= Wealth-variable Legendre dual
{title2=$J(z)=\sup_{x>0}\{v(x)-xz\},\quad J^\prime=-x$}
For an increasing strictly <concave> wealth value $v$, the wealth-variable dual is the convex function
$$
J(z)=\sup_{x>0}\{v(x)-xz\}.
$$
At a unique interior maximizer where $v$ is differentiable, $z=v'(x)$ and $J'(z)=-x$, whenever the maximizer gives a differentiable dual. Thus $v(x)=J(z)-zJ'(z)$. If, additionally, $v$ is twice continuously differentiable and $v''(x)<0$ at that maximizer, the <inverse function theorem> yields
$$
J''(z)=-\frac1{v''(x)}>0.
$$
Strict concavity alone does not ensure nonzero curvature. The <Inada utility with vanishing curvature> is an explicit example where the dual has a first derivative but fails to have a finite second derivative at the corresponding price.
This definition is the negative of the <concave Legendre dual>. Under the stated smooth nonzero-curvature hypotheses, it can convert the optimized portfolio term of a <Hamilton-Jacobi-Bellman equation> into a linear second derivative. Boundary optimizers require the appropriate one-sided or subgradient formulation instead of the interior derivative identities.
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