Theory of Probability and Mathematical Statistics
Ergodicity of jump diffusions driven by multivariate nonlinear Hawkes processes
C. Dion, S. Lemler, E. Loecherbach
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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.
Keywords: Diffusions with jumps, nonlinear Hawkes process, piecewise deterministic Markov processes (PDMP), ergodicity, Foster--Lyapunov drift criteria, $\beta$-mixing
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