(i) Explain briefly what it means to say that a stochastic process $\left\{W_{t}, t \geqslant 0\right\}$ is a standard Brownian motion.

Let $\left\{W_{t}, t \geqslant 0\right\}$ be a standard Brownian motion and let $a, b$ be real numbers. What condition must $a$ and $b$ satisfy to ensure that the process $e^{a W_{t}+b t}$ is a martingale? Justify your answer carefully.

(ii) At the beginning of each of the years $r=0,1, \ldots, n-1$ an investor has income $X_{r}$, of which he invests a proportion $\rho_{r}, 0 \leqslant \rho_{r} \leqslant 1$, and consumes the rest during the year. His income at the beginning of the next year is

$$X_{r+1}=X_{r}+\rho_{r} X_{r} W_{r}$$

where $W_{0}, \ldots, W_{n-1}$ are independent positive random variables with finite means and $X_{0} \geqslant 0$ is a constant. He decides on $\rho_{r}$ after he has observed both $X_{r}$ and $W_{r}$ at the beginning of year $r$, but at that time he does not have any knowledge of the value of $W_{s}$, for any $s>r$. The investor retires in year $n$ and consumes his entire income during that year. He wishes to determine the investment policy that maximizes his expected total consumption

$$\mathbb{E}\left[\sum_{r=0}^{n-1}\left(1-\rho_{r}\right) X_{r}+X_{n}\right]$$

Prove that the optimal policy may be expressed in terms of the numbers $b_{0}, b_{1}, \ldots$, $b_{n}$ where $b_{n}=1, b_{r}=b_{r+1}+\mathbb{E} \max \left(b_{r+1} W_{r}, 1\right)$, for $r<n$, and determine the optimal expected total consumption.