Pré-Publication, Document De Travail Année : 2025

Nonparametric estimation of the stationary density for Hawkes-diffusion systems with known and unknown intensity

Résumé

We investigate the nonparametric estimation problem of the density π, representing the stationary distribution of a two-dimensional system (Z t ) t∈[0,T ] = (X t , λ t ) t∈[0,T ] . In this system, X is a Hawkes-diffusion process, and λ denotes the stochastic intensity of the Hawkes process driving the jumps of X. Based on the continuous observation of a path of (X t ) over [0, T ], and initially assuming that λ is known, we establish the convergence rate of a kernel estimator π (x * , y * ) of π (x * , y * ) as T → ∞. Interestingly, this rate depends on the value of y * influenced by the baseline parameter of the Hawkes intensity process. From the rate of convergence of π (x * , y * ), we derive the rate of convergence for an estimator of the invariant density λ. Subsequently, we extend the study to the case where λ is unknown, plugging an estimator of λ in the kernel estimator and deducing new rates of convergence for the obtained estimator. The proofs establishing these convergence rates rely on probabilistic results that may hold independent interest. We introduce a Girsanov change of measure to transform the Hawkes process with intensity λ into a Poisson process with constant intensity. To achieve this, we extend a bound for the exponential moments for the Hawkes process, originally established in the stationary case, to the non-stationary case. Lastly, we conduct a numerical study to illustrate the obtained rates of convergence of our estimators.

Fichier principal
Vignette du fichier
Hawkes_new.pdf (550.44 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-05290064 , version 1 (30-09-2025)

Licence

Identifiants

  • HAL Id : hal-05290064 , version 1

Citer

Chiara Amorino, Charlotte Dion-Blanc, Arnaud Gloter, Sarah Lemler. Nonparametric estimation of the stationary density for Hawkes-diffusion systems with known and unknown intensity. 2025. ⟨hal-05290064⟩
203 Consultations
162 Téléchargements

Partager

  • More