Remarks on the Poisson stochastic process (III) (On a property of the homogeneous Poisson process)
C. Ryll-Nardzewski (1954)
Studia Mathematica
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C. Ryll-Nardzewski (1954)
Studia Mathematica
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Francisco Jiménez Gómez, Mariano J. Valderrama Bonnet (1992)
Extracta Mathematicae
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A. Dermoune (1997)
Annales mathématiques Blaise Pascal
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Dražen Pantić, Predrag Peruničić (1991)
Publications de l'Institut Mathématique
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Nicolas Privault (2012)
Annales de l'I.H.P. Probabilités et statistiques
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We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai ( (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of...
Klaus Krickeberg (1980)
Banach Center Publications
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Sznitman, Alain-Sol (2009)
Electronic Journal of Probability [electronic only]
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K. Urbanik (1997)
Colloquium Mathematicum
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The paper deals with nonnegative stochastic processes X(t,ω)(t ≤ 0) not identically zero with stationary and independent increments right-continuous sample functions and fulfilling the initial condition X(0,ω)=0. The main aim is to study the moments of the random functionals for a wide class of functions f. In particular a characterization of deterministic processes in terms of the exponential moments of these functionals is established.