= Solution
Let the <infinitesimal generator> be the second-order operator
$$
Lg(x)=\frac12\sum_{i,j=1}^d a_{ij}(x)\partial_{ij}g(x)+\sum_{i=1}^d b_i(x)\partial_i g(x),
$$
where $a(x)$ is symmetric positive semidefinite and $b(x)$ is a vector. In the standard bounded-coefficient setting take $a,b$ bounded and <measurable>. An \b[<L-diffusion>] is a continuous <adapted process> $X$ for which, for every $g\in C_b^2(\mathbb R^d)$,
$$
\boxed{g(X_t)-g(X_0)-\int_0^t Lg(X_s)ds\text{ is a martingale}.}
$$
Here $C_b^2$ includes boundedness of the function and its derivatives through order two. This is the true-<martingale> version of the <diffusion martingale problem>. The matrix $a$ is the diffusivity, and $b$ is the drift. An <Itô diffusion> with coefficient $\sigma$ has $a=\sigma\sigma^{\mathsf T}$.
A more general local <martingale problem> may permit unbounded coefficients and require only a <local martingale>. That convention is weaker and must not silently replace the true <martingale> property used here and in part (b). The bounded-coefficient convention guarantees all finite-horizon integrability needed below; a sufficient more general replacement is $\mathbb E\int_0^T(\sum_i|b_i(X_s)|+\sum_i a_{ii}(X_s))ds<\infty$ for every $T$.
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