I am preparing for my exams and I can't get my head around the following question.
I know there exists a general method for solving these problems but I don't know where to start. I would greatly appreciate it if someone can help me get started or explain how the method works.
Solve $E\left( \sum\limits_{k=0}^{31} u_k^{1/2} + x_{32}^{1/2} \right) \rightarrow \max$, such that $x_0 \geq 0, 0 \leq u_k \leq x_k$ and $x_{k+1} = Y_k(x_k - u_k)$. Where $Y_k$ are independent positive identically distributed random variables for which $E(Y^{1/2})$ is defined.
I know that I first need to make a Bellman equation. My attempt at doing so:
$V(k, x_k) = \max\ E(u_k^{1/2}+x_{32}^{1/2} ) + V(k+1, Y_k(x_k - u_k)) $
But I have no idea where to go from here.