Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 28K (Optimization and Control)

Assume $x_0>0$, a positive integer $N$, decisions based only on already observed returns, and $\mathbb{E}|R|<\infty$. The last is a sufficient well-posedness condition absent from the print; without moment assumptions the objectives may be infinite. Put $r=1/N$. Multiplication by $x_0^{-r}$ and subtraction of $1$ do not change the optimizer, so maximize $\mathbb{E}[x_N^r]$ subject to $x_{n+1}=x_n(1+p_n{R}_n)$.

The Bellman equation for the value function isThe expectation uses a fresh independent return, since the action is chosen before it is observed. Zero wealth remains zero.

Let $M={\max }_{p\in [0,1]}\mathbb{E}(1+pR{)}^r$ and choose an optimizer $p^{\ast }$. Existence follows from continuity and compactness: the integrands are dominated by a constant times $1+|R|$. Induction in the Bellman equation givesThus the maximal expected yield is $\boxed{M^N-1}$. Neither current wealth nor time changes the maximizing fraction; there may be several optimizers.

The right derivative at $p=0$ is $r\mathbb{E}R$, using the finite first moment and dominated differentiation near zero. If $\mathbb{E}R>0$, increasing $p$ slightly improves on $p=0$, so $\boxed{p^{\ast }>0}$. No explicit value of $p^{\ast }$ is needed.

For expected final capital, the Bellman equation is linear in wealth, with one-period factor $\mathbb{E}(1+pR)=1+p\mathbb{E}R$. Therefore invest everything when the mean return is positive, nothing when it is negative, and any fraction when it is zero. For $N=1$ this agrees with the yield objective.

For $N>1$, the root objective is concave and penalizes losses, so its optimal fraction need not be $1$ even when the mean is positive. For example, if $\mathbb{P}(R=-1)>0$ and $\mathbb{E}R>0$, the left derivative of $\mathbb{E}(1+pR{)}^r$ at $1$ is $-\infty$, while its right derivative at $0$ is positive; the optimum lies strictly between $0$ and $1$. If $\mathbb{E}R\le 0$, Jensen inequality gives $\mathbb{E}(1+pR{)}^r\le (1+p\mathbb{E}R{)}^r\le 1$, so $p=0$ is optimal. Thus the root objective can favor partial investment over the all-in policy.