 Solution

The mark-to-market value of holdings $H_t$ in the asset-price vector is the dot product $X_t=H_t\cdot P_t$. A self-financing strategy pays for every rebalancing from within the portfolio. Over a short interval, the gains on the currently held assets are $H_t\cdot d{P}_t$, while consumption removes $c_{t}dt$ units of wealth. ThereforeThe holdings are predictable and integrable against the price semimartingale; the consumption rate is nonnegative and suitably measurable. This is the continuous-time self-financing-with-consumption convention.