 Solution

$X_t^{x,H}$ is the wealth delivered by the holdings chosen before time $t$, after receiving their time-$t$ dividends. The investor then chooses $H_{t+1}$, whose market value is $H_{t+1}\cdot P_t$, and consumes the remainder $C_t^{x,H}$. A numéraire portfolio is a strategy $\eta$ with zero consumption and strictly positive wealth $N_t=X_t^{\nu ,\eta }$ at every time, where $\nu ={\eta }_1\cdot P_0$.