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

$X_t^{x,H}$ 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 $\nu ={\eta }_1\cdot P_0$.

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.

Putand defineBecause the numéraire strategy is self-financing,Moreover,This also proves the required wealth formula.

Letwhere the sum is componentwise, and setThenAlso $K_{t+1}\cdot P_t=H_{t+1}\cdot {\tilde{P}}_t$, so subtracting the new holdings value gives $C_t^{x,K}={\tilde{C}}_t^{x,H}$.