## The Annals of Applied Probability

### Limit theorems for nearly unstable Hawkes processes

#### Abstract

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the $L^{1}$ norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox–Ingersoll–Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65–77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 600-631.

Dates
First available in Project Euclid: 19 February 2015

https://projecteuclid.org/euclid.aoap/1424355125

Digital Object Identifier
doi:10.1214/14-AAP1005

Mathematical Reviews number (MathSciNet)
MR3313750

Zentralblatt MATH identifier
1319.60101

#### Citation

Jaisson, Thibault; Rosenbaum, Mathieu. Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25 (2015), no. 2, 600--631. doi:10.1214/14-AAP1005. https://projecteuclid.org/euclid.aoap/1424355125

#### References

• [1] Adamopoulos, L. (1976). Cluster models for earthquakes: Regional comparisons. Journal of the International Association for Mathematical Geology 8 463–475.
• [2] Aït-Sahalia, Y., Cacho-Diaz, J. and Laeven, R. J. (2010). Modeling financial contagion using mutually exciting jump processes. Technical report, National Bureau of Economic Research, Cambridge, MA.
• [3] Alaya, M. B. and Kebaier, A. (2012). Parameter estimation for the square-root diffusions: Ergodic and nonergodic cases. Stoch. Models 28 609–634.
• [4] Bacry, E., Dayri, K. and Muzy, J.-F. (2012). Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data. Eur. Phys. J. B 85 1–12.
• [5] Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2012). Scaling limits for Hawkes processes and application to financial statistics. Preprint. Available at arXiv:1202.0842.
• [6] Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13 65–77.
• [7] Bacry, E. and Muzy, J.-F. (2013). Hawkes model for price and trades high-frequency dynamics. Preprint. Available at arXiv:1301.1135.
• [8] Barczy, M., Ispány, M. and Pap, G. (2011). Asymptotic behavior of unstable $\mathrm{INAR}(p)$ processes. Stochastic Process. Appl. 121 583–608.
• [9] Bauwens, L. and Hautsch, N. (2004). Dynamic latent factor models for intensity processes. Working paper, UCL-CORE Center for Operations Research and Econometrics.
• [10] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• [11] Bouchaud, J.-P., Gefen, Y., Potters, M. and Wyart, M. (2004). Fluctuations and response in financial markets: The subtle nature of “random” price changes. Quant. Finance 4 176–190.
• [12] Bowsher, C. G. (2007). Modelling security market events in continuous time: Intensity based, multivariate point process models. J. Econometrics 141 876–912.
• [13] Brémaud, P. and Massoulié, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Probab. 38 122–135.
• [14] Chavez-Demoulin, V., Davison, A. C. and McNeil, A. J. (2005). Estimating value-at-risk: A point process approach. Quant. Finance 5 227–234.
• [15] Cox, J. C., IngersollJr, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385–407.
• [16] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
• [17] Embrechts, P., Liniger, T. and Lin, L. (2011). Multivariate Hawkes processes: An application to financial data. J. Appl. Probab. 48A 367–378.
• [18] Errais, E., Giesecke, K. and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1 642–665.
• [19] Filimonov, V. and Sornette, D. (2012). Quantifying reflexivity in financial markets: Toward a prediction of flash crashes. Phys. Rev. E (3) 85 056108.
• [20] Filimonov, V. and Sornette, D. (2013). Apparent criticality and calibration issues in the Hawkes self-excited point process model: Application to high-frequency financial data. Preprint. Available at arXiv:1308.6756.
• [21] Hardiman, S. J., Bercot, N. and Bouchaud, J.-P. (2013). Critical reflexivity in financial markets: A Hawkes process analysis. Preprint. Available at arXiv:1302.1405.
• [22] Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 33 438–443.
• [23] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 83–90.
• [24] Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Probab. 11 493–503.
• [25] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 327–343.
• [26] Hewlett, P. (2006). Clustering of order arrivals, price impact and trade path optimisation. In Workshop on Financial Modeling with Jump Processes, 68 September 2006. Ecole Polytechnique, France.
• [27] Jacod, J. (1974/75). Multivariate point processes: Predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253.
• [28] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
• [29] Jaisson, T. (2013). Market impact as anticipation of the order flow imbalance. Working paper.
• [30] Jaisson, T. and Rosenbaum, M. (2014). Limit theorems for nearly unstable Hawkes processes: Version with technical appendix. Technical Report 1607, Laboratoire de Probabilités et Modèles Aléatoires, Univ. Pierre et Marie Curie.
• [31] Jakubowski, A., Mémin, J. and Pagès, G. (1989). Convergence en loi des suites d’intégrales stochastiques sur l’espace $\mathbf{D}^{1}$ de Skorokhod. Probab. Theory Related Fields 81 111–137.
• [32] Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing. Mathematics and Its Applications 413. Kluwer, Dordrecht.
• [33] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
• [34] Large, J. (2007). Measuring the resiliency of an electronic limit order book. Journal of Financial Markets 10 1–25.
• [35] Lillo, F. and Farmer, J. D. (2004). The long memory of the efficient market. Stud. Nonlinear Dyn. Econom. 8 3.
• [36] Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Ann. Inst. Statist. Math. 30 243–261.
• [37] Ogata, Y. (1983). Likelihood analysis of point processes and its applications to seismological data. Bull. Inst. Internat. Statist. 50 943–961.
• [38] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
• [39] Reynaud-Bouret, P. and Schbath, S. (2010). Adaptive estimation for Hawkes processes; application to genome analysis. Ann. Statist. 38 2781–2822.
• [40] Shah, P. V. and Jana, R. K. (2013). Results on generalized Mittag–Leffler function via Laplace transform. Appl. Math. Sci. (Ruse) 7 567–570.
• [41] Wyart, M., Bouchaud, J.-P., Kockelkoren, J., Potters, M. and Vettorazzo, M. (2008). Relation between bid–ask spread, impact and volatility in order-driven markets. Quant. Finance 8 41–57.
• [42] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 760–771.