Exponential ergodicity for diffusions with jumps driven by a Hawkes process
Probability
2020-01-09 v3
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 --mixing.
Cite
@article{arxiv.1904.06051,
title = {Exponential ergodicity for diffusions with jumps driven by a Hawkes process},
author = {Charlotte Dion and Sarah Lemler and Eva Löcherbach},
journal= {arXiv preprint arXiv:1904.06051},
year = {2020}
}