Asymptotics of First-Passage Time Over a One-Sided Stochastic Boundary

We study the asymptotic behavior of the first-passage times for Brownian motion, Levy processes and continuous martingales over one-sided increasing stochastic, as well as deterministic, boundaries. In particular, we study the firstpassage time of a Brownian motion over the increasing function of its local time, give necessary and sufficient conditions for t –1/2 asymptotics, and obtain exact asymptotics for linear functions.