Some results on the Brownian meander with drift

In this paper we study the drifted Brownian meander, that is a Brownian motion starting from $u$ and subject to the condition that $ \min_{ 0\leq z \leq t} B(z)> v $with$ u > v $\mathrm{.}Thelimitingprocessfor$ u \downarrow v $ is analyzed and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case.