The Annals of Probability

Stability of nonlinear Hawkes processes

Pierre Brémaud and Laurent Massoulié

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Abstract

We address the problem of the convergence to equilibrium of a general class of point processes, containing, in particular, the nonlinear mutually exciting point processes, an extension of the linear Hawkes processes, and give general conditions guaranteeing the existence of a stationary version and the convergence to equilibrium of a nonstationary version, both in distribution and in variation. We also give a new proof of a result of Kerstan concerning point processes with bounded intensity and general nonlinear dynamics satisfying a Lipschitz condition.

Article information

Source
Ann. Probab. Volume 24, Number 3 (1996), 1563-1588.

Dates
First available in Project Euclid: 9 October 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1065725193

Digital Object Identifier
doi:10.1214/aop/1065725193

Mathematical Reviews number (MathSciNet)
MR1411506

Zentralblatt MATH identifier
0870.60043

Subjects
Primary: 60G55: Point processes 60H20: Stochastic integral equations

Keywords
Stochastic processes point processes stochastic intensity stationary point processes mutually exciting point processes Hawkes processes

Citation

Brémaud, Pierre; Massoulié, Laurent. Stability of nonlinear Hawkes processes. Ann. Probab. 24 (1996), no. 3, 1563--1588. doi:10.1214/aop/1065725193. http://projecteuclid.org/euclid.aop/1065725193.


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