Martingale problem for superprocesses with non-classical branching functional - Leduc, Guillaume (2006)
The martingale problem for superprocesses with parameters (, Ф, ) is studied where () 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 [8] 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 (, Ф, ) -superprocesses “approximating” the (, Ф, )-superprocess, where () has the form λ (,). Using an argument in [9], applied to the (, Ф, )-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.