Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

E. Löcherbach ()
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E. Löcherbach: Université de Cergy-Pontoise

Journal of Theoretical Probability, 2019, vol. 32, issue 1, 131-162

Abstract: Abstract We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Löcherbach (Stoch Process Appl 2017, http://arxiv.org/abs/1512.00265 ) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high-dimensional chain of differential equations driven by 2-dimensional Brownian motion. We study the large population, i.e., small noise limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.

Keywords: Hawkes processes; Piecewise deterministic Markov processes; Diffusion approximation; Sample path large deviations for degenerate diffusions; Control theory for degenerate diffusions; 60G17; 60G55; 60J60 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-017-0789-6

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