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

(b) Let $\Delta_{0}, \Delta_{1}, \ldots$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ with $\mathbb{E}\left[\left|\Delta_{i}\right|\right]<\infty$ and $\mathbb{E}\left[\Delta_{i}\right]=0, i \geqslant 0$. Further, let

$$X_{0}=\Delta_{0} \quad \text { and } \quad X_{n+1}=X_{n}+\Delta_{n+1} f_{n}\left(X_{0}, \ldots, X_{n}\right), \quad n \geqslant 0$$

where

$$f_{n}\left(x_{0}, \ldots, x_{n}\right)=\frac{1}{n+1} \sum_{i=0}^{n} x_{i}$$

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

(c) State and prove the optional stopping theorem for a bounded stopping time $\tau$.