Stochastic convex (conave) functions vs. convex (concave) function

Question

Can someone help me understand the difference beween stochastic convex (conave) functions and convex (concave) function

Answer 1

The definition is surely explained in the book or lecture note you're reading. A stochastic function is a function from $\Omega$ to a function space. Here is an example : let $X : \Omega \to \mathbb{R}$ be a random variable. Then the following defines a random function : $$ F : \omega \longrightarrow \Big( f_{\omega} : x \to X(\omega) + x \Big)$$

A stochastic convex function is a random variable taking values in a space of convex functions. In the example given above, $F$ would be a stochastic convex function if for every $\omega$, $f_{\omega}$ is convex.