Generator of an SDE diffusion ($L=b\cdot \nabla +\frac{1}{2}{\sum }_k{\sigma }_k{\sigma }_k^T:D^2$)

The Itô formula identifies the operator of an SDE with drift $b$ and noise vector fields ${\sigma }_k$ as the displayed second-order operator. With bounded measurable coefficients, every solution satisfies the true martingale identity for $C_b^2$ test functions, hence is an L-diffusion in the martingale-problem sense.

// Target: probability-and-statistics.bigb