The last zero-crossing of an iterated brownian motion with drift

In this paper, we consider the iterated Brownian motion μ1μ2I(t)=Bμ11(∣∣Bμ22(t)∣∣) where Bμjj,j=1,2 are two independent Brownian motions with drift μj. Here, we study the last zero crossing before the maximum time span travelled by the inner process of μ1μ2I(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 μ after t. All these results permit us to derive explicit formulas for IμT0=sup{s