Recently I was thinking in general on how to maximize the expectation of a function (not necessarily a utility function, but apparently this is a common case). To give an idea of the kind of problem, here there is a possible setting.
- $F (x, y, z, k)$ integrable function
- $y := \phi (z, k)$
- $x \in [0,1]$
- $k \in K$.
$$ \max_{x} \int_{K} F (x, y, z, k) \ dk.$$
Now, how do we deal with a problem like this one?
I thought that maybe Calculus of Variations (or optimal control theory) should be the thing. However, looking on the Internet I found a question named "Maximizing Expected Utility" on MO, where a user answers that not necessarily we have to use calculus of variations for maximizing expected utility.
QUESTIONS:
1) How do we deal with a problem like this one?
2) How do we deal with this problem without using the Calculus of Variations?
Is there somebdoy that can shed some light on it?
Thank you for your time.