Prove that, for (a>S_{0}), and (t leq T) [mathbb{P}left(T_{a}(S)>T mid mathcal{F}_{t} ight)=mathbb{1}_{left{max _{s leq t} S_{s}

Prove that, for \(a>S_{0}\), and \(t \leq T\)

\[\mathbb{P}\left(T_{a}(S)>T \mid \mathcal{F}_{t}\right)=\mathbb{1}_{\left\{\max _{s \leq t} S_{s}

with

\[\begin{aligned}d_{1} & =\frac{1}{\sigma \sqrt{T-t}}\left(\ln \left(\frac{a}{S_{t}}\right)-\left(r-\delta-\frac{\sigma^{2}}{2}\right)\right) \\d_{2} & =\frac{1}{\sigma \sqrt{T-t}}\left(\ln \left(\frac{S_{t}}{a}\right)-\left(r-\delta-\frac{\sigma^{2}}{2}\right)\right) .\end{aligned}\]