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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.

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Bichteler, K.; Gravereaux, J.B.; Jacod, J., Malliavin calculus for processes with jumps, stochastic monographs vol. 2, (1987), Gordon and Breach Science Publishers London · Zbl 0706.60057
[2] Buckdahn, R., Anticipative Girsanov transformation, Probab. theory related fields, 89, 211-238, (1991) · Zbl 0722.60059
[3] Buckdahn, R., Enchev, O., 1989. Nonlinear transformations on the abstract Wiener space. Technical Report 240 (Neue Folge), Humboldt Univ, Berlin.
[4] Cameron, R.H.; Martin, W.T., Transformation of Wiener integral under translation, Ann. math., 45, 386-396, (1944) · Zbl 0063.00696
[5] Carlen, E., Pardoux, E., 1990. Differential calculus and integration by parts on Poisson space. In: Albeverio, S., Blanchard, Ph., Testard, D. (Eds.), Stochastics, Algebra and Analysis in Classical and Quantum Dynamics. Kluwer, Dordrecht, pp. 63-73. · Zbl 0685.60056
[6] Dellacherie, C.; Meyer, P.A., Probabilités et potentiel tome II, (1980), Hermann Paris
[7] Enchev, O.; Stroock, D.W., Anticipative diffusions and related change of measure, J. funct. anal., 116, 449-477, (1996) · Zbl 0804.60072
[8] Girsanov, I.V., On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory probab. appl., 5, 285-301, (1960) · Zbl 0100.34004
[9] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, (1984), Springer Berlin · Zbl 0830.60025
[10] Nualart, D., The Malliavin calculus and related topics, (1995), Springer Berlin · Zbl 0837.60050
[11] Nualart, D., Vives, J., 1990. Anticipative calculus for the Poisson process based on the Fock space. In: Séminaire de Probabilité XXIV, Lecture Notes in Mathematics, Vol. 1426, Springer, Berlin. · Zbl 0701.60048
[12] Picard, J., Formules de dualité sur l’espace de Poisson, Ann. inst. Henri Poincaré, 32, 4, 509-548, (1996) · Zbl 0859.60045
[13] Picard, J., Transformations et équations anticipantes pour LES processus de Poisson, Ann. math. blaise Pascal, 3, 1, 111-123, (1996) · Zbl 0856.60059
[14] Privault, N., Girsanov theorem for anticipative shifts on the Poisson space, Probab. theory related fields, 104, 1, 61-76, (1996) · Zbl 0838.60038
[15] Rockner, M., 1998. Stochastic analysis on configuration space, basic ideas and recent results in New directions in Dirichlet forms. Studies in Advanced Mathematics, Vol. 8, Amer. Math. Soc., Providence, RI, pp. 157-231. · Zbl 1037.58026
[16] Ustunel, A.S.; Zakai, M., Transformation of measure on Wiener space, (2000), Springer Berlin · Zbl 0938.46045
[17] Ustunel, A.S.; Zakai, M., Transformation of the Wiener measure under anticipative flows, Probab. theory related fields, 93, 91-136, (1992) · Zbl 0767.60046
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