×

Stability for Hawkes processes with inhibition. (English) Zbl 1439.60048

Summary: We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G57 Random measures
60J25 Continuous-time Markov processes on general state spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Brémaud, P., Massoulié, L. Stability of nonlinear Hawkes processes. The Annals of Probability 24(3), (1996), 1563-1588. · Zbl 0870.60043
[2] Clinet, S., Yoshida, N. Statistical inference for ergodic point processes and application to Limit Order Book. Stoch. Proc. Appl. 127, (2017), 1800-1839. · Zbl 1373.60086 · doi:10.1016/j.spa.2016.09.014
[3] Costa, M., Graham, C., Marsalle, L., Tran, Viet Chi. Renewal in Hawkes processes with self-excitation and inhibition. arXiv:1801.04645. · Zbl 1473.60081
[4] Delattre, S., Fournier, N., Hoffmann, M. Hawkes processes on large networks. Ann. Appl. Probab. 26, (2016), 216-261. · Zbl 1334.60082 · doi:10.1214/14-AAP1089
[5] Ditlevsen, S., Löcherbach, E. Multi-class oscillating systems of interacting neurons. Stoch. Proc. Appl. 127, (2017), 1840-1869. · Zbl 1367.92024 · doi:10.1016/j.spa.2016.09.013
[6] Hawkes, A. G., Oakes, D. A cluster process representation of a self-exciting process. J. Appl. Probab. 11, (1974), 493-503. · Zbl 0305.60021 · doi:10.2307/3212693
[7] Höpfner, R., Löcherbach, E. Statistical models for Birth and Death on a Flow: Local absolute continuity and likelihood ratio processes. Scandinavian Journal of Statistics 26(1), (1999), 107-128. · Zbl 0933.60086
[8] Ikeda, N., Nagasawa, M., Watanabe, S. A construction of Markov processes by piecing out. Proc. Japan Acad. 42(4), (1966), 370-375. · Zbl 0178.53401 · doi:10.3792/pja/1195522037
[9] Kaplan, N. The multitype Galton-Watson process with immigration. The Annals of Probability 6, (1973), 947-953. · Zbl 0292.60135 · doi:10.1214/aop/1176996802
[10] Löcherbach, E., Loukianova, D. On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stoch. Proc. Appl. 118, (2008), 1301-1321. · Zbl 1202.60122 · doi:10.1016/j.spa.2007.09.003
[11] Meyn, S. P., Tweedie, R. L. Stability of Markovian processes I: Criteria for discrete-time chains. Adv. Appl. Probab. 24, (1992), 542-574. · Zbl 0757.60061 · doi:10.2307/1427479
[12] Meyn, S. · Zbl 0781.60053 · doi:10.2307/1427522
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.