I'm working on a problem involving stochastic dominance and ``minimums'' of sets of random variables.
For concreteness, consider two distributions with cdfs $F(x)$ and $G(x)$. We say that $F$ first-order stochastically dominates $G$ if $F(x) \leq G(x)$ for all $x$.
Consider the set $S$ of non-negative random variables that all have the same median $m$. Even though this set has two properties that I'm interested in
it is very ``natural'', in the sense that it contains all distributions coherent with some partial information.
it has a ``minimum'' under stochastic dominance: the discrete random variable which assigns probability $50 \%$ to 0 and probability $50 \%$ to $m$. This random variable belongs in $S$, and is dominated by all other random variables in $S$.
My question is, are there other ``natural'' sets that have a minimum under stochastic dominance? For example, if I take the set $T(\mu,\sigma)$ of all positive random variables with mean $\mu$ and variance $\sigma^2$, does this set have a minimum under stochastic dominance?