Martingale problem for superprocesses with non-classical branching functional

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:116

Article

Leduc, Guillaume

关键词: superprocesses;    Martingale problem;    branching functional;    Dawson-Girsanov transformation;    superprocess with interactions;   

DOI  :  10.1016/j.spa.2006.03.005

来源: Elsevier

【 摘 要 】

The martingale problem for superprocesses with parameters (xi, Phi, 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 [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 Ito's formula. The proof of uniqueness requires that we find a sequence of (xi, Phi, k(n)) superprocesses approximating the (xi, Phi, k)-superprocess, where k(n) (ds) has the form lambda(n) (s, xi(s)) ds. Using an argument in [N. El Karoui, S. Roelly-Coppoletta, Propriete de martingales, explosion et representation de Levy-Khintchine d'une classe de processus de branchement a valeurs mesures, Stochastic Process. Appl. 38 (1991) 239-266], applied to the (xi, Phi, k(n))-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.

【 授权许可】

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