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Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Lingjiong Zhu
Lingjiong Zhu*
Affiliation:
New York University
*
∗ Postal address: School of Mathematics, University of Minnesota - Twin Cities, 206 Church Street S. E., Minneapolis, MN-55455, USA. Email address: ling@cims.nyu.edu
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Abstract

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In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

Keywords

Cox-Ingersoll-Ross processpoint processHawkes processself-exciting processcentral limit theoremlarge deviations

MSC classification

Primary: 60G07: General theory of processes 60G55: Point processes 60F05: Central limit and other weak theorems 60F10: Large deviations

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 699 - 712
Copyright
© Applied Probability Trust 

References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13, 65–77.Google Scholar
Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Scaling limits for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499.Google Scholar
Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.Google Scholar
Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.Google Scholar
Dahl, M. (2004). Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insurance Math. Econom. 35, 113–136.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
Duffie, D. (2005). Credit risk modeling with affine processes. J. Banking Finance 29, 2751–2802.Google Scholar
Errais, E., Giesecke, K. and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1, 642–665.Google Scholar
Feller, W. (1951). Two singular diffusion problems. Ann. Math. (2) 54, 173–182.CrossRefGoogle Scholar
Hairer, M. (2010). Convergence of Markov processes. Lecture Notes, University of Warwick. Available at http://www.hairer.org/notes/Convergence.pdf.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Peng, X. and Kou, S. (2009). Default clustering and valuation of collateralized debt obligations. Working Paper, Columbia University.Google Scholar
Stabile, G. and Torrisi, G. L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodol. Comput. Appl. Prob. 12, 415–429.CrossRefGoogle Scholar
Varadhan, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Prob. 50, 760–771.Google Scholar
Zhu, L. (2013). Moderate deviations for Hawkes processes. Statist. Prob. Lett. 83, 885–890.CrossRefGoogle Scholar
Zhu, L. (2013). Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims. Insurance Math. Econom. 53, 544–550.Google Scholar
Zhu, L. (2014). Large deviations for Markovian nonlinear Hawkes processes. To appear in Ann. Appl. Prob. Google Scholar
Zhu, L. (2014). Process-level large deviations for nonlinear Hawkes point processes. Ann. Inst. H. Poincaré Prob. Statist. 50, 845–871.Google Scholar