(from Azma and Yor [38].) Let (X) be a BES ({ }^{3}) starting from 0 . Prove...

(from Azéma and Yor [38].) Let \(X\) be a BES \({ }^{3}\) starting from 0 . Prove that \(1 / X\) is a local martingale, but not a martingale. Establish that, for \(u<1\),

\[\mathbb{E}\left(\left.\frac{1}{X_{1}} \rightvert\, \mathcal{R}_{u}\right)=\frac{1}{X_{u}} \sqrt{\frac{2}{\pi}} \Phi\left(\frac{X_{u}}{1-u}\right)\]

where \(\Phi(a)=\int_{0}^{a} d y e^{-y^{2} / 2}\). Such a formula " measures" the non-martingale property of the local martingale \(\left(1 / X_{t}, t \leq 1\right)\). In general, the quantity \(\mathbb{E}\left(Y_{t} \mid \mathcal{F}_{s}\right) / Y_{s}\) for \(s