We consider strictly stationary infinitely divisible processes and first extend the mixing conditions given in Maruyama [Theory Probab. Appl. 15 (1970) 1-22] and Rosiński and Żak [Stoc. Proc. Appl. 61 (1996) 277-288] from the univariate to the d-dimensional case. Thereafter, we show that multivariate Lévy-driven mixed moving average processes satisfy these conditions and hence a wide range of well-known processes such as superpositions of Ornstein - Uhlenbeck (supOU) processes or (fractionally integrated) continuous time autoregressive moving average (CARMA) processes are always mixing. Finally, mixing of the log-returns and the integrated volatility process of a multivariate supOU type stochastic volatility model, recently introduced in Barndorff - Nielsen and Stelzer [Math. Finance 23 (2013) 275-296], is established.
Mots-clés : infinitely divisible process, mixing, mixed moving average process, supOU process, stochastic volatility model, codifference
@article{PS_2013__17__455_0,
author = {Fuchs, Florian and Stelzer, Robert},
title = {Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the {supOU} stochastic volatility model},
journal = {ESAIM: Probability and Statistics},
pages = {455--471},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011158},
mrnumber = {3070886},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2011158/}
}
TY - JOUR AU - Fuchs, Florian AU - Stelzer, Robert TI - Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model JO - ESAIM: Probability and Statistics PY - 2013 SP - 455 EP - 471 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2011158/ DO - 10.1051/ps/2011158 LA - en ID - PS_2013__17__455_0 ER -
%0 Journal Article %A Fuchs, Florian %A Stelzer, Robert %T Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model %J ESAIM: Probability and Statistics %D 2013 %P 455-471 %V 17 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2011158/ %R 10.1051/ps/2011158 %G en %F PS_2013__17__455_0
Fuchs, Florian; Stelzer, Robert. Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 455-471. doi : 10.1051/ps/2011158. http://www.numdam.org/articles/10.1051/ps/2011158/
[1] and , Topics in Stochastic Processes, Prob. Math. Stat., vol. 27. Academic Press, New York (1975). | MR | Zbl
[2] , Superposition of Ornstein-Uhlenbeck type processes. Teor. Veroyatnost. i Primenen. 45 (2000) 289-311. | MR | Zbl
[3] and , Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167-241. | MR | Zbl
[4] and , Multivariate supOU processes. Ann. Appl. Probab. 21 (2011) 140-182. | MR | Zbl
[5] and , The multivariate supOU stochastic volatility model. Math. Finance 23 (2013) 275-296. | MR | Zbl
[6] , and , Finite variation of fractional Lévy processes. J. Theor. Probab. 25 (2012) 595-612. | MR | Zbl
[7] , Lévy-driven continuous-time ARMA processes, in Handbook of Financial Time Series, edited by T.G. Andersen, R. Davis, J.-P. Kreiß and T. Mikosch. Springer, Berlin (2009) 457-480. | Zbl
[8] , and , Chaotic behavior of infinitely divisible processes. Stud. Math. 115 (1995) 109-127. | MR | Zbl
[9] and , Financial Modelling with Jump Processes. CRC Financial Mathematics Series. Chapman & Hall, London (2004). | MR | Zbl
[10] , and , Ergodic Theory, Grundlehren der mathematischen Wissenschaften, vol. 245. Springer-Verlag, New York (1982). | MR | Zbl
[11] and , Extremes of supOU processes, in Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin (2007) 340-359. | MR | Zbl
[12] , , , , and , From the bird's eye to the microscope: a survey of new stylized facts of the intra-daily foreign exchange markets. Finance Stoch. 1 (1997) 95-129. | Zbl
[13] , Large sample properties of generalized method of moments estimators. Econometrica 50 (1982) 1029-1054. | MR | Zbl
[14] , Ergodic Theorems, de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter, Berlin (1985). | MR | Zbl
[15] , A note on Maruyama's mixing theorem. Theory Probab. Appl. 54 (2010) 322-324. | MR
[16] , Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 (2006) 1099-1126. | MR | Zbl
[17] and , Multivariate CARMA processes. Stoc. Proc. Appl. 117 (2007) 96-120. | MR | Zbl
[18] , Infinitely divisible processes. Theory Probab. Appl. 15 (1970) 1-22. | MR | Zbl
[19] , The Lévy-Itô decomposition of an independently scattered random measure. MaPhySto research report 2, MaPhySto and University of ?rhus. Available from http://www.maphysto.dk (2003).
[20] , Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge, UK (1983). | MR | Zbl
[21] and , A Multivariate Ornstein-Uhlenbeck Type Stochastic Volatility Model. Available from http://www.uni-ulm.de/mawi/finmath.html (2009).
[22] and , Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82 (1989) 451-487. | MR | Zbl
[23] and , Simple conditions for mixing of infinitely divisible processes. Stoch. Proc. Appl. 61 (1996) 277-288. | Zbl
[24] and , The equivalence of ergodicity and weak mixing for infinitely divisible processes. J. Theor. Probab. 10 (1997) 73-86. | Zbl
[25] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge, UK (1999). | Zbl
[26] , , and , Stable mixed moving averages. Probab. Theory Relat. Fields 97 (1993) 543-558. | MR | Zbl
[27] and , Moment based estimation of supOU processes and a related stochastic volatility model. In preparation (2011).
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