The problem, as stated, is under specified. First, if upward jumps are allowed, pretty much all bets are off. Second, even with continuous sample paths, there are delicate issues with the infinite time horizon. This is also true in a Black-Scholes economy, depending in part on the magnitude of the (P measure) expected rate of return on the underlying. Finally, I can think of no circumstance where the (Q measure) hitting time probability equals one; it certainly won't under GBM. For whatever it is worth, the problem is more interesting with r>0.