Let (W) be a (mathbb{P})-Brownian motion and (d mathbb{Q}||_{mathcal{F}_{t}}=left.e^{W_{t}-t / 2} d mathbb{P} ight|_{mathcal{F}_{t}}). Let (tau=inf left{t:

Let \(W\) be a \(\mathbb{P}\)-Brownian motion and \(d \mathbb{Q}||_{\mathcal{F}_{t}}=\left.e^{W_{t}-t / 2} d \mathbb{P}\right|_{\mathcal{F}_{t}}\). Let \(\tau=\inf \left\{t: W_{t}=-m\right\}\) for \(m>0\). Compute \(\mathbb{P}(\tau<\infty)\) and \(\mathbb{Q}(\tau<\infty)\). Hint: \(\mathbb{P}(\tau<\infty)=1\), and using results on hitting times of BM \(\mathbb{Q}(\tau<\infty)=e^{-m} \mathbb{E}_{\mathbb{P}}\left(e^{-\tau / 2}\right)=e^{-2 m}\).