= Solution
Let the <value function> at time $n$ be the greatest conditional expected terminal utility attainable from wealth $x$. Independence of the innovations gives the <Bellman equation>
$$
\boxed{
V(N,x)=U(x),\qquad
V(n,x)=\sup_{\theta\in\mathbb R^d}
\mathbb E\left[V\left(n+1,(1+r)x+\theta^T\xi_{n+1}\right)\right]
}
$$
for $0\leq n<N$. A policy obtained from the maximizing choices is admissible because each time-$n$ holding is a <predictable process>, hence is known before $\xi_n$ is observed.
We prove the claimed properties by <backward induction>. They hold at $n=N$ because $V(N,\cdot)=U$ is increasing and concave. Suppose they hold at time $n+1$. For fixed $\theta$, the map
$$
x\longmapsto
\mathbb E\left[V\left(n+1,(1+r)x+\theta^T\xi_{n+1}\right)\right]
$$
is increasing because $1+r>0$, and taking a supremum preserves this inequality.
For concavity, let $\theta_x$ and $\theta_y$ be optimal at wealths $x$ and $y$. For $0\leq t\leq1$, use the admissible portfolio
$$
\theta_t=t\theta_x+(1-t)\theta_y.
$$
The next wealth is the same <convex combination> of the next wealths generated from $(x,\theta_x)$ and $(y,\theta_y)$. Therefore
$$
\begin{aligned}
V(n,tx+(1-t)y)
&\geq\mathbb E\left[V\left(n+1,(1+r)(tx+(1-t)y)+\theta_t^T\xi_{n+1}\right)\right]\\
&\geq tV(n,x)+(1-t)V(n,y).
\end{aligned}
$$
This completes the induction and proves <monotonicity and concavity of a portfolio value function>.
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