(a) Consider a family $\left(X_{n}: n \geqslant 0\right)$ of independent, identically distributed, positive random variables and fix $z_{0}>0$. Define inductively

$$z_{n+1}=z_{n} X_{n}, \quad n \geqslant 0$$

Compute, for $n \in\{1, \ldots, N\}$, the conditional expectation $\mathbb{E}\left(z_{N} \mid z_{n}\right)$.

(b) Fix $R \in[0,1)$. In the setting of part (a), compute also $\mathbb{E}\left(U\left(z_{N}\right) \mid z_{n}\right)$, where

$$U(x)=x^{1-R} /(1-R), \quad x \geqslant 0 .$$

(c) Let $U$ be as in part (b). An investor with wealth $w_{0}>0$ at time 0 wishes to invest it in such a way as to maximise $\mathbb{E}\left(U\left(w_{N}\right)\right)$ where $w_{N}$ is the wealth at the start of day $N$. Let $\alpha \in[0,1]$ be fixed. On day $n$, he chooses the proportion $\theta \in[\alpha, 1]$ of his wealth to invest in a single risky asset, so that his wealth at the start of day $n+1$ will be

$$w_{n+1}=w_{n}\left\{\theta X_{n}+(1-\theta)(1+r)\right\}$$

Here, $\left(X_{n}: n \geqslant 0\right)$ is as in part (a) and $r$ is the per-period riskless rate of interest. If $V_{n}(w)=\sup \mathbb{E}\left(U\left(w_{N}\right) \mid w_{n}=w\right)$ denotes the value function of this optimization problem, show that $V_{n}\left(w_{n}\right)=a_{n} U\left(w_{n}\right)$ and give a formula for $a_{n}$. Verify that, in the case $\alpha=1$, your answer is in agreement with part (b).