  Motivated by an approximation problem from mathematical finance, we analyse
the stability of the boundary crossing probability for the multivariate
Brownian motion process, with respect to small changes of the boundary. Under
broad assumptions on the nature of the boundary, including the Lipschitz
condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an
analogue of the Borovkov and Novikov (2005) upper bound for the difference
between boundary hitting probabilities for "close boundaries" in the univariate
case. We also obtained upper bounds for the first boundary crossing time
densities.
