Abstract
The martingale problem for superprocesses with parameters (𝛏, Ф, 𝑘) is studied where 𝑘(𝒹𝑠) 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 (𝛏, Ф, 𝑘𝑛) -superprocesses “approximating” the (𝛏, Ф, 𝑘)-superprocess, where 𝑘𝑛(𝒹𝑠) has the form λ𝑛 (𝑠,𝛏𝑠)𝒹𝑠. Using an argument in [9], applied to the (𝛏, Ф, 𝑘𝑛)-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.
