# Transformation of random variables and second order stochastic dominance

Suppose $$X$$ and $$Y$$ are two random variables where $$X$$ has SOSD (second order stochastic dominance) over $$Y$$. Let $$g(\cdot)$$ be a monotonic function and $$X' = g(X)$$ and $$Y' = g(Y)$$.

Under what conditions of $$g(\cdot)$$ $$X'$$ has SOSD over $$Y'$$? I know that if $$g$$ is linear, SOSD property is reserved. Is there any sufficient and necessary conditions of $$g$$ that assures SOSD property?

As long as $$g(\cdot)$$ is Nondecreasing and Concave
Definition: For any lotteries $$F$$ and $$G$$, $$F$$ second-order stochastically dominates $$G$$ if and only if the decision maker weakly prefers $$F$$ to $$G$$ under every weakly increasing concave utility function $$u$$. Link