×

zbMATH — the first resource for mathematics

Some limit theorems for Hawkes processes and application to financial statistics. (English) Zbl 1292.60032
Summary: In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval \([0,T]\) when \(T\to \infty \). We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh \(\varDelta\) over \([0,T]\) up to some further time shift \(\tau\). The behaviour of this functional depends on the relative size of \(\varDelta\) and \(\tau\) with respect to \(T\) and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in E. Bacry et al. [Quant. Finance 13, No. 1, 65–77 (2013; Zbl 1280.91073)] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead-lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.

MSC:
60F05 Central limit and other weak theorems
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. Abergel, N. Huth, High frequency lead/lag relationship-empirical facts, 2011, Arxiv Preprint.arXiv:1111.7103v1.
[2] Abergel, F.; Pomponio, F., Trade-throughs: empirical facts and application to lead – lag measures, (Abregel, F.; Chakrabarti, B. K.; Chakrabart, A.; Mitra, M., Econophysics of Order-Driven Markets, New Economic Windows, (2011), Springer Berlin, Heidelberg)
[3] Y. Ait-Sahalia, J. Cacho-Diaz, R. Laeven, Modeling financial contagion using mutually exciting jump processes, Working Paper, 2011.
[4] Ait-Sahalia, Y.; Mykland, P. A.; Zhang, L., How often to sample a continuous-time process in the presence of market microstructure noise, The Review of Financial Studies, 18, 351-416, (2005)
[5] Ait-Sahalia, Y.; Mykland, P. A.; Zhang, L., Ultra high frequency volatility estimation with dependent microstructure noise, Journal of Econometrics, 160, 160-175, (2011) · Zbl 1441.62577
[6] Andersen, T.; Bollerslev, T.; Diebold, F. X.; Labys, P., (understanding, optimizing, using and forecasting) realized volatility and correlation, (“Great Realizations”, Risk, (2000)), 105-108
[7] Bacry, E.; Delattre, S.; Hoffmann, M.; Muzy, J. F., Modelling microstructure noise with mutually exciting point processes, Quantitative Finance, 13, 65-77, (2013) · Zbl 1280.91073
[8] Barndorff-Nielsen, O.; Hansen, P.; Lunde, A.; Stephard, N., Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise, Econometrica, 76, 6, 1481-1536, (2008) · Zbl 1153.91416
[9] Bartlett, M. S., The spectral analysis of point processes, Journal of the Royal Statistical Society: Series B, 25, 264-296, (1963) · Zbl 0124.08504
[10] Bauwens, L.; Hautsch, N., Modelling financial high frequency data using point processes, (Mikosch, T.; Kreiss, J.-P.; Davis, R. A.; Andersen, T. G., Handbook of Financial Time Series, (2009), Springer Berlin, Heidelberg) · Zbl 1178.91218
[11] Brémaud, P.; Massoulié, L., Power spectra of general shot noises and Hawkes point processes with a random excitation, Advances in Applied Probability, 34, 205-222, (2002) · Zbl 1002.60029
[12] Daley, D. J.; Vere-Jones, D., (An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, Probability and its Applications (New York), (2003), Springer-Verlag New York) · Zbl 1026.60061
[13] K. Dayri, E. Bacry, J.F. Muzy, Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data, 2011. http://arxiv.org/abs/1112.1838.
[14] Embrechts, P.; Liniger, J. T.; Lu, L., Multivariate Hawkes processes: an application to financial data, Journal of Applied Probability, 48, 367-378, (2011) · Zbl 1242.62093
[15] Epps, T. W., Comovements in stock prices in the very short run, Journal of the American Statistical Association, 74, 291-298, (1979)
[16] Hawkes, A. G., Point spectra of some mutually exciting point processes, Journal of the Royal Statistical Society: Series B, 33, 438-443, (1971) · Zbl 0238.60094
[17] Hawkes, A. G., Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58, 83-90, (1971) · Zbl 0219.60029
[18] Hawkes, A. G., Spectra of some mutually exciting point processes with associated variables, (Lewis, P. A.W., Stochastic Point Processes, (1972), Wiley New York) · Zbl 0262.60036
[19] Hawkes, A. G.; Oakes, D., A cluster process representation of a self-exciting process, Journal of Applied Probability, 11, 493-503, (1974) · Zbl 0305.60021
[20] Hewlett, P., Clustering of order arrivals, price impact and trade path optimisation, (Workshop on Financial Modeling with Jump Processes, (2006), Ecole Polytechnique)
[21] Hoffmann, M.; Rosenbaum, M.; Yoshida, N., Estimation of the lead – lag effect from nonsynchronous data, Bernoulli, 19, 426-461, (2013) · Zbl 06168759
[22] Jacod, J., Multivariate point processes: predictable projection, radon – nikodým derivatives, representation of martingales, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31, 235-253, (1974-1975)
[23] Jacod, J.; Shiryaev, A. N., (Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 288, (1987), Springer-Verlag Berlin)
[24] Ogata, Y., The asymptotic behaviour of maximum likelihood estimators for stationary point processes, Annals of the Institute of Statistical Mathematics, 30, 243-261, (1978) · Zbl 0451.62067
[25] Reynaud-Bouret, P.; Schbath, S., Adaptive estimation for Hawkes processes; application to genome analysis, Annals of Statistics, 38, 2781-2822, (2010) · Zbl 1200.62135
[26] Robert, C. Y.; Rosenbaum, M., A new approach for the dynamics of ultra high frequency data: the model with uncertainty zones, Journal of Financial Econometrics, 9, 344-366, (2011)
[27] Rosenbaum, M., A new microstructure noise index, Quantitative Finance, 6, 883-899, (2011) · Zbl 1217.91212
[28] Zhuang, J.; Ogata, Y.; Vere-Jones, D., Stochastic declustering of space – time earthquake occurrences, Journal of the American Statistical Association, 97, 369-380, (2002) · Zbl 1073.62558
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.