Title :

Renewal in Hawkes processes with self-excitation and inhibition

Author(s) :

Costa, Manon [Auteur]
Institut de Mathématiques de Toulouse UMR5219 [IMT]
Graham, Carl [Auteur]
Analyse d’interactions stochastiques intelligentes et coopératives [ASCII]
École polytechnique [X]
Marsalle, Laurence [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Tran, Chi [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Advances in Applied Probability

Pages :

879-915

Publisher :

Applied Probability Trust

Publication date :

2020-09-24

ISSN :

0001-8678

English keyword(s) :

Point processes
self-excitation
inhibition
renewal theory
ergodic limit theo- rems
concentration inequalities
Galton-Watson trees
M/G/∞ queues

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Statistiques [math.ST]

English abstract : [en]

This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction ...
Show more >This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities , and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/∞ queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL