(a) Let $\left(B_{t}\right)_{t \geqslant 0}$ be a real-valued random process.

(i) What does it mean to say that $\left(B_{t}\right)_{t \geqslant 0}$ is a Brownian motion?

(ii) State the reflection principle for Brownian motion.

(b) Suppose that $\left(B_{t}\right)_{t \geqslant 0}$ is a Brownian motion and set $M_{t}=\sup _{s \leqslant t} B_{s}$ and $Z_{t}=M_{t}-B_{t}$.

(i) Find the joint distribution function of $B_{t}$ and $M_{t}$.

(ii) Show that $\left(M_{t}, Z_{t}\right)$ has a joint density function on $[0, \infty)^{2}$ given by

$$\mathbb{P}\left(M_{t} \in d y \text { and } Z_{t} \in d z\right)=\frac{2}{\sqrt{2 \pi t}} \frac{(y+z)}{t} e^{-(y+z)^{2} /(2 t)} d y d z$$

(iii) You are given that two of the three processes $\left(\left|B_{t}\right|\right)_{t \geqslant 0},\left(M_{t}\right)_{t \geqslant 0}$ and $\left(Z_{t}\right)_{t \geqslant 0}$ have the same distribution. Identify which two, justifying your answer.