Wealth-variable Legendre dual ($J(z)={\sup }_{x>0}\{v(x)-xz\},J^{\prime }=-x$)

For an increasing strictly concave wealth value $v$, the wealth-variable dual is the convex functionAt a unique interior maximizer where $v$ is differentiable, $z=v^{\prime }(x)$ and $J^{\prime }(z)=-x$, whenever the maximizer gives a differentiable dual. Thus $v(x)=J(z)-z{J}^{\prime }(z)$. If, additionally, $v$ is twice continuously differentiable and $v^{\prime \prime }(x)<0$ at that maximizer, the inverse function theorem yieldsStrict 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.