(i) What does it mean to say that a process $\left(M_{t}\right)_{t \geqslant 0}$ is a martingale? What does the martingale convergence theorem tell us when applied to positive martingales?

(ii) What does it mean to say that a process $\left(B_{t}\right)_{t \geqslant 0}$ is a Brownian motion? Show that $\sup _{t \geqslant 0} B_{t}=\infty$ with probability one.

(iii) Suppose that $\left(B_{t}\right)_{t \geqslant 0}$ is a Brownian motion. Find $\mu$ such that

$$S_{t}=\exp \left(x_{0}+\sigma B_{t}+\mu t\right)$$

is a martingale. Discuss the limiting behaviour of $S_{t}$ and $\mathbb{E}\left(S_{t}\right)$ for this $\mu$ as $t \rightarrow \infty$.