(1) Let (f) a Borel function satisfying (0 0) a.s., but (mathbb{P}left(Z_{infty}^{f}=0 ight)=1 / 2). (2) Prove...

(1) Let \(f\) a Borel function satisfying \(0<\int_{0}^{\infty} f^{2}(u) d u<\infty\). Compute, for any \(t, \mathbb{P}\left(\int_{0}^{\infty} f(s) d W_{s}>0 \mid \mathcal{F}_{t}\right)=: Z_{t}^{f}\). Prove that, as a consequence \(Z_{t}^{f}>0\) a.s., but \(\mathbb{P}\left(Z_{\infty}^{f}=0\right)=1 / 2\).

(2) Prove that there exist pairs \((\mathbb{Q}, \mathbb{P})\) of probabilities that are locally equivalent, but \(\mathbb{Q}\) is not equivalent to \(\mathbb{P}\) on \(\mathcal{F}_{\infty}\).