= Compound Poisson transition law of a driftless square-root diffusion
{title2=$N\sim\operatorname{Poisson}(2x/h),\quad Y_j\sim\operatorname{Exp}(2/h)$}
The elapsed-time $h>0$ transition distribution of a <driftless square-root diffusion> starting at $x$ is the <compound Poisson distribution>
$$
X_h\ \stackrel{d}{=}\ \sum_{j=1}^{N}Y_j,\qquad N\sim\operatorname{Poisson}(2x/h),\quad Y_j\sim\operatorname{Exp}(2/h),
$$
where all variables on the right are independent and the empty sum is zero. Here the <exponential distribution> parameter is its rate. The <Laplace transform> of this sum is
$$
\exp\left[\frac{2x}{h}\left(\frac{2/h}{2/h+u}-1\right)\right]=\exp\left(-\frac{xu}{1+uh/2}\right),
$$
which is the <Laplace transform> of the <driftless square-root diffusion>; uniqueness of the <Laplace transform> proves the distributional identity. In particular, for $0\leq v<2/h$, the <exponential moment> is
$$
\mathbb E_x[e^{vX_h}]=\exp\left(\frac{xv}{1-vh/2}\right).
$$
For $x>0$ this <exponential moment> is infinite when $v\geq2/h$, since the event $N=1$ has positive probability and one exponential jump already has an infinite moment there. At $x=0$ the process remains zero and every such moment is one.
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