Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 5 / b / Solution

Let $A=⟨X⟩$ be the quadratic variation. We use bilinear quadratic covariation for the complex martingale: $⟨M,X⟩=⟨\mathrm{Re}M,X⟩+i⟨\mathrm{Im}M,X⟩$. The Itô formula givesBy the Itô product rule, the finite-variation part of $d(e^{-2i\theta X}M)$ isPart (a) says the product is a martingale, so uniqueness of the continuous semimartingale decomposition makes this finite-variation part zero. For $\theta \ne 0$, division givesFor $\theta =0$, $M\equiv 1$ and both sides are zero, so the identity holds without exception.