What is Brownian motion $\left(B_{t}\right)_{t \geqslant 0}$ ? Briefly explain how Brownian motion can be considered as a limit of simple random walks. State the Reflection Principle for Brownian motion, and use it to derive the distribution of the first passage time $\tau_{a} \equiv \inf \left\{t: B_{t}=a\right\}$ to some level $a>0$.

Suppose that $X_{t}=B_{t}+c t$, where $c>0$ is constant. Stating clearly any results to which you appeal, derive the distribution of the first-passage time $\tau_{a}^{(c)} \equiv \inf \left\{t: X_{t}=a\right\}$ to $a>0$.

Now let $\sigma_{a} \equiv \sup \left\{t: X_{t}=a\right\}$, where $a>0$. Find the density of $\sigma_{a}$.