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2019 Functional inequalities for marked point processes
Ian Flint, Nicolas Privault, Giovanni Luca Torrisi
Electron. J. Probab. 24: 1-40 (2019). DOI: 10.1214/19-EJP369

Abstract

In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincaré inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the law of marked temporal point processes with a Papangelou conditional intensity, and extends a related inequality which is known to hold on a general Poisson space. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Our results are shown to apply to classes of renewal, nonlinear Hawkes and Cox point processes.

Citation

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Ian Flint. Nicolas Privault. Giovanni Luca Torrisi. "Functional inequalities for marked point processes." Electron. J. Probab. 24 1 - 40, 2019. https://doi.org/10.1214/19-EJP369

Information

Received: 10 October 2018; Accepted: 28 September 2019; Published: 2019
First available in Project Euclid: 11 October 2019

zbMATH: 07142910
MathSciNet: MR4029419
Digital Object Identifier: 10.1214/19-EJP369

Subjects:
Primary: 60G55, 60H07

JOURNAL ARTICLE
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