= 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 $H_t\cdot dP_t$, while <consumption> removes $c_tdt$ units of wealth. Therefore
$$
\boxed{X_t=H_t\cdot P_t,\qquad dX_t=H_t\cdot dP_t-c_tdt.}
$$
The 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.
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