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

Put $F_t=\exp (-i\theta X_t-\frac{1}{2}{\theta }^2{A}_t)$. The Itô formula givesThe explicit decreasing exponential contributes half the finite-variation term; the other half is the quadratic correction from $e^{-i\theta X_t}$. Now the Itô product rule and part (b) yieldThe finite-variation terms cancel because $-i\theta (i\theta )={\theta }^2$. This proves the product is a local martingale. Moreover $|F_t|=e^{-{\theta }^2{A}_t/2}\le 1$ and $|M_t|\le 1$. The bounded local martingale criterion therefore proves the product is a true martingale.