Define the $\tau_x=inf\{t:W_t = x\}$, where $W_t$ is a brownian motion. I know the distribution of $\tau_x$ is $$f_{\tau_x}(t)=\frac{|x|}{\sqrt{2\pi}}t^{-1.5}e^{\frac{-x^2}{2t}}$$, which is an inverse gaussian dist. I want to prove $\mathbb{E}(e^{-\alpha\tau_x})=e^{-|x|\sqrt{2\alpha}}$, i.e. the MGF, from scratch, but i am struggling with the integration. How to derive this MGF, or in general, the MGF of inverse gaussian?
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See this question: http://math.stackexchange.com/q/600518/ – saz Apr 08 '15 at 16:48