Exercise: Reweighting a Two-State Model

Prerequisites: Change of Measure

Problem

Let $\Omega=\{u,d\}$, $\mathbb{P}(u)=0.7$, $\mathbb{P}(d)=0.3$, $\mathbb{Q}(u)=0.4$, and $\mathbb{Q}(d)=0.6$. Compute $Z=d\mathbb{Q}/d\mathbb{P}$ in each state and verify $\mathbb{E}^{\mathbb{P}}[Z]=1$.

Hint