Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 1 / b / Solution

Write $V_t=X_t^{x,H}-C_t^{x,H}=H_{t+1}\cdot P_t>0$. Set $a_0=1$ and recursivelyThe factors are positive and adapted, so $\eta$ is previsible. For $t\ge 1$,while the same identity at $t=0$ defines $\nu ={\eta }_1\cdot P_0$. Thus consumption is zero and wealth is strictly positive, so $\eta$ is a numéraire portfolio.