Deflated wealth equation with consumption
= Deflated wealth equation with consumption
{title2=$d(YX)=H\cdot d(YP)-Yc\,dt$}
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.