Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 35 / 3 / c / Solution

Localize to compact subintervals of $(0,\infty )$, so that the Itô formula can be applied to $f(z)=\sqrt{z}$. Its derivatives are $f^{\prime }(z)=1/(2\sqrt{z})$ and $f^{\prime \prime }(z)=-1/(4z^{3/2})$, while $d[Z{]}_t=4Z_{t}dt$. Thus, before $\zeta$,Hence the Bessel process equation isThe inverse-power drift is interpreted only while $R>0$; its boundary behaviour is justified next, not assumed in this calculation.