= Strict positive covariance with an increasing payoff
{title2=$\operatorname{Cov}(g(Z),Z)>0$}
If $Z$ is nondegenerate, $g$ is strictly increasing and the relevant products are <integrable>, an independent copy $Z'$ gives $2\operatorname{Cov}(g(Z),Z)=\mathbb E[(g(Z)-g(Z'))(Z-Z')]$. The integrand is positive exactly when $Z\ne Z'$, so the <covariance> is strictly positive. The same argument applies within each conditional law and yields a positive terminal <stock> hedge for an attainable increasing payoff.
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