Write $V_t=X_t^{x,H}-C_t^{x,H}=H_{t+1}\cdot P_t>0$. Set $a_0=1$ and recursively
$$
a_t=a_{t-1}\frac{X_t^{x,H}}{V_t},
\qquad
\eta_{t+1}=a_tH_{t+1}.
$$
The factors are positive and adapted, so $\eta$ is previsible. For $t\geq1$,
$$
X_t^{\nu,\eta}=a_{t-1}X_t^{x,H}
=a_tV_t=\eta_{t+1}\cdot P_t,
$$
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>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-211/1/b/solution