Deflated wealth equation with consumption ($d(YX)=H\cdot d(YP)-Ycdt$)

A self-financing investor with consumption satisfies $X=H\cdot P$ and $dX=H\cdot dP-cdt$. The Itô product rule and $d[Y,X]=H\cdot d[Y,P]$ give the displayed equation. For a positive local martingale deflator $Y$, the first term is a stochastic integral against the vector of deflated asset-price local martingales, not their current levels.

If wealth and consumption are nonnegative, this local martingale is bounded below by minus the finite initial deflated capital. Adding that capital gives a nonnegative local martingale, hence a supermartingale by the Conditional Fatou lemma. Consequently expected discounted consumption cannot exceed initial deflated wealth. The consumption sign and predictable stochastic integrability are part of the hypotheses.