Sharp estimates of transition probability density for Bessel process in half-line

TL;DR

This paper derives precise estimates for the transition probability density of a Bessel process with non-zero index, killed upon reaching a positive level, across all relevant space and time parameters.

Contribution

It provides sharp, comprehensive estimates of the transition density for Bessel processes with non-zero index, extending understanding of their probabilistic behavior.

Findings

Sharp estimates valid for all space parameters x,y > a

Results applicable for all positive times t > 0

Enhanced understanding of Bessel process transition densities

Abstract

In this paper we study the Bessel process R_t^{(\mu)} with index \mu\neq 0 starting from x>0 and killed when it reaches a positive level a, where x>a>0. We provide sharp estimates of the transition probability density p_a^{(\mu)}(t,x,y) for the whole range of space parameters x,y>a and every t>0.