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

Applying the Itô product rule to $YB$ makes its drift vanish automatically. For $YS$, the drift isThus both deflated prices are local martingales whenSince self-financing gives $dX=\varphi dB+\pi dS$, another application of the product rule, including $d[X,Y]$, cancels the drift and yields