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, where the jumps are driven by a multivariate linear Hawkes process, and 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, λ), where X denotes the diffusion process and λ 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. This paper is the foundation for a second paper [11] in which we carry a statistical study of diffusions driven by Hawkes jumps, with a view towards applications in neuroscience.
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Submitted on : Wednesday, April 10, 2019 - 10:28:36 AM
Last modification on : Sunday, April 14, 2019 - 1:25:58 AM

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  • HAL Id : hal-02094514, version 1
  • 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-02094514⟩

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