Dual ruin boundary with debt service
= Dual ruin boundary with debt service
{title2=$J(z_*)=J'(z_*)=0$}
For the <wealth-variable Legendre dual> $J(z)=\sup_{w\geq0}[V(w)-zw]$, a finite slope $z_*=V'(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\geq z_*$. Matching gives $J(z_*)=J'(z_*)=0$. It does not generally give $J''(z_*^-)=0$: killing at ruin permits the dual curvature to jump.