Martingale Problems and Path Properties of Solutions

Existence and uniqueness of solutions of martingale problems, not only in the class of r.c.l.l. or continuous process, but also in the class of progressively measurable processes has become important. In this article we give several examples which show that existence or uniqueness in any one class need not imply the same in another class. We also give an example of a well-posed martingale problem where the operator is not a core for the generator of the corresponding Markov process. AMS 2000 subject classification: Primary 60J25, Secondary 60G17, 60J35.