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Martingale problem for superprocesses with non-classical branching functional

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  • Leduc, Guillaume

Abstract

The martingale problem for superprocesses with parameters ([xi],[Phi],k) 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 [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 ([xi],[Phi],kn)-superprocesses "approximating" the ([xi],[Phi],k)-superprocess, where has the form . 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 ([xi],[Phi],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.

Suggested Citation

  • Leduc, Guillaume, 2006. "Martingale problem for superprocesses with non-classical branching functional," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1468-1495, October.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:10:p:1468-1495
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    References listed on IDEAS

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    1. El Karoui, Nicole & Roelly, Sylvie, 1991. "Propriétés de martingales, explosion et représentation de Lévy--Khintchine d'une classe de processus de branchement à valeurs mesures," Stochastic Processes and their Applications, Elsevier, vol. 38(2), pages 239-266, August.
    2. Dawson, Donald A. & Fleischmann, Klaus, 1994. "A super-Brownian motion with a single point catalyst," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 3-40, January.
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