Exercise 3.7. Theorem 3.6.2 provides the Laplace transform of the density of the first passage time for Brownian motion. This problem derives the analogous formula for Brownian motions with drift. Let $W$ be a Brownian motion. Fix $m>0$ and $\mu \in \mathbb{R}$. For $0 \leq t<\infty$, define
$$
\begin{aligned}
X(t) & =\mu t+W(t), \\
\tau_m & =\min \{t \geq 0 ; X(t)=m\} .
\end{aligned}
$$
As usual, we set $\tau_m=\infty$ if $X(t)$ never reaches the level $m$. Let $\sigma$ be a positive number and set
$$
Z(t)=\exp \left\{\sigma X(t)-\left(\sigma \mu+\frac{1}{2} \sigma^2\right) t\right\} .
$$
(i) Show that $Z(t), t \geq 0$, is a martingale.
(ii) Use (i) to conclude that
$$
\mathbb{E}\left[\exp \left\{\sigma X\left(t \wedge \tau_m\right)-\left(\sigma \mu+\frac{1}{2} \sigma^2\right)\left(t \wedge \tau_m\right)\right\}\right]=1, \quad t \geq 0 .
$$
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3 Brownian Motion
(iii) Now suppose $\mu \geq 0$. Show that, for $\sigma>0$,
$$
\mathbf{E}\left[\exp \left\{\sigma m-\left(\sigma \mu+\frac{1}{2} \sigma^2\right) \tau_m\right\} \mathbf{I}_{\left\{\tau_m<\infty\right\}}\right]=1 .
$$
Use this fact to show $\mathbb{P}\left\{\tau_m<\infty\right\}=1$ and to obtain the Laplace transform
$$
\mathbb{E} e^{-\alpha \tau_m}=e^{m \mu-m \sqrt{2 \alpha+\mu^2}} \text { for all } \alpha>0 .
$$
(iv) Show that if $\mu>0$, then $\mathbb{E} \tau_m<\infty$. Obtain a formula for $\mathbb{E} \tau_m$. (Hint: Differentiate the formula in (iii) with respect to $\alpha$.)
(v) Now suppose $\mu<0$. Show that, for $\sigma>-2 \mu$,
$$
\mathbb{E}\left[\exp \left\{\sigma m-\left(\sigma \mu+\frac{1}{2} \sigma^2\right) \tau_m\right\} \mathbb{I}_{\left\{\tau_m<\infty\right\}}\right]=1 .
$$
Use this fact to show that $\mathbb{P}\left\{\tau_m<\infty\right\}=e^{-2 x|\mu|}$, which is strictly less than one, and to obtain the Laplace transform
$$
\mathbf{E} e^{-\alpha \tau_m}=e^{m \mu-m \sqrt{2 \alpha+\mu^2}} \text { for all } \alpha>0 \text {. }
$$