1^{{\mu }_1}(|B_2^{{\mu }_2}(t)|)$ 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 ${}_{{\mu }_2}^{{\mu }_1}I(t)$ 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 ${}_{\mu }^I{T}_0=\sup \{s<{\max }_{0\le z\le t}|B_2(z)|:B_1^{\mu }(s)=0\}$. 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.