I was trying to find a transition density function for a squared Bessel process. In the book "Continuous martingale and Brownian motion" by Revuz and Yor, I find a Corollary on page 441 that says:
(1.4) Corollary. For $\delta>0$, the semi-group of $\mathrm{BES}Q^\delta$ has a density in $y$ equal to
$$ q_t^\delta(x, y)=\frac{1}{2}\left(\frac{y}{x}\right)^{y / 2} \exp (-(x+y) / 2 t) I_v(\sqrt{x y} / t), \quad t>0, x>0, $$
where $v$ is the index corresponding to $\delta$ and $I_v$ is the Bessel function of index $v$.
However in https://hsrm-mathematik.de/WS201516/master/option-pricing/Bessel-Processes.pdf equation (20.12) has an extra $1/t$ multiplier. I reached the same result by applying an RV transformation to a non-central chi-square distribution.
The book by Revuz and Yor is way above my level, so I probably misunderstand this corollary. My guess is both of them are right but the Corollary is not actually talking about the density function I am looking for -- am I right?
