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Martingale problem characterization for superprocesses. (English) Zbl 0854.60044

Summary: 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. We show that for any process \(X\) which partially solves a certain martingale problem, an extended form of the liftings defined by E. B. Dynkin, S. E. Kuznetsov and A. V. Skorokhod [Probab. Theory Relat. Fields 99, No. 1, 55-96 (1994; Zbl 0813.60076)] can be found, which permits the full martingale problem to be well defined. A sequence of \((\xi, \Phi, k^n)\)-superprocesses approximating the \((\xi, \Phi, k)\)-superprocess can be found, where \(k^n (ds)\) has the form \(\varphi (s, \xi_s) ds\). Using an argument of S. Roelly-Coppoletta [Stochastics 17, 43-65 (1986; Zbl 0598.60088)], applied to the \((\xi, \Phi, k^n)\)-superprocesses, we derive that the full martingale problem is well posed.

MSC:

60G44 Martingales with continuous parameter
60G57 Random measures
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