= Driftless square-root diffusion
{title2=$dX_t=\sqrt{X_t}\,dW_t$}
A driftless square-root <diffusion process> is the nonnegative solution of $dX_t=\sqrt{X_t}\,dW_t$, with zero an absorbing boundary. For elapsed time $h>0$, starting from $x\geq0$, its <Laplace transform> is
$$
\mathbb E_x[e^{-uX_h}]=\exp\left(-\frac{xu}{1+uh/2}\right),\qquad u\geq0.
$$
To prove this, fix the final time $h$ and set $F(t,x)=\exp[-xu/(1+u(h-t)/2)]$. Differentiation gives $F_t+\tfrac12xF_{xx}=0$, while $F(h,x)=e^{-ux}$. By <Itô formula>, $F(t,X_t)$ is a <local martingale>; it takes values in $[0,1]$, so localization and bounded convergence make it a true <martingale>. Taking expectations at the two endpoints proves the formula. The <Markov property> gives the same formula conditionally at any starting time.
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