Inside the nature of squared Bessel process

TL;DR

This paper introduces a new stochastic process called the squared Bessel process with special stochastic time, extending fundamental properties of Brownian motion and providing novel results on its hitting times, joint distributions, and related processes.

Contribution

It presents a new squared Bessel process with special stochastic time, along with analogues of Brownian motion properties, a time inversion result, and a method for calculating hitting time densities.

Findings

Established an analogue of the strong Markov property for the new process.

Derived the joint distribution of two correlated squared Bessel processes.

Developed a new method for finding the density of the first hitting time.

Abstract

A new stochastic process is introduced and considered - squared Bessel process with special stochastic time. The analogues of fundamental properties for Brownian motion are deduced for squared Bessel process. In particular an analogue of the celebrated strong Markov construction of Brownian motion independent of a given sigma field is presented and proved. This result has strong consequences. It allows for deeper understanding the nature of first hitting time of squared Bessel process. The joint distribution of two correlated squared Bessel processes is presented. For squared Bessel process it is also established a new interesting time inversion result. It is presented a general formula that ties squared Bessel process, geometric Brownian motion and its additive functional and that is generalization of the known Lamperti's relation. The new introduced process enables to find the conditional distribution of the first hitting time of a squared Bessel process with nonnegative index. Finally a completely new method of finding the density of first hitting time of squared Bessel process with nonnegative index is presented.