Power variation and time change

O. E. Barndorff-Nielsen   ; N. Shephard

Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 115-130

Résumé

This paper provides limit distribution results for power variation, that is, sums of powers of absolute increments under nonequidistant subdivisions of time and for certain types of time-changed Brownian motion and $\alpha$-stable processes. Special cases of these processes are stochastic volatility models used extensively in financial econometrics.

Détail

Keywords: power variation, realized variance, semimartingales, stochastic volatility, time change.
Mots-clés : $r$-variation

Bibliographie

[1] Andersen T. G., Bollerslev T., Diebold F. X., “Parametric and nonparametric volatility measurement”, Handbook of Financial Econometrics, eds. Y. Ait-Sahalia and L. P. Hansen, North-Holland, Amsterdam, 2005 (to appear) | Zbl

[2] Barndorff-Nielsen O. E., “Superposition of Ornstein–Uhlenbeck type processes”, Teoriya veroyatn. i ee primen., 45:2 (2000), 289–311 | MR | Zbl

[3] Barndorff-Nielsen O. E., Graversen S. E., Jacod J., Podolskij M., Shephard N., A central limit theorem for realized power and bipower variations of continuous semimartingales, Economics working paper 2004-W29, Nuffield College, Oxford, 2004

[4] Barndorff-Nielsen O. E., Nicolato E., Shephard N., “Some recent developments in stochastic volatility modelling”, Quantitative Finance, 2:1 (2002), 11–23 | DOI | MR

[5] Barndorff-Nielsen O. E., Shephard N., “Modelling by Lévy processes for financial econometrics”, Lévy Processes. Theory and Applications, eds. O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick, Birkhäuser, Boston, 2001, 283–318 | MR | Zbl

[6] Barndorff-Nielsen O. E., Shephard N., “Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics”, J. Roy. Statist. Soc., Ser. B, 63:2 (2001), 167–241 | DOI | MR | Zbl

[7] Barndorff-Nielsen O. E., Shephard N., “Econometric analysis of realized volatility and its use in estimating stochastic volatility models”, J. Roy. Statist. Soc., Ser. B, 64:2 (2002), 253–280 | DOI | MR | Zbl

[8] Barndorff-Nielsen O. E., Shephard N., “Realized power variation and stochastic volatility”, Bernoulli, 9:2 (2003), 243–265 ; Corrections: ibid. 6, 1109–1111 | DOI | MR | Zbl | MR | Zbl

[9] Barndorff-Nielsen O. E., Shephard N., “Econometric analysis of realized covariation: high frequency covariance, regression, and correlation in financial economics”, Econometrica, 72:3 (2004), 885–925 | DOI | MR | Zbl

[10] Bass R. F., Hambly B. M., Lyons T. J., “Extending the Wong–Zakai theorem to reversible Markov processes”, J. Eur. Math. Soc., 4 (2002), 237–269 | DOI | MR | Zbl

[11] Dudley R. M., “Fréchet differentiability, $p$-variation and uniform Donsker classes”, Ann. Probab., 20:4 (1992), 1968–1982 | DOI | MR | Zbl

[12] Dudley R. M., Norvaiša R., An introduction to $p$-variation and Young integrals; with emphasis on sample functions of stochastic processes, MaPhySto Lecture Notes No 1998-1, Aarhus Univ., MaPhySto, Aarhus, 1998 | Zbl

[13] Dudley R. M., Norvaiša R., “Product integrals, Young integrals and $p$-variation”, Lecture Notes in Math., 1703, 1999, 73–208 | MR

[14] Dudley R. M., Norvaiša R., Qian J., “Bibliographies on $p$-variations and $\phi$-variation”, Lecture Notes in Math., 1703, 1999, 241–261 | MR

[15] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984, 751 pp.

[16] Fristedt B., Taylor S. J., “Strong variation for the sample functions of a stable process”, Duke Math. J., 40 (1973), 259–278 | DOI | MR | Zbl

[17] Gnedenko B. V., Kolmogorov A. N., Predelnye raspredeleniya dlya summ nezavisimykh sluchainykh velichin, Gostekhizdat, M., L., 1949, 264 pp. | MR

[18] Hudson W. N., Mason J. D., “Variational sums for additive processes”, Proc. Amer. Math. Soc., 55:2 (1976), 395–399 | DOI | MR | Zbl

[19] Jacod J., Limit of random measures associated with the increments of a Brownian semimartingale, Preprint No 120, Laboratoire de Probabilites, Universite P. et M. Curie, Paris, 1994

[20] Jacod J., Protter P., “Asymptotic error distributions for the Euler method for stochastic differential equations”, Ann. Probab., 26 (1998), 267–307 | DOI | MR | Zbl

[21] Kallsen J., Shiryaev A. N., “Time change representation of stochastic integrals”, Teoriya veroyatn. i ee primen., 46:3 (2001), 579–585 | MR | Zbl

[22] Lépingle D., “La variation d'ordre $p$ des semi-martingales”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 36:4 (1976), 295–316 | DOI | MR | Zbl

[23] Loève M., Probability Theory, v. I, Springer-Verlag, New York, 1977, 425 pp. | MR | Zbl

[24] Lyons T., “Differential equations driven by rough signals. I. An extension of an inequality by L. C. Young”, Math. Res. Lett., 1:4 (1994), 451–464 | MR | Zbl

[25] Mikosch T., Norvaiša R., “Stochastic integral equations without probability”, Bernoulli, 6:3 (2000), 401–434 | DOI | MR | Zbl

[26] Mykland P. A., Zhang L., “ANOVA for diffusions”, Ann. Statist., 33 (2005) (to appear)

[27] Sato K., Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, 1999, 486 pp. | MR

[28] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 1, 2, FAZIS, M., 2004, 512 pp.; 544 с.

[29] Woerner J., “Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models”, Statist. Decisions, 21:1 (2003), 47–68 | DOI | MR | Zbl

[30] Young L. C., “An inequality of the Hölder type, connected with Stieltjes integration”, Acta Math., 67 (1936), 251–282 | DOI | MR