Lemma A.1 — $F_0(q)=f_0(q,-q)=-2H_v(-q)$ is non-decreasing

Let $f_0(q,\gamma):=q|\hat\sigma(\gamma)|^2+\hat c_2(\gamma)$ be the total cost of volatility borne by the producer when the unit cost of volatility is $q$ and the payment rate for volatility reduction is $\gamma$ ((A.10)), and let

Then for every real $q$,

and $F_0$ is non-decreasing on $\mathbb R$.

The lemma eliminates the volatility payment $\gamma$ from the producer's problem: the optimal $\gamma$ equals minus the producer's unit cost of volatility, and what remains is a monotone function of that cost. This is what turns the producer's HJB equation (A.11) into a scalar minimisation over $z$.

Formalization Note. The statement is for every real $q$, without a sign hypothesis; for $q<0$ the point $-q$ lies outside $\{\gamma\le0\}$, but $\hat b(-q)=\hat b(0)$ so the identity still holds.