(a) What does it mean to say that $\left(M_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale?

(b) Let $Y_{1}, Y_{2}, \ldots$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ with $Y_{i}>0 \mathbb{P}$-a.s. and $\mathbb{E}\left[Y_{i}\right]=1, i \geqslant 1$. Further, let

$$M_{0}=1 \quad \text { and } \quad M_{n}=\prod_{i=1}^{n} Y_{i}, \quad n \geqslant 1$$

Show that $\left(M_{n}\right)_{n \geqslant 0}$ is a martingale with respect to the filtration $\mathcal{F}_{n}=\sigma\left(Y_{1}, \ldots, Y_{n}\right)$.

(c) Let $X=\left(X_{n}\right)_{n \geqslant 0}$ be an adapted process with respect to a filtration $\left(\mathcal{F}_{n}\right)_{n \geqslant 0}$ such that $\mathbb{E}\left[\left|X_{n}\right|\right]<\infty$ for every $n$. Show that $X$ admits a unique decomposition

$$X_{n}=M_{n}+A_{n}, \quad n \geqslant 0,$$

where $M=\left(M_{n}\right)_{n \geqslant 0}$ is a martingale and $A=\left(A_{n}\right)_{n \geqslant 0}$ is a previsible process with $A_{0}=0$, which can recursively be constructed from $X$ as follows,

$$A_{0}:=0, \quad A_{n+1}-A_{n}:=\mathbb{E}\left[X_{n+1}-X_{n} \mid \mathcal{F}_{n}\right]$$

(d) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a super-martingale. Show that the following are equivalent:

(i) $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale.

(ii) $\mathbb{E}\left[X_{n}\right]=\mathbb{E}\left[X_{0}\right]$ for all $n \in \mathbb{N}$.