Investment with fixed debt service ($dw=\sigma \theta dW+[rw+(\mu -r)\theta -h-c]dt$)

A fixed loan principal contributes a constant interest outflow $h$. If the objective stops when available portfolio wealth reaches zero, the value function has an absorbing boundary $V(0)=0$. This differs from requiring the portfolio to finance the debt service forever.

For the wealth-variable Legendre dual $J(z)={\sup }_{w\ge 0}[V(w)-zw]$, a finite slope $z_{\ast }=V^{\prime }(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\ge z_{\ast }$. Matching gives $J(z_{\ast })=J^{\prime }(z_{\ast })=0$. It does not generally give $J^{\prime \prime }(z_{\ast }^{-})=0$: killing at ruin permits the dual curvature to jump.