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Let $X_t$ be a generalized Wiener process with drift rate $\mu$ and variance $\sigma^2$, and let $\tau$ be the stopping time

$$\tau:=\inf \left\{ t\geq0: X_t= b\right\}, \quad b\geq0 $$

Can anyone give me some insights on how to compute the generating function:

$$ E[\mathrm{e}^{-\lambda\tau}], \quad \lambda\geq0 $$

Many thanks in advance.

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Here's a solution, perhaps not in the most formal language (i.e. I may be skipping some technical details), and hopefully without any major errors.

If $\tau_b$ is the time until $X_t=b\ge0$ is reached, let us define $$F_\lambda(b)=\text{E}\left[e^{-\lambda\tau_b}\right]$$ for $\lambda>0$, with the convention that $e^{-\lambda\tau_b}=0$ if $X_t=b$ is never reached.

First, we note that $F_\lambda(0)=1$, and that $F_\lambda(b)$ is decreasing in $b$.

If $b>0$, $X_T=x$ and $X_s

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mejor te escribo en castellano no? Tengo problemas con el ejercicio 2 apartados 1 y 3 del tema 1, así como con el ejercicio 1 del tema 2 y el ejercicio 2 apartado d del tema 2. ¿Podrías ayudarme? Respecto al que me pides, mira este link: http://www.columbia.edu/~hz2244/teaching/Lec9.pdf A ver si hay suerte!!! Saludos

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    Thank you VMS. Do you have an email address where I can send you some hints?2012-09-14
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    If you want, I can provide you more solutions.2012-09-14
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    Where are you RCA? Please, give me a reply....2012-09-14
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    +1 for the link but -1 for the miscellaneous discussion that doesn't relate to the question.2012-09-23