WIAS Preprint No. 319, (1997)

Martingale problem for (Xi, Phi, k) - superprocesses


  • Leduc, Guillaume
    ORCID: 0000-0003-3800-8211




The martingale problem for superprocesses with parameters (ξ,ɸ,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 [11] 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,ɸ,kn)-superprocesses "approximating" the (ξ,ɸ,k)-superprocess, where kn(ds) has the form λn (s,ξs)ds. Using an argument in [12], applied to the (ξ,ɸ,kn)-superprocesses, we derive, passing to the limit, that the full martingale problem has a unique solution. This result is apply to construct superprocesses with interactions via a Dawson-Girsanov transformation.

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