= Wealth-variable Legendre dual
{title2=$J(z)=\sup_{x>0}\{v(x)-xz\},\quad J^\prime=-x$}
For increasing strictly <concave> wealth value $v$, the dual $J(z)=\sup_x[v(x)-xz]$ is convex. At an interior optimum $z=v^\prime(x)$, $J^\prime=-x$, $J^{\prime\prime}=-1/v^{\prime\prime}>0$, and $v=J-zJ^\prime$. This sign convention is the negative of the <concave Legendre dual>. It can turn the optimized portfolio term of a <Hamilton-Jacobi-Bellman equation> into a linear second derivative.
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