= Laplace transform of symmetric Brownian interval-exit time
{c}
{title2=$\mathbb E e^{-\lambda\tau_x}=\operatorname{sech}(x\sqrt{2\lambda})$}
For standard <Brownian motion> started at zero and first exit $\tau_x$ from $(-x,x)$, the <Laplace transform> is $1/\cosh(x\sqrt{2\lambda})$ for $\lambda>0$. The <stochastic process> $e^{-\lambda t}\cosh(\sqrt{2\lambda}B_t)$ is the average of two copies of the <Exponential martingale for Brownian motion>. Its values stopped at $t\wedge\tau_x$ are bounded, so bounded-time application of the <optional stopping theorem> followed by <dominated convergence> evaluates the transform without assuming <uniform integrability> of an unstopped exponential <martingale> on the infinite horizon.
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