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Anticipative Markovian transformations on the Poisson space. (English) Zbl 1059.60071
Summary: We study anticipative transformations on the Poisson space in the framework introduced by J. Picard [Ann. Inst. Henri Poincaré, Probab. Stat. 32, 509–548 (1996; Zbl 0859.60045)]. Those are stochastic transformations that add particles to an initial condition or remove particles to it; they may be seen as a perturbation of the initial state with respect to the finite difference gradient \(D\) introduced by D. Nualart and J. Vives [in: Séminaire de probabilités XXIV 1988/89. Lect. Notes Math. 1426, 154–165 (1990; Zbl 0701.60048)]. We study here an analogue of the anticipative flows on the Wiener space, which is in our context a Markov process taking its values in the Poisson space \(\Omega \) and look for some criterion ensuring that the image of the Poisson probability \(\mathbb P\) under the transformation is absolutely continuous with respect to \(\mathbb P\). We obtain results which are close to the results of O. Enchev and D. W. Stroock [J. Funct. Anal. 116, 449–477 (1996; Zbl 0804.60072)] founded in the Wiener space case.

60H07 Stochastic calculus of variations and the Malliavin calculus
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text: DOI
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