American-option superhedge with a funded reserve ($d{D}_t=(r{D}_t+a(S_t))dt$)

If a nonnegative value $V$ dominates the payoff and $a=rV-{\mathcal{L}}_{r}V\ge 0$, delta holdings $V\prime (S)$ combined with a reserve satisfying $dD=(rD+a(S))dt$ give a self-financing portfolio of wealth $V(S)+D$. The reserve retains the surplus instead of consuming it, so the wealth dominates the payoff at every exercise time.