Laplace transform of a Brownian exit time

HardStochastic Processes~12m

Let $W_t$ be a standard Brownian motion and $T_a = \inf\{t > 0 : |W_t| > a\}$ the first time it exits $(-a, a)$ , for $a > 0$ . Find $\mathbb{E}[e^{-\lambda T_a}]$ , then evaluate it at $\lambda = 4$ , $a = \ln 2$ .

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Use the martingale $e^{-\lambda t}\cosh(\sqrt{2\lambda}\,W_t)$ and optional stopping at $T_a$ .
By symmetry $W_{T_a}=\pm a$ , so $\mathbb{E}[e^{-\lambda T_a}]=1/\cosh(a\sqrt{2\lambda})$ .

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Laplace transform of a Brownian exit time

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