Projections of scaled Bessel processes

TL;DR

This paper investigates the projection of scaled squared Bessel processes onto the filtration of one component, revealing conditions under which the projection is a supermartingale and characterizing its finite-variation part.

Contribution

It provides a detailed analysis of the projection of scaled squared Bessel processes onto a sub-filtration, including conditions for supermartingale behavior and the structure of the Doob-Meyer decomposition.

Findings

Projection is a strict supermartingale if and only if m<2.

Finite-variation term charges the support of the local time at zero.

Characterizes the dynamics of the projected process.

Abstract

Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.