= Wealth-cap investment boundary
{title2=$J^\prime(z_*)=-1,\quad J^{\prime\prime}(z_*)=0$}
A wealth-cap investment boundary prevents wealth from exceeding a fixed historical maximum. In the <wealth-variable Legendre dual> for wealth normalized by that maximum, $J^\prime(z_*)=-1$ and vanishing optimal portfolio volatility gives $J^{\prime\prime}(z_*)=0$. For power utility and a high-water tax, the smooth tax-paying solution has boundary dual curvature $(1-\alpha\tau)/(Rz_*)$, so it cannot be admissible when $\alpha\tau>1$.
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