Exponential ergodicity for diffusions with jumps driven by a Hawkes process

Charlotte Dion (1, 2) , Sarah Lemler (3, 4) , Eva Löcherbach (5, 6)

1 SU - Sorbonne Université
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
3 CentraleSupélec
4 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
5 UP1 - Université Paris 1 Panthéon-Sorbonne
6 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)

In this paper, we introduce a new class of processes which are diffusions with jumps, where the jumps are driven by a multivariate linear Hawkes process, and study their long-time behavior. In the case of exponential memory kernels for the underlying Hawkes process, we establish conditions for the positive Harris recurrence of the couple (X, λ), where X denotes the diffusion process and λ the stochastic intensity of the driving Hawkes. As a direct consequence of the Harris recurrence, we obtain the ergodic theorem for X. Furthermore, we provide sufficient conditions under which the process is exponentially β−mixing. This paper is the foundation for a second paper [11] in which we carry a statistical study of diffusions driven by Hawkes jumps, with a view towards applications in neuroscience.

Mots clés: 60F99; 60H99; 60G55 Keywords: Diffusions with jumps; Diffusions with jumps; Linear Hawkes process; Piecewise deterministic Markov processes (PDMP); Ergodicity; Foster-Lyapunov drift criteria; β-mixing; AMS Classification: 60J35

Domaines: Probabilités [math.PR]

Charlotte Dion, Sarah Lemler, Eva Löcherbach. Exponential ergodicity for diffusions with jumps driven by a Hawkes process. 2019. ⟨hal-02094514v1⟩