= 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 $Z$, so
$$
d[Z,X]_t=\sum_iH_t^i\,d[Z,P^i]_t.
$$
Using $X=H\cdot P$ and the wealth equation,
$$
\begin{aligned}d(ZX)&=Z\,dX+X\,dZ+d[Z,X]\\&=H\cdot(Z\,dP+P\,dZ+d[Z,P])-Zc\,dt.
\end{aligned}
$$
Thus the <deflated wealth equation with consumption> is
$$
\boxed{d(Z_tX_t)=H_t\cdot d(Z_tP_t)-Z_tc_t\,dt.}
$$
No <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.
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