Electronic Journal of Probability

Functional inequalities for marked point processes

Ian Flint, Nicolas Privault, and Giovanni Luca Torrisi

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 116, 40 pp.

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

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Digital Object Identifier

Primary: 60G55: Point processes 60H07: Stochastic calculus of variations and the Malliavin calculus

Clark-Ocone formula Malliavin calculus marked point processes Poincaré inequality transportation cost inequalities variational representation

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Flint, Ian; Privault, Nicolas; Torrisi, Giovanni Luca. Functional inequalities for marked point processes. Electron. J. Probab. 24 (2019), paper no. 116, 40 pp. doi:10.1214/19-EJP369. https://projecteuclid.org/euclid.ejp/1570759241

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