# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.06051