  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.

Probability