= Solution
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 give
$$
d(YX)=YH\cdot dP-Yc\,dt+X\,dY+d[Y,X].
$$
The <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 gives
$$
\boxed{d(Y_tX_t)=H_t\cdot d(Y_tP_t)-Y_tc_t\,dt.}
$$
The 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>.
Back to article page
