Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 6 / a

For the diffusion operatora continuous adapted process solves the diffusion martingale problem when, for every $f\in C_b^2({\mathbb{R}}^d)$,Here $C_b^2$ means that the function and its derivatives through order two are bounded and continuous. An equivalent compact-support/local convention is often used for the general martingale problem; the bounded-coefficient setting lets us use the true-martingale convention explicitly. The matrix $a$ is interpreted as a covariance matrix in the diffusion formulation, hence is symmetric nonnegative; the displayed operator in any case uses only its symmetric part.

A well-posed martingale problem has existence and uniqueness in law on continuous path space for each specified starting point, or for each initial law in the initial-law formulation. This is uniqueness of the process law, not an assertion of pathwise uniqueness on a prescribed Brownian probability space.