Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 4 / a / Solution

Applying the Itô formula to $f(x)=x^2$ and the semimartingale $B$ givesHenceThe first term is a continuous local martingale and the second has finite variation. By uniqueness of the continuous semimartingale decomposition, $X$ could be a local martingale only if the finite-variation term were constant. Its derivative $B_s^2-1$ is not zero almost everywhere, so $X$ is not a local martingale.