= 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 $x_{n+1}=x_n(1+p_nR_n)$.
The <Bellman equation> for the <value function> is
$$
\boxed{J_N(x)=x^r,\qquad J_n(x)=\max_{0\leq p\leq1}\mathbb E[J_{n+1}(x(1+pR))].}
$$
The <expectation> uses a fresh <independent> return, since the action is chosen before it is observed. Zero wealth remains zero.
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