Exponential ergodicity for diffusions with jumps driven by a Hawkes process
Authors:
Charlotte Dion, Sarah Lemler and Eva Löcherbach
Journal:
Theor. Probability and Math. Statist. 102 (2020), 97-115
MSC (2020):
Primary 60J35, 60F99, 60H99, 60G55
DOI:
https://doi.org/10.1090/tpms/1129
Published electronically:
March 29, 2021
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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.
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References
- D. Applebaum, Lévy processes and Stochastic Calculus, Cambridge University Press, 2009. MR 2512800
- E. Bacry, S. Delattre, M. Hoffmann, and J-F Muzy, Some limit theorems for Hawkes processes and application to financial statistics, Stochastic Process. Appl., 123(2013), no. 7, 2475–2499. MR 3054533
- E. Bacry and J-F Muzy, Second order statistics characterization of Hawkes processes and non-parametric estimation, arXiv preprint arXiv:1401.0903, 2014. MR 3480107
- M. Benaïm, S. Le Borgne, F. Malrieu, and P-A Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. H. Poincaré Probab. Statist., 51 (2015), no. 3, 1040–1075. MR 3365972
- M. Bonde Raad, Renewal time points for Hawkes processes, arXiv, 2019.
- P. Brémaud and L. Massoulié, Stability of nonlinear Hawkes processes, Ann. Probab., 1996, 1563–1588. MR 1411506
- S. Chen, A. Shojaie, E. Shea-Brown, and D. Witten. The multivariate Hawkes process in high dimensions: Beyond mutual excitation, arXiv preprint arXiv:1707.04928, 2017.
- S. Clinet and N. Yoshida, Statistical inference for ergodic point processes and application to limit order book, Stochastic Process. Appl., 127 (2017), no. 6, 1800–1839. MR 3646432
- D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes: volume II: general theory and structure, Springer Science & Business Media, 2007. MR 2371524
- S. Delattre, N. Fournier, and M Hoffmann, Hawkes processes on large networks, Ann. Appl. Probab., 26 (2016), no. 1, 216–261. MR 3449317
- C. Dion and S. Lemler, Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process, HAL, 2019. MR 4136702
- R. Douc, G. Fort, and A. Guillin, Subgeometric rates of convergence of f-ergodic strong markov processes, Stochastic Process. Appl., 119 (2009), no. 3, 897–923. MR 2499863
- A. Duarte, E. Löcherbach, and G. Ost, Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels, arXiv preprint arXiv:1610.03300, 2016. MR 4044609
- E. Gobet, LAN property for ergodic diffusions with discrete observations, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), no. 5, 711–737. MR 1931584
- C. Graham, Regenerative properties of the linear Hawkes process with unbounded memory, arXiv, 2019.
- R. Z. Has’minskii, Stochastic stability of differential equations, Sijthoff & Noordhoff, 1980.
- A.G Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), no. 1, 83–90. MR 278410
- P. Hodara and E. Löcherbach, Hawkes processes with variable length memory and an infinite number of components, Adv. in Appl. Probab., 49 (2017), no. 1, 84–107. MR 3631217
- R. Höpfner and E. Löcherbach, Statistical models for Birth and Death on a Flow: Local absolute continuity and likelihood ratio processes, Scand. J. Stat., 26 (1999), no. 1, 107–128. MR 1685305
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- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, Elsevier Science, 2014. MR 1011252
- K. Ito and H. P. McKean, Diffusion Processes and their Sample Paths, 1965. MR 0199891
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- A. M. Kulik, Introduction to Ergodic Rates for Markov Chains and Processes with Applications to Limit Theorems, Potsdam University Press, 2015. MR 3617049
- H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1992. MR 1070361
- E. Löcherbach, Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps, Stochastic Process. Appl., 128 (2018), 1797–1829. MR 3797644
- E. Löcherbach and D. Loukianova, On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions, Stochastic Process. Appl., 118 (2008), 1301–1321. MR 2427041
- E. Löcherbach and D. Loukianova, Polynomial deviation bounds for recurrent Harris processes having general state space, ESAIM Probab. Stat., 17 (2013), 195–218. MR 3021315
- H. Masuda, Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps, Stochastic Process. Appl., 117 (2007), no. 1, 35–56. MR 2287102
- S. P. Meyn and R. L. Tweedie, Stability of Markovian processes I: Criteria for discrete-time chains, Adv. Appl. Probab., 24 (1992), 542–574. MR 1174380
- S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes, Adv. Appl. Probab., 25 (1993), 487–548. MR 1234295
- A. Millet and M Sanz-Solé, A simple proof of the support theorem for diffusion processes, Séminaire de probabilités de Strasbourg, 28 (1994), 36–48. MR 1329099
- D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer Science & Business Media, 2013. MR 3184878
- P. Reynaud-Bouret, V. Rivoirard, and C. Tuleau-Malot, Inference of functional connectivity in neurosciences via Hawkes processes, In Global Conference on Signal and Information Processing (GlobalSIP), IEEE, 2013, pp. 317–320.
- D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Probability Theory, University of California Press, Berkeley, Calif., 1972, pp. 333–359. MR 0400425
- P. Tankov and E. Voltchkova, Jump-diffusion models: a practitioner’s guide, Banque et Marchés, 99 (2009), no. 1, 24.
- A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stochastic Process. Appl., 70 (1997), no. 1, 115–127. MR 1472961
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Additional Information
Charlotte Dion
Affiliation:
Sorbonne Université, UMR CNRS 8001, LPSM, 75005 Paris, France
Email:
charlotte.dion@upmc.fr
Sarah Lemler
Affiliation:
Université Paris-Saclay, École CentraleSupélec, MICS, 91190 Gif-sur-Yvette, France
Email:
sarah.lemler@centralesupelec.fr
Eva Löcherbach
Affiliation:
Université Paris 1, SAMM EA 4543, 75013 Paris, France
Email:
locherbach70@gmail.com
Keywords:
Diffusions with jumps,
nonlinear Hawkes process,
piecewise deterministic Markov processes (PDMP),
ergodicity,
Foster–Lyapunov drift criteria,
$\beta$-mixing.
Received by editor(s):
July 20, 2019
Published electronically:
March 29, 2021
Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv
