 Solution

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.