Let \(W\) be a standard Brownian motion, \(a>1\), and \(\tau\) the stopping time \(\tau=\inf \left\{t: e^{W_{t}-t / 2}>a\right\}\). Prove that, \(\forall \lambda \geq 1 / 2\),

\[\mathbb{E}\left(\mathbb{1}_{\{\tau<\infty\}} \exp \left(\lambda W_{\tau}-\frac{1}{2} \lambda^{2} \tau\right)=1\right.\]

\[\mathbb{E}\left(\mathbb{1}_{\tau<\infty} \exp \left(\lambda W_{\tau}-\frac{1}{2} \lambda^{2} \tau\right)\right)=\mathbf{W}^{(\lambda)}(\tau<\infty)\]

The process \(\left(W_{t}-\frac{1}{2} t, t \geq 0\right)\) is, under \(\mathbf{W}^{(\lambda)}\), a BM with drift \(\lambda-\frac{1}{2}\).