A pursuit problem for squared Bessel processes

TL;DR

This paper investigates the probability that two independent squared Bessel processes do not cross over a long period, revealing a power decay related to hypergeometric functions and determining the crossing location distribution.

Contribution

It provides a novel analysis of non-crossing probabilities for squared Bessel processes and links these probabilities to hypergeometric function zeros.

Findings

Non-crossing probability decays as a power law.

Decay rate is determined by the first zero of a hypergeometric function.

Distribution of crossing location is explicitly computed.

Abstract

In this note, we are interested in the probability that two independent squared Bessel processes do not cross for a long time. We show that this probability has a power decay which is given by the first zero of some hypergeometric function. We also compute along the way the distribution of the location where the crossing eventually occurs.