The last zero crossing of an iterated Brownian motion with drift

Abstract

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.