Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 2 / a / 1 / Solution

Interpret positivity of $Y$ as strict positivity almost surely. Suppose the stated bounded positive pricing variable exists. For a vector $H$ satisfying the two sign restrictions,A nonnegative random variable with zero expectation vanishes almost surely. Since $Y>0$, this gives $H\cdot P=0$ almost surely and $H\cdot p=0$. The supplied nondegeneracy condition now yields $\boxed{H=0}$. This proves the direction from a one-period martingale deflator to the separating-portfolio condition; no asset-pricing theorem has been invoked.