Find expected value $\mathbb E[e^{\lambda\tau_a}]$

Asked May 24 '21 at 09:26

Active May 24 '21 at 09:56

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Let $\tau_a = \inf\{t: W_t > a\}$, where $W_t$ is a Wiener Process. I am interested in expected value $\mathbb E \left[e^{-\lambda\tau_a}\right]$, where $\lambda \in \mathbb R$.

I think that I should use the fact that $M_t = e^{\lambda W_t-\frac{\lambda^2}{2}t}$ is a martingale, but no I idea how to go from there.

Thanks for any hints.

edited May 24 '21 at 09:56

asked May 24 '21 at 09:26

I.Kiaan

1

$M_t$ is not a martingale, but $e^{M_t}$ is. Hint: use the optional sampling theorem. – user159517 May 24 '21 at 09:41
Sure, my mistake, corrected it. – I.Kiaan May 24 '21 at 09:56

Find expected value $\mathbb E[e^{\lambda\tau_a}]$

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