= Solution
Use the local convention for the <martingale problem>, which is appropriate for arbitrary measurable coefficients. A continuous <adapted process> $X$ solves $\mathbf M(a,b)$ if the coefficient integrals are locally finite and, for every $f\in C_c^\infty(\mathbb R)$,
$$
N_t^f=f(X_t)-f(X_0)-\int_0^t
\left(\frac12a(X_s)f''(X_s)+b(X_s)f'(X_s)\right)ds
$$
is a <local martingale>. Here locally finite means $\int_0^T(|a(X_s)|+|b(X_s)|)ds<\infty$ <almost surely> for each $T<\infty$. Suitable localization upgrades the identities to true <martingale> identities. With bounded coefficients and bounded test derivatives, this distinction disappears on finite horizons.
A <Markov diffusion> generated by $L$ is a time-homogeneous <Markov process> with continuous paths whose <infinitesimal generator> on the stated test domain is $L$. Its transition operators satisfy the generator limit, and <Dynkin formula> gives precisely the preceding <martingale problem> identities. Some terminology uses <L-diffusion> for the test-function identities alone; the Markov property is an additional assertion unless uniqueness supplies it.
Suppose now $a=\sigma^2>0$. Use smooth compactly supported functions agreeing with $x$ and $x^2$ on $[-R,R]$, and stop before exiting this interval. Removing these stops shows that
$$
N_t=X_t-X_0-\int_0^tb(X_s)ds
$$
is a <continuous local martingale>. Consequently $X$ is a <continuous semimartingale>. The <martingale problem> applied to the second cutoff function says that
$$
X_t^2-X_0^2-\int_0^t\bigl(a(X_s)+2X_sb(X_s)\bigr)ds
$$
is a <local martingale>. The <Itô formula> for $X^2$ also says that
$$
X_t^2-X_0^2-2\int_0^tX_sb(X_s)ds-[N]_t
$$
is a <local martingale>. Subtracting, the continuous finite-variation process $[N]_t-\int_0^ta(X_s)ds$ is a <local martingale> and starts at zero. The theorem that a <continuous finite-variation local martingale is constant> gives
$$
[N]_t=\int_0^t\sigma^2(X_s)ds.
$$
Define
$$
B_t=\int_0^t\frac1{\sigma(X_s)}\,dN_s.
$$
The integrand is predictable and is stochastically integrable, since its integral against the bracket is exactly $t$. Thus $B$ is a <continuous local martingale> with $B_0=0$ and $[B]_t=t$. The <Lévy characterization of Brownian motion> states that these conditions make $B$ a <Brownian motion> relative to the given filtration. Associativity of <stochastic integrals> recovers $N=\int\sigma(X)\,dB$, and therefore
$$
\boxed{dX_t=b(X_t)dt+\sigma(X_t)dB_t.}
$$
The original process is a <weak solution of a stochastic differential equation>, with its driving <Brownian motion> constructed on the original space.
Conversely, the <Itô formula> applied to any weak solution with locally integrable coefficients gives the <martingale problem> identities. A <Markov diffusion> with generator $L$ gives them by <Dynkin formula>. A <well-posed martingale problem>, with the usual measurable path-space setup, identifies a unique family of diffusion laws and yields the <Strong Markov property>. Without uniqueness, the test-function identities alone do not assert that every chosen solution is Markov. This specifies both the equivalence with weak equations and the role of uniqueness in identifying a diffusion.
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