(a) 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. If $M_{0}=\hat{M}_{0}$, show that $\mathbb{E}\left(M_{T}\right) \geqslant \mathbb{E}\left(\hat{M}_{T}\right)$ for any bounded stopping time $T$. [If you use a general result about supermartingales, you must prove it.]

(b) Consider a market with one stock with time- $n$ price $S_{n}$ and constant interest rate $r$. Explain why a self-financing investor's wealth process $\left(X_{n}\right)_{n \geqslant 0}$ satisfies

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

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

(c) Given an initial wealth $X_{0}$, an investor seeks to maximize $\mathbb{E}\left[U\left(X_{N}\right)\right]$, 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 copies of a random variable $\xi$. Let $V$ be defined inductively by

$$V(n-1, x)=\sup _{t \in \mathbb{R}} \mathbb{E}[V(n,(1+r) x+t(1+r-\xi))]$$

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

Show that the process $\left(V\left(n, X_{n}\right)\right)_{0 \leqslant n \leqslant N}$ is a supermartingale for any trading strategy $\left(\theta_{n}\right)_{1 \leqslant n \leqslant N}$. Suppose that the trading strategy $\left(\theta_{n}^{*}\right)_{1 \leqslant n \leqslant N}$ with corresponding wealth process $\left(X_{n}^{*}\right)_{0 \leqslant n \leqslant N}$ are such that the process $\left(V\left(n, X_{n}^{*}\right)\right)_{0 \leqslant n \leqslant N}$ is a martingale. Show that $\left(\theta_{n}^{*}\right)_{1 \leqslant n \leqslant N}$ is optimal.