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

For a deterministic $t\le T$, conditional symmetry makes the conditional characteristic function of $X_T-X_t$ invariant under $\theta \mapsto -\theta$. The bounded real and imaginary parts of the exponential are legitimate test functions. HenceThe right side is a bounded complex martingale. Both sides have continuous versions by the assumptions; equality on rational times and continuity make them indistinguishable. Thus $e^{-2i\theta X_t}{M}_t$ is a martingale on $[0,T]$. Complex martingale assertions mean the corresponding assertions for both real and imaginary parts.