 Solution

The marked-to-market value of the current holdings is $X_t=H_t\cdot P_t$. In a self-financing portfolio, purchases are paid for by sales within the portfolio, so rebalancing itself changes no net wealth. Asset price changes contribute $H_t\cdot d{P}_t$, and nonnegative consumption removes $c_{t}dt$ from wealth. HenceThe holdings must be predictable and stochastically integrable. The second identity is the self-financing definition with withdrawals; it is not obtained by incorrectly setting the changes in the holdings themselves to zero.