Martingale problem for superprocesses with non-classical branching functional

The martingale problem for superprocesses with parameters ( ξ , Φ , k ) is studied where k ( d s ) may not be absolutely continuous with respect to the Lebesgue measure. This requires a generalization of the concept of martingale problem: we show that for any process X which partially solves the martingale problem, an extended form of the liftings defined in [E.B. Dynkin, S.E. Kuznetsov, A.V. Skorohod, Branching measure-valued processes, Probab. Theory Related Fields 99 (1995) 55–96] exists; these liftings are part of the statement of the full martingale problem, which is hence not defined for processes X who fail to solve the partial martingale problem. The existence of a solution to the martingale problem follows essentially from Itô’s formula. The proof of uniqueness requires that we find a sequence of ( ξ , Φ , k n ) -superprocesses “approximating” the ( ξ , Φ , k ) -superprocess, where k n ( d s ) has the form λ n ( s , ξ s ) d s . Using an argument in [N. El Karoui, S. Roelly-Coppoletta, Propriété de martingales, explosion et représentation de Lévy–Khintchine d’une classe de processus de branchement à valeurs mesures, Stochastic Process. Appl. 38 (1991) 239–266], applied to the ( ξ , Φ , k n ) -superprocesses, we prove, passing to the limit, that the full martingale problem has a unique solution. This result is applied to construct superprocesses with interactions via a Dawson–Girsanov transformation.