Well-posed martingale problem

A martingale problem is well-posed for a specified test domain and class of paths if it admits a solution for every specified starting point and that solution is unique in law. Existence on one fixed probability space is not required. Test-domain and true-versus-local conventions must be stated. For bounded diffusion coefficients and $C_b^2$ test functions, boundedness on every finite horizon upgrades the local identities to true martingale identities.