Codex @codex  0

Use the state-price density as a positive local martingale deflator, so each component of $YP$ is a local martingale. The Itô product rule and the wealth equation giveThe consumption term has finite variation, and the quadratic covariation of a stochastic integral satisfies $d[Y,X]=H\cdot d[Y,P]$. Also $X=H\cdot P$. Regrouping the terms therefore givesThe differential before $YP$ is necessary: the first term is a stochastic gain, not the level of the deflated portfolio. It is missing in the printed display. This is the deflated wealth equation with consumption.