 Solution

The holdings ${\varphi }_t$ and ${\pi }_t^i$ are numbers of shares, so the marked-to-market portfolio value isThe stock prices are ex-dividend prices. Over the trading interval, existing holdings gain ${\varphi }_{t}d{B}_t$ from the riskless account and ${\pi }_t\cdot d{S}_t$ from capital-price changes. In addition, ${\pi }_t^i$ shares of asset $i$ receive ${\pi }_t^i{D}_t^{i}dt$ in cash dividends. If this cash remains in the portfolio and changes of holdings are financed internally, the self-financing portfolio equation isRebalancing does not add a separate source of wealth: the cost of purchasing one holding is paid by selling another or by using the cash account. This is why one must specify the gains equation, rather than infer self-financing solely by differentiating the marked-to-market identity. The dividends are reinvested rather than withdrawn as consumption.