 Banking stock dividends versus reinvesting them

For a stock with constant continuous dividend yield $\delta$, holding one fixed share and depositing each dividend in a bank account at rate $r$ gives the displayed wealth process. Since $d{A}_t=r{A}_{t}dt+\delta S_{t}dt$, this is a self-financing portfolio: its only cash inflow to the bank is the dividend earned by its stock holding. A claim paying $n{X}_T$ is replicated by $n$ copies and costs $n{X}_t$, including the known accrued-dividend balance at time $t$.

A different claim paying only $n{S}_T$ costs $n{S}_t{e}^{-\delta (T-t)}$ in the Black-Scholes model with dividends. It is replicated by $n{e}^{-\delta (T-t)}$ shares with dividends reinvested in new shares: the share count grows at rate $\delta$ and reaches $n$ at expiry. Thus banking dividends and reinvesting them produce distinct terminal payoffs and distinct current values, even though both strategies are self-financing.