Tails of passage-times and an application to stochastic processes with boundary reflection in wedges

In this paper we obtain lower bounds for the tails of the distributions of the first passage-times for some stochastic processes. We consider first discrete parameter processes with asymptotically small drifts taking values in R+ and prove for them a general result giving lower bounds for these tails. As an application of the obtained results, we obtain lower bounds for the tails of the distributions of the first passage-times for reflected random walks in a quadrant with zero-drift in the interior. The latter bounds are then used to get explicit conditions for the finiteness or not of the moments of the first passage-time to the origin for a Brownian motion with oblique reflection in a wedge.