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.
Highlights
Point processes with a Papangelou conditional intensity ([6], [9], [18], [20], [21]) constitute an important class of point process models, which generalizes the Poisson process
We emphasize that the Poincare inequality proved in this paper concerns one-dimensional marked point processes and it holds under different conditions than the Poincare inequality for Gibbs point processes provided in [13]
Such exponential moments are controlled by a stochastic convex inequality for functionals of marked point processes with a Papangelou conditional intensity (Proposition 4.2), which is based on the Clark-Ocone formula and generalizes the corresponding result in [12]
Summary
Point processes with a Papangelou conditional intensity ([6], [9], [18], [20], [21]) constitute an important class of point process models, which generalizes the Poisson process. A key ingredient in the proofs is a new Clark-Ocone formula for square-integrable functionals of marked point processes with a Papangelou conditional intensity (Theorem 3.19), which generalizes the corresponding formula in [8] in two directions. The proof of the transportation cost inequality exploits its characterization via exponential moments proved in [10] Such exponential moments are controlled by a stochastic convex inequality for functionals of marked point processes with a Papangelou conditional intensity (Proposition 4.2), which is based on the Clark-Ocone formula and generalizes the corresponding result in [12]. The proof of the Clark-Ocone formula is based on the representation theorem for square-integrable martingales, on the use of an integration by parts formula for functionals of point processes with a Papangelou conditional intensity, and on an isometry formula for point processes with stochastic intensity, which allows us to identify the integrand appearing in the representation theorem. We include an appendix, where we prove some technical lemmas and propositions
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