Let (M) be a positive martingale, such that (M_{0}=1) and (lim _{t ightarrow infty} M_{t}=0). Let

Let \(M\) be a positive martingale, such that \(M_{0}=1\) and \(\lim _{t \rightarrow \infty} M_{t}=0\). Let \(a \in\left[0,1\left[\right.\right.\) and define \(G_{a}=\sup \left\{t: M_{t}=a\right\}\). Prove that

\[\mathbb{P}\left(G_{a} \leq t \mid \mathcal{F}_{t}\right)=\left(1-\frac{M_{t}}{a}\right)^{+}\]

Assume that, for every \(t>0\), the law of the r.v. \(M_{t}\) admits a density \(\left(m_{t}(x), x \geq 0\right)\), and \((t, x) \rightarrow m_{t}(x)\) may be chosen continuous on \((0, \infty)^{2}\) and that \(d\langle Mangle_{t}=\sigma_{t}^{2} d t\), and there exists a jointly continuous function \((t, x) \rightarrow \theta_{t}(x)=\mathbb{E}\left(\sigma_{t}^{2} \mid M_{t}=x\right)\) on \((0, \infty)^{2}\). Prove that

\[\mathbb{P}\left(G_{a} \in d t\right)=\left(1-\frac{M_{0}}{a}\right) \delta_{0}(d t)+\mathbb{1}_{\{t>0\}} \frac{1}{2 a} \theta_{t} (a) m_{t} (a) d t\]