What is a martingale? What is a supermartingale? What is a stopping time?

Let $M=\left(M_{n}\right)_{n \geqslant 0}$ be a martingale and $\hat{M}=\left(\hat{M}_{n}\right)_{n \geqslant 0}$ a supermartingale with respect to a common filtration. If $M_{0}=\hat{M}_{0}$, show that $\mathbb{E} M_{T} \geqslant \mathbb{E} \hat{M}_{T}$ for any bounded stopping time $T$.

[If you use a general result about supermartingales, you must prove it.]

Consider a market with one stock with prices $S=\left(S_{n}\right)_{n \geqslant 0}$ and constant interest rate $r$. Explain why an investor's wealth $X$ satisfies

$$X_{n}=(1+r) X_{n-1}+\pi_{n}\left[S_{n}-(1+r) S_{n-1}\right]$$

where $\pi_{n}$ is the number of shares of the stock held during the $n$th period.

Given an initial wealth $X_{0}$, an investor seeks to maximize $\mathbb{E} U\left(X_{N}\right)$ where $U$ is a given utility function. Suppose the stock price is such that $S_{n}=S_{n-1} \xi_{n}$ where $\left(\xi_{n}\right)_{n} \geqslant 1$ is a sequence of independent and identically distributed random variables. Let $V$ be defined inductively by

$$V(n, x, s)=\sup _{p \in \mathbb{R}} \mathbb{E} V\left[n+1,(1+r) x-p s\left(1+r-\xi_{1}\right), s \xi_{1}\right]$$

with terminal condition $V(N, x, s)=U(x)$ for all $x, s \in \mathbb{R}$.

Show that the process $\left(V\left(n, X_{n}, S_{n}\right)\right)_{0 \leqslant n \leqslant N}$ is a supermartingale for any trading strategy $\pi$.

Suppose $\pi^{*}$ is a trading strategy such that the corresponding wealth process $X^{*}$ makes $\left(V\left(n, X_{n}^{*}, S_{n}\right)\right)_{0} \leqslant n \leqslant N$ a martingale. Show that $\pi^{*}$ is optimal.