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

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  • E. Löcherbach

    (Université de Cergy-Pontoise)

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.

Suggested Citation

  • E. Löcherbach, 2019. "Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes," Journal of Theoretical Probability, Springer, vol. 32(1), pages 131-162, March.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:1:d:10.1007_s10959-017-0789-6
    DOI: 10.1007/s10959-017-0789-6
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    References listed on IDEAS

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    2. Sylvain Delattre & Christian Y. Robert & Mathieu Rosenbaum, 2013. "Estimating the efficient price from the order flow: a Brownian Cox process approach," Papers 1301.3114, arXiv.org, revised Apr 2013.
    3. Benaim, Michel & Hirsch, Morris W., 1999. "Mixed Equilibria and Dynamical Systems Arising from Fictitious Play in Perturbed Games," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 36-72, October.
    4. Delattre, Sylvain & Robert, Christian Y. & Rosenbaum, Mathieu, 2013. "Estimating the efficient price from the order flow: A Brownian Cox process approach," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2603-2619.
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