(a) What does it mean to say that a stochastic process $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale with respect to a filtration $\left(\mathcal{F}_{n}\right)_{n \geqslant 0}$ ?

(b) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a martingale, and let $\xi_{n}=X_{n}-X_{n-1}$ for $n \geqslant 1$. Suppose $\xi_{n}$ takes values in the set $\{-1,+1\}$ almost surely for all $n \geqslant 1$. Show that $\left(X_{n}\right)_{n \geqslant 0}$ is a simple symmetric random walk, i.e. that the sequence $\left(\xi_{n}\right)_{n \geqslant 1}$ is $\operatorname{IID}$ with $\mathbb{P}\left(\xi_{1}=1\right)=1 / 2=$ $\mathbb{P}\left(\xi_{1}=-1\right) .$

(c) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a martingale and let the bounded process $\left(H_{n}\right)_{n \geqslant 1}$ be previsible.

Let $\hat{X}_{0}=0$ and

$$\hat{X}_{n}=\sum_{k=1}^{n} H_{k}\left(X_{k}-X_{k-1}\right) \text { for } n \geqslant 1$$

Show that $\left(\hat{X}_{n}\right)_{n \geqslant 0}$ is a martingale.

(d) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk with $X_{0}=0$, and let

$$T_{a}=\inf \left\{n \geqslant 0: X_{n}=a\right\}$$

where $a$ is a positive integer. Let

$$\hat{X}_{n}= \begin{cases}X_{n} & \text { if } n \leqslant T_{a} \\ 2 a-X_{n} & \text { if } n>T_{a}\end{cases}$$

Show that $\left(\hat{X}_{n}\right)_{n \geqslant 0}$ is a simple symmetric random walk.

(e) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk with $X_{0}=0$, and let $M_{n}=\max _{0 \leqslant k \leqslant n} X_{k}$. Compute $\mathbb{P}\left(M_{n}=a\right)$ for a positive integer $a$.