Limit theorems for functionals of higher order differences of Brownian semi-stationary processes

被引：9

作者：

Barndorff-Nielsen, Ole E. ^{[1
]}

Corcuera, José Manuel ^{[2
]}

Podolskij, Mark ^{[3
]}

机构：

[1] Department of Mathematics, University of Aarhus, DK-8000 Aarhus C, Ny Munkegade

[2] Universitat de Barcelona, 08007 Barcelona

[3] Department of Mathematics, University of Heidelberg, 69120 Heidelberg

来源：

Springer Proceedings in Mathematics and Statistics | 2013年 / 33卷

关键词：

Brownian semi-stationary processes; Central limit theorem; Gaussian processes; High frequency observations; Higher order differences; Multipower variation; Stable convergence;

D O I：

10.1007/978-3-642-33549-5_4

中图分类号：

学科分类号：

摘要：

We present some new asymptotic results for functionals of higher order differences of Brownian semi-stationary processes. In an earlier work [8] we have derived a similar asymptotic theory for first order differences. However, the central limit theorems were valid only for certain values of the smoothness parameter of a Brownian semi-stationary process, and the parameter values which appear in typical applications, e.g. in modeling turbulent flows in physics, were excluded. The main goal of the current paper is the derivation of the asymptotic theory for the whole range of the smoothness parameter by means of using second order differences. We present the law of large numbers for the multipower variation of the second order differences of Brownian semi-stationary processes and show the associated central limit theorem. Finally, we demonstrate some estimation methods for the smoothness parameter of a Brownian semi-stationary process as an application of our probabilistic results. © Springer-Verlag Berlin Heidelberg 2013.

引用

页码：69 / 96

页数：27

共 23 条

[1]

Aldous D.J., Eagleson G.K., On mixing and stability of limit theorems, Ann. Probab., 6, 2, pp. 325-331, (1978)

[2]

Barndorff-Nielsen O.E., Schmiegel J., Brownian semistationary processes and volatility/ intermittency, Advanced Financial Modelling. Radon Series of Computational and Applied Mathematics, 8, pp. 1-26, (2009)

[3]

Barndorff-Nielsen O.E., Shephard N., Power and bipower variation with stochastic volatility and jumps (with discussion), J. Financ. Econom., 2, pp. 1-48, (2004)

[4]

Barndorff-Nielsen O.E., Shephard N., Econometrics of testing for jumps in financial economics using bipower variation, J. Financ. Econom., 4, pp. 1-30, (2006)

[5]

Barndorff-Nielsen O.E., Graversen S.E., Jacod J., Podolskij M., Shephard N., A central limit theorem for realised power and bipower variations of continuous semimartingales, From Stochastic Calculus to Mathematical Finance: Festschrift in Honour, (2006)

[6]

Barndorff-Nielsen O.E., Shephard N., Winkel M., Limit theorems for multipower variation in the presence of jumps, Stoch. Process. Appl., 116, pp. 796-806, (2006)

[7]

Barndorff-Nielsen O.E., Corcuera J.M., Podolskij M., Power variation for Gaussian processes with stationary increments, Stoch. Process. Appl., 119, pp. 1845-1865, (2009)

[8]

Barndorff-Nielsen O.E., Corcuera J.M., Podolskij M., Multipower variation for Brownian semistationary processes, Bernoulli, 17, pp. 1159-1194, (2011)

[9]

Barndorff-Nielsen O.E., Corcuera J.M., Podolskij M., Woerner J.H.C., Bipower variation for Gaussian processes with stationary increments, J. Appl. Probab., 46, pp. 132-150, (2009)

[10]

Basse A., Gaussian moving averages and semimartingales, Electron. J. Probab., 13, 39, pp. 1140-1165, (2008)

← 1 2 3 →