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Martingale problem for superprocesses with non-classical branching functional
Leduc, Guillaume
Leduc, Guillaume
Date
2006
Author
Advisor
Type
Peer-Reviewed
Article
Published version
Article
Published version
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Abstract
The martingale problem for superprocesses with parameters (π, Π€, π) is studied where π(πΉπ ) 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 ItΓ΄βs formula. The proof of uniqueness requires that we find a sequence of (π, Π€, ππ) -superprocesses βapproximatingβ the (π, Π€, π)-superprocess, where ππ(πΉπ ) has the form Ξ»π (π ,ππ )πΉπ . Using an argument in [9], applied to the (π, Π€, ππ)-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 DawsonβGirsanov transformation.