 Solution

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