Let (T_{a}^{*}=inf left{t geq 0:left|W_{t} ight|=a ight}). Using the fact that the process (left(e^{-lambda^{2} t / 2}

Let \(T_{a}^{*}=\inf \left\{t \geq 0:\left|W_{t}\right|=a\right\}\). Using the fact that the process \(\left(e^{-\lambda^{2} t / 2} \cosh \left(\lambda W_{t}\right), t \geq 0\right)\) is a martingale, prove that

\[\mathbb{E}\left(\exp \left(-\lambda^{2} T_{a}^{*} / 2\right)\right)=[\cosh (a \lambda)]^{-1}\]