A bit of a basic question maybe, but I am confused how to write down the Bellman of this recursive utility function based on Epstein-Zin-Weil preferences and some stochastic constraints.
$$ U_t = \left \{(1-\beta) C_t^{1-\frac{1}{\eta}} + \beta[E_t(U_{t+1}^{1-\gamma})]^{\frac{1-\frac{1}{\eta}}{1-\gamma}}\right \}^{\frac{1}{1-\frac{1}{\eta}}} $$
It already looks like a Bellman equation, Any pointers?
What you have there are the preferences under an arbitrary policy -- what some call the prevalue function. The only thing missing is the max operator. Written with maximization (and making the state and choice explicit), the Bellman equation is
\begin{align} U(K_t, \epsilon_t) =& \max_{C_t} \{ (1-\beta) C_t^{1 - \frac{1}{\eta}} + \beta [E_t(U(K_{t+1}, \epsilon_{t+1})^{1-\gamma})]^{\frac{1 - \frac{1}{\eta}}{1 - \gamma}} \}^{\frac{1}{1 - \frac{1}{\eta}}} \\ \text{s.t. } & C_t \in F(K_t, \epsilon_t) \\ & K_{t+1} = G(K_t, C_t, \epsilon_t), \\ & \epsilon \sim \Phi \end{align}
where $K_t$ is the state, $F(K_t)$ is the choice set, $\epsilon_t$ is some shock with CDF $\Phi$ (assumed to be revealed before choosing $C_t$; it doesn't need to be IID), and $G(K_t, C_t)$ is the law of motion.
Maximization with recursive preferences is pretty much the same as with standard time-separable preferences, and even plays well with dynamic programming methods (e.g. if you have some VFI code set up you can make the preferences recursive by just changing the prevalue function). The only real change is that we now have different aggregators over states and over time on the RHS of the prevalue function. These notes on recursive preferences may help, in particular section 5 ("Optimization and the Bellman Equation").
Making the constraints stochastic won't change the details of your problem -- it'll be similar to how you would formulate and solve the time-separable equivalent problem with a stochastic constraint -- unless you have some specific structure in mind which would change things for recursive preferences differently than for time-separable preferences. I can't think of such a structure off the top of my head, but I'm sure one could find them with some digging.
One thing that's different with recursive preferences is the utility scaling -- the period utility is scaled by $(1-\beta)$. This is discussed a bit on page 3 of the attached notes. The scaling allows utility to be measured in units of consumption, so that the utility of a sequence of sure-thing constant consumption levels is that same consumption level, i.e. $U(c,c,...) = c$. It's a good exercise to try showing this symmetry without the scaling to see how it comes out (if you do it with the risk aggregator, remember that the certainty equivalent of a sure thing is the sure thing itself).
