Let $\{X_t:t\geq 0\}$ be a Brownian motion with drift $\mu>0$ and define a stopping time $\tau$ by $$\tau=\inf\{t\geq 0:X_t=a\}.$$ Now I want to show that $$\mathbb{E}(e^{-\lambda\tau})=e^{(\mu-\sqrt{\mu^2+2\lambda})a}$$ for $\lambda>0$. Now as a hint I know that I need to use the martingale $M_t=e^{\alpha X_t-\alpha\mu t-\frac{1}{2}\alpha^2t}$. Obviously I need to use Doobs optional stopping theorem but I do not know how. Anyone has a suggestion?
Brownian Motion – Expectation Stopped Brownian Motion with Drift
brownian motionmartingalesstopping-times
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