Extremes of Brownian Decision Trees

Abstract

We consider a Brownian motion with linear drift that splits at fixed time points into a fixed number of branches, which may depend on the branching point. For this process, which we shall refer to as the Brownian decision tree, we investigate the exact asymptotics of high exceedance probabilities in finite time horizon, including: the probability that at least one branch exceeds some high threshold, the probability that the largest distance between branches gets large and the probability that all branches simultaneously exceed some high barrier. Additionally, we find the asymptotics for the probability that all branches of at least one of $M$ independent Brownian decision trees exceed a high threshold.