= Solution
An $L$-diffusion solves the <martingale problem> for
$$
Lf(x)=b(x)f'(x)+\frac12\sigma(x)^2f''(x):
$$
for every $f\in C_c^2(\mathbb R)$,
$$
f(X_t)-f(X_0)-\int_0^tLf(X_s)\,ds
$$
is a local martingale. Applying this to cutoff approximations of $x$ and $x^2$ shows that
$$
M_t=X_t-X_0-\int_0^tb(X_s)\,ds
$$
is a continuous local martingale with
$$
[M]_t=\int_0^t\sigma(X_s)^2\,ds.
$$
Part b gives $M=\int|\sigma(X_s)|\,dW_s$. 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 $M=\int\sigma(X_s)\,dB_s$. Thus
$$
dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t.
$$
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