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

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