Martingale problem for superprocesses with non-classical branching functional
- 作者：Guillaume Leduc
- 关键词：Superprocesses ; Martingale problem ; Branching functional ; Dawson– ; Girsanov transformation ; Superprocess with interactions
- 刊名：Stochastic Processes and their Applications
- 出版年：2006
- 期刊代码：57_03044149
- 类别：mt
- 出版时间：October 2006
- 卷：116
- 期：10
- 页码：1468-1495
- 文件大小：430 K
摘要
The martingale problem for superprocesses with parameters (ξ,Φ,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 (ξ,Φ,kn)-superprocesses “approximating” the (ξ,Φ,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 (ξ,Φ,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.