Applying the <Itô formula> to $f(x)=x^2$ and the semimartingale $B$ gives
$$
B_t^2=2\int_0^tB_s\,dB_s+t.
$$
Hence
$$
X_t=-2\int_0^tB_s\,dB_s+\int_0^t(B_s^2-1)\,ds.
$$
The 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/4/a/solution