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Exponential ergodicity for diffusions with jumps driven by a Hawkes process. (English) Zbl 1477.60075

Summary: 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.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
60F05 Central limit and other weak theorems
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