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Bipower Variation for Gaussian Processes with Stationary Increments

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ole E. Barndorff-Nielsen ,
José Manuel Corcuera ,
Mark Podolskij  and
Jeannette H. C. Woerner
Ole E. Barndorff-Nielsen*
Affiliation:
University of Aarhus
José Manuel Corcuera*
Affiliation:
University of Barcelona
Mark Podolskij*
Affiliation:
University of Aarhus and CREATES
Jeannette H. C. Woerner*
Affiliation:
University of Göttingen
*
∗Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: [email protected]
∗∗Postal address: Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain. Email address: [email protected]
∗∗∗Current address: Department of Mathematics, ETH Zürich, HG G32.2, 8092 Zürich, Switzerland. Email address: [email protected]
∗∗∗∗Postal address: Institut für Mathematische Stochastik, Maschmühlenweg 8-10, 37073 Göttingen, Germany. Email address: [email protected]
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Abstract

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Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.

Keywords

Bipower variationcentral limit theoremchaos expansionGaussian processmultiple Wiener–Itô integrals

MSC classification

Secondary: 60G15: Gaussian processes 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 132 - 150
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Prob. 6, 325–331.CrossRefGoogle Scholar
[2] Barndorff-Nielsen, O. E. and Shephard, N. (2003). Realized power variation and stochastic volatility models. Bernoulli 9, 243–265.CrossRefGoogle Scholar
[3] Barndorff-Nielsen, O. E. and Shephard, N. (2004). Econometric analysis of realized covariation: high frequency covariance, regression, and correlation in financial economics. Econometrica 72, 885–925.Google Scholar
[4] Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and Jumps (with discussion). J. Financial Econometrics 2, 1–48.CrossRefGoogle Scholar
[5] Barndorff-Nielsen, O. E. and Shephard, N. (2006). Impact of Jumps on returns and realised variances: econometric analysis of time-deformed Lévy processes. J. Econometrics 131, 217–252.CrossRefGoogle Scholar
[6] Barndorff-Nielsen, O. E. and Shephard, N. (2007). Variation, Jumps, market frictions and high frequency data in financial econometrics. In Advances in Economics and Econometrics, eds Blundell, R. et al., 9th World Congress, Cambridge University Press, pp. 328–372.Google Scholar
[7] Barndorff-Nielsen, O. E., Corcuera, J. M. and Podolskij, M. (2009). Power variation for Gaussian processes with stationary increments. To appear in Stoch. Process. Appl. CrossRefGoogle Scholar
[8] Barndorff-Nielsen, O. E., Shephard, N. and Winkel, M. (2006). Limit theorems for multipower variation in the presence of Jumps. Stoch. Process. Appl. 116, 796–806.CrossRefGoogle Scholar
[9] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J. and Shephard, N. (2006). Limit theorems for bipower variation in financial econometrics. Econometric Theory 22, 677–719.Google Scholar
[10] Barndorff-Nielsen, O. E. et al. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In Stochastic Calculus to Mathematical Finance, eds Kabanov, Yu. et al. Springer, Berlin, pp. 33–68.Google Scholar
[11] Bender, C., Sottinen, T. and Valkeyla, E. (2007). Arbitrage with fractional Brownian motion? Theory Stoch. Process. 13, 23–34.Google Scholar
[12] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[13] Comte, F. and Renault, E. (1998). Long memory in continous-time stochastic volatility models. Math. Finance 8, 291–323.Google Scholar
[14] Corcuera, J. M., Nualart, D. and Woerner, J. H. C. (2006). Power variation of some integral fractional processes. Bernoulli 12, 713–735.Google Scholar
[15] Cultand, N. J., Kopp, P. E. and Willinger, W. (1995). Stock price returns and the Joseph effect: a fractional version of the Black–Scholes model. In Seminar of Stochastic Analysis, Random Fields and Applications (Ascona 1993; Progress Prob. 36), eds Bolthausen, E. et al. Birkhäuser, Berlin, pp. 327–351.Google Scholar
[16] Guyon, L. and Leon, J. (1989). Convergence en loi des H-variations d'un processus gaussien stationnaire sur {R}. Ann. Inst. H. Poincaré Prob. Statist. 25, 265–282.Google Scholar
[17] Hu, Y. and Nualart, D. (2005). Renormalized self-intersection local time for fractional Brownian motion. Ann. Prob. 33, 948–983.Google Scholar
[18] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stoch. Process. Appl. 118, 517–559.Google Scholar
[19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
[20] Jacod, J. and Todorov, V. (2009). Testing for common arrival of Jumps in discretely-observed multidimensional processes. To appear in Ann. Statist. Google Scholar
[21] Kinnebrock, S. and Podolskij, M. (2008). A note on the central limit theorem for bipower variation of general functions. Stoch. Process. Appl. 118, 1056–1070.Google Scholar
[22] Nourdin, I. and Peccati, G. (2009). Non-central convergence of multiple integrals. To appear in Ann. Prob. Google Scholar
[23] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin.Google Scholar
[24] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Prob. 33, 177–193.Google Scholar
[25] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Process. Appl. 118, 614–628.Google Scholar
[26] Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), eds Emery, M. et al. Springer, Berlin, pp. 247–262.Google Scholar
[27] Rényi, A. (1963). On stable sequences of events. Sankhyā A 25, 293–302.Google Scholar
[28] Veraart, A. (2009). Inference for the Jump part of quadratic variation of Itô semimartingales. To appear in Econometric Theory.Google Scholar
[29] Woerner, J. H. C. (2003). Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models. Statist. Decisions 21, 47–68.Google Scholar
[30] Woerner, J. H. C. (2005). Estimation of integrated volatility in stochastic volatility models. Appl. Stoch. Models Business Industry 21, 27–44.Google Scholar
[31] Woerner, J. H. C. (2006). Power and multipower variation: inference for high frequency data. In Stochastic Finance, eds Shiryaev, A. N. et al. Springer, New York, pp. 343–364.Google Scholar
[32] Woerner, J. H. C. (2007). Inference in Lévy-type stochastic volatility models. Adv. Appl. Prob. 39, 531–549.Google Scholar
[33] Woerner, J. H. C. (2008). Volatility estimates for high frequency data: market microstructure noise versus fractional Brownian motion models. Working paper.Google Scholar
[34] Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.Google Scholar