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

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.

Enlarge the space by an independent Brownian motion $W$ and defineThe two terms have zero cross-variation andThe Lévy characterization of Brownian motion makes $B$ a Brownian motion. The residual $\int 1_{\{A=0\}}dX$ has zero quadratic variation and is therefore constant, so

An $L$-diffusion solves the martingale problem forfor every $f\in C_c^2(\mathbb{R})$,is a local martingale. Applying this to cutoff approximations of $x$ and $x^2$ shows thatis a continuous local martingale withPart b gives $M=\int |\sigma (X_s)|d{W}_s$. Changing the sign of $W$ predictably where $\sigma <0$, and filling the zero set with independent Brownian noise, produces a Brownian motion $B$ such that $M=\int \sigma (X_s)d{B}_s$. Thus