I need help with the following problem: Let us denote the water level in a dam at time $t$ by $X(t)$, where $t$ is measured in months. We will assume that, at least until the first time that the dam gets empty (i.e. $X(t) = 0$), $X$ can be modeled as a Brownian motion with drift $ \mu = +1 $ and variance $ \sigma^2 = 1 $. If the initial water level is $X_0 = 200$, find the probability that the water level reaches 210 before dropping below 100.
Any hints/pointers would be great!
Let $a<x_0<b$ and $\mu$ be given real numbers. Let $B(t)$ be a standard Brownian motion with $B(0)=x_0$ a.s., we consider the process $X(t)=B(t)+\mu t$. Define the stopping time $$T=\inf\bigl\{t >0, \; X(t)\in\{a,b\}\bigr\}.$$ We are trying to find the value of $\mathbb{P}\bigl(X(T)=b\bigr)$.
To begin with, it should be clear that $T$ is indeed an almost surely finite stopping time associated to $(\mathscr{F}_t)$ the natural filtration of the Brownian motion. Then we know that for any $\lambda$, $$\exp\left(\lambda B(t)-\frac{\lambda^2 t}{2}\right)$$ is a $(\mathscr{F}_t)$-martingale (see for instance [1], Ch. IV, Prop. 3.4). A direct consequence is that $\exp\left(-2\mu X(t)\right)$ is a $(\mathscr{F}_t)$-martingale as well. Now, on one hand, the optional stopping theorem (see [1], Ch. II, Th. 3.2) yields $$\mathbb{E}\left[\mathrm{e}^{-2\mu W(T)}\right]=\mathbb{E}\left[\mathrm{e}^{-2\mu W(0)}\right]=\mathrm{e}^{-2\mu x_0}$$ and on the other hand, $$\mathbb{E}\left[\exp(-2\mu W(T))\right]=\exp(-2\mu a)\mathbb{P}\left(W(T)=a\right)+\exp(-2\mu b)\mathbb{P}\left(W(T)=b\right).$$ Since $\mathbb{P}\left(W(T)=a\right)+\mathbb{P}\left(W(T)=b\right)=1$, we just proved that $$\mathbb{P}\left(W(T)=b\right)=\frac{\mathrm{e}^{-2\mu x_0}-\mathrm{e}^{-2\mu a}}{\mathrm{e}^{-2\mu b}-\mathrm{e}^{-2\mu a}}.$$
In our specific setting, $x_0=200$, $a=100$, $b=210$ and $\mu=1$, which gives a probability very close to $1.0$.
[1]: Revuz & Yor, Continuous Martingales and Brownian Motion, Third edition
