Extremes of Brownian Decision Trees
Krzysztof Dȩbicki, Pavel Ievlev, Nikolai Kriukov
TL;DR
The paper introduces the Brownian decision tree, a drifted Brownian motion that branches at fixed times into multiple offspring, and derives exact finite-horizon tail asymptotics for several extreme-event regimes.A combination of Gaussian process tail theory, covariance/variance structure analysis, and quadratic programming yields explicit asymptotics for single-branch exceedance, inter-branch diameter, and simultaneous all-branch exceedance, with a Korshunov-Wang-type refinement providing tight bounds and constants.Extensions to forests of independent Brownian decision trees are developed via a dominance principle, showing that the overall ruin probability is asymptotically the sum of the dominating trees’ probabilities, each described by the main asymptotics.The results illuminate the interplay between branching structure, drift, and extremal events in a Brownian-branching setting and provide tractable formulas and techniques applicable to related Gaussian-extremes problems.
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.