 Solution

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.