The last zero crossing of an iterated Brownian motion with drift

TL;DR

This paper investigates the last zero crossing of an iterated Brownian motion with drift, deriving explicit distributions and formulas for zero-crossings and passage times, extending to arbitrary nesting levels.

Contribution

It introduces explicit formulas for last zero crossings and passage times of drifted Brownian motions within an iterated framework, including arbitrary nesting levels.

Findings

Derived the last zero-crossing distribution for drifted Brownian motion.

Obtained joint distribution of last zero crossing and first passage time.

Extended analysis to arbitrary nesting levels of zero-crossings.

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.