Extremes of Brownian Decision Trees

TL;DR

This paper analyzes the extreme behavior of a Brownian decision tree process, deriving precise asymptotics for high exceedance probabilities of branches over finite time horizons.

Contribution

It provides exact asymptotic formulas for various high exceedance probabilities in Brownian decision trees, including joint and maximum exceedances, extending understanding of such stochastic processes.

Findings

Derived asymptotics for the probability that at least one branch exceeds a high threshold

Established asymptotics for the maximum distance between branches

Analyzed the probability that all branches in multiple trees exceed high barriers

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.