Exponential ergodicity for diffusions with jumps driven by a Hawkes process

TL;DR

This paper studies a new class of diffusion processes with jumps driven by multivariate nonlinear Hawkes processes, establishing conditions for their long-term stability, ergodicity, and exponential mixing.

Contribution

It introduces a novel class of diffusions with jumps driven by Hawkes processes and provides conditions for their exponential ergodicity and mixing properties.

Findings

Established positive Harris recurrence for the process

Proved ergodic theorem for the diffusion component

Derived conditions for exponential β-mixing

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 $\beta$--mixing.