The proposed variable satisfies
$$
\mathbb E(ZP_1)
=\mathbb E(P_1P_1^T)Q^{-1}P_0=P_0.
$$
No arbitrage and the <fundamental theorem of asset pricing> provide a strictly positive one-period deflator $Y$ with $\mathbb E(YP_1)=P_0$. Part c makes such a deflator unique in a complete market, so $Z=Y>0$ almost surely.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-211/3/d/solution