(a) What is a Brownian motion?

(b) State the Brownian reflection principle. State the Cameron-Martin theorem for Brownian motion with constant drift.

(c) Let $\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion. Show that

$$\mathbb{P}\left(\max _{0 \leqslant s \leqslant t}\left(W_{s}+a s\right) \leqslant b\right)=\Phi\left(\frac{b-a t}{\sqrt{t}}\right)-e^{2 a b} \Phi\left(\frac{-b-a t}{\sqrt{t}}\right)$$

where $\Phi$ is the standard normal distribution function.

(d) Find

$$\mathbb{P}\left(\max _{u \geqslant t}\left(W_{u}+a u\right) \leqslant b\right)$$