High-water mark tax boundary condition ($V_{\bar{w}}-\tau V_w=0$)

At an active maximum-raising boundary in high-water mark investment taxation, the finite-variation part of the Itô formula is $(V_{\bar{w}}-\tau V_w)d\bar{w}$, so smooth optimality sets this coefficient to zero. If the maximizing policy avoids increasing the maximum, the corresponding inequality $V_{\bar{w}}-\tau V_w\le 0$ is compatible with a wealth-cap investment boundary instead.