A pursuit problem for squared Bessel processes

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, as well as some Cram{\'e}r's estimates for squared Bessel processes perturbed by a power drift.