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On Maximal Inequalities for Purely Discontinuous Martingales in Infinite Dimensions

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Séminaire de Probabilités XLVI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2123))

Abstract

The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.

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Notes

  1. 1.

    Just to avoid (unlikely) confusion, we note that \(\mathbb{E}(\cdots \,)^{\alpha }\) always stands for the expectation of (⋯ )α, and not for \([\mathbb{E}(\cdots \,)]^{\alpha }\).

  2. 2.

    The subscript ⋅  c means “with compact support”, and C b 2(H) denotes the set of twice continuously differentiable functions \(\varphi: H \rightarrow \mathbb{R}\) such that \(\varphi\), \(\varphi ^{{\prime}}\) and \(\varphi ^{{\prime\prime}}\) are bounded.

  3. 3.

    One can verify that the proof in [22] goes through without any change also for Hilbert-space-valued martingales.

  4. 4.

    The definition of class D is not standard and it is introduced just for the sake of concision.

  5. 5.

    It should be mentioned that there are discrete-time real-valued analogs of BJ 2, p , p ≥ 2, that go under the name of Burkholder-Rosenthal (in alphabetical but reverse chronological order: Rosenthal [38] proved it for sequences of independent random variables in 1970, then Burkholder [5] extended it to discrete-time (real) martingales in 1973), and some authors speak of continuous-time Burkholder-Rosenthal inequalities. One may then also propose to use the expression Burkholder-Rosenthal-Novikov inequality, that, however, seems too long.

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Acknowledgements

A large part of the work for this paper was carried out during visits of the first-named author to the Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn, invited by S. Albeverio. The second-named author is supported by the DFG through the SFB 701.

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Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Carlo Marinelli

  2. Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany

    Michael Röckner

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  1. Carlo Marinelli
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Correspondence to Carlo Marinelli .

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  1. Université de Versailles-St-Quentin Laboratoire de Mathématiques, Versailles, France

    Catherine Donati-Martin

  2. INRIA, Nancy-Université, Vandoeuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, Versailles, France

    Alain Rouault

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Marinelli, C., Röckner, M. (2014). On Maximal Inequalities for Purely Discontinuous Martingales in Infinite Dimensions. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_10

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