 Solution

Apply the Itô product rule and the quadratic covariation rule for a stochastic integral. The finite variation consumption term has zero quadratic covariation with $Z$, soUsing $X=H\cdot P$ and the wealth equation,Thus the deflated wealth equation with consumption isNo finite variation assumption on $H$ is needed: the self-financing equation and the stochastic-integral quadratic covariation rule already incorporate the portfolio's financing constraint.