= Solution
Let $a(x)=\sigma(x)\sigma(x)^{\mathsf T}$. The multidimensional <Itô formula> gives the second-order <diffusion generator>
$$
\boxed{\mathcal L f(x)=\sum_{i=1}^d b_i(x)\partial_i f(x)+\frac12\sum_{i,j=1}^d a_{ij}(x)\partial_i\partial_jf(x).}
$$
Indeed,
$$
f(X_t)-\int_0^t(\mathcal Lf)(X_s)\,ds=f(X_0)+\int_0^t\nabla f(X_s)^{\mathsf T}\sigma(X_s)\,dW_s.
$$
The integrand is locally bounded along the continuous path after stopping on compact sets, because the coefficients and derivatives are continuous. Thus the final <stochastic integral> is a <local martingale>. No global growth or uniqueness assumption on this already-given <weak solution of a stochastic differential equation> is needed.
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