Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries

We calculate several hitting time probabilities for a correlated multidimensional Brownian bridge process, where the boundaries are hyperplanes that move linearly with time. We compute the probability that a Brownian bridge will cross a moving hyperplane if the endpoints of the bridge lie on the same side of the hyperplane at the starting and ending times, and we derive the distribution of the hitting time if the endpoints lie on opposite sides of the moving hyperplane. Our third result calculates the probability that this process remains between two parallel hyperplanes, and we extend this result in the independent case to a hyperrectangle with moving faces. To derive these quantities, we rotate the coordinate axes to transform the problem into a one-dimensional calculation.