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Article Dans Une Revue Theory of Probability and Mathematical Statistics Année : 2021

Exponential ergodicity for diffusions with jumps driven by a Hawkes process

Résumé

In this paper, we introduce a new class of processes which are diffusions with jumps driven by a multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. In the case of exponential memory kernels for the underlying Hawkes process we establish conditions for the positive Harris recurrence of the couple (X, Y), where X denotes the diffusion process and Y the piecewise deterministic Markov process (PDMP) defining the stochastic intensity of the driving Hawkes. As a direct consequence of the Harris recurrence, we obtain the ergodic theorem for X. Furthermore, we provide sufficient conditions under which the process is exponentially β−mixing.
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Dates et versions

hal-02094514 , version 1 (10-04-2019)
hal-02094514 , version 2 (18-07-2019)
hal-02094514 , version 3 (20-12-2019)

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Charlotte Dion, Sarah Lemler, Eva Löcherbach. Exponential ergodicity for diffusions with jumps driven by a Hawkes process. Theory of Probability and Mathematical Statistics, 2021, ⟨10.1090/tpms/1129⟩. ⟨hal-02094514v3⟩
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