Exponential ergodicity for diffusions with jumps driven by a Hawkes process

Source 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 ) (X,Y ) , where X X denotes the diffusion process and Y 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 . X. Furthermore, we provide sufficient conditions under which the process is exponentially β − \beta - mixing.