Exponential ergodicity for diffusions with jumps driven by a Hawkes process

Abstract : 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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Submitted on : Thursday, July 18, 2019 - 5:43:11 PM
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  • HAL Id : hal-02094514, version 2
  • ARXIV : 1904.06051

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Charlotte Dion, Sarah Lemler, Eva Löcherbach. Exponential ergodicity for diffusions with jumps driven by a Hawkes process. 2019. ⟨hal-02094514v2⟩

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