= Banking stock dividends versus reinvesting them
{title2=$A_t=\int_0^t\delta S_u e^{r(t-u)}du,\quad X_t=S_t+A_t$}
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 $dA_t=rA_tdt+\delta S_tdt$, this is a <self-financing portfolio>: its only cash inflow to the bank is the dividend earned by its stock holding. A claim paying $nX_T$ is replicated by $n$ copies and costs $nX_t$, including the known accrued-dividend balance at time $t$.
A different claim paying only $nS_T$ costs $nS_te^{-\delta(T-t)}$ in the <Black-Scholes model> with dividends. It is replicated by $ne^{-\delta(T-t)}$ 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.
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