Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 5 / b / Solution

The product rule and self-financing identity show that $XY$ is the stochastic integral of $H$ against the vector of deflated prices $YP$, hence is a local martingale. It is nonnegative by admissibility and positivity of $Y$. Every nonnegative local martingale is a supermartingale, so