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Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model

Published online by Cambridge University Press:  03 June 2013

Florian Fuchs  and
Robert Stelzer
Florian Fuchs
Affiliation:
TUM Institute for Advanced Study & Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, 85748 Garching, Germany. ffuchs@ma.tum.de; www-m4.ma.tum.de
Robert Stelzer
Affiliation:
Institute of Mathematical Finance, Ulm University, Helmholtzstraße 18, 89081 Ulm, Germany; robert.stelzer@uni-ulm.de; www.uni-ulm.de/mawi/finmath.html
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Abstract

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.

Keywords

Infinitely divisible processmixingmixed moving average processsupOU processstochastic volatility modelcodifference
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 455 - 471
Copyright
© EDP Sciences, SMAI, 2013

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References

R.B. Ash and M.F. Gardner, Topics in Stochastic Processes, Prob. Math. Stat., vol. 27. Academic Press, New York (1975).
Barndorff-Nielsen, O.E., Superposition of Ornstein-Uhlenbeck type processes. Teor. Veroyatnost. i Primenen. 45 (2000) 289–311. Google Scholar
Barndorff-Nielsen, O.E. and Shephard, N., 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. Google Scholar
Barndorff-Nielsen, O.E. and Stelzer, R., Multivariate supOU processes. Ann. Appl. Probab. 21 (2011) 140–182. Google Scholar
Barndorff-Nielsen, O.E. and Stelzer, R., The multivariate supOU stochastic volatility model. Math. Finance 23 (2013) 275–296. Google Scholar
Bender, C., Lindner, A. and Schicks, M., Finite variation of fractional Lévy processes. J. Theor. Probab. 25 (2012) 595–612. Google Scholar
P.J. Brockwell, 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.
Cambanis, S., Podgórski, K. and Weron, A., Chaotic behavior of infinitely divisible processes. Stud. Math. 115 (1995) 109–127. Google Scholar
R. Cont and P. Tankov, Financial Modelling with Jump Processes. CRC Financial Mathematics Series. Chapman & Hall, London (2004).
I.P. Cornfeld, S.V. Fomin and Y.G. Sinaǐ, Ergodic Theory, Grundlehren der mathematischen Wissenschaften, vol. 245. Springer-Verlag, New York (1982).
V. Fasen and C. Klüppelberg, 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.
Guillaume, D.M., Dacorogna, M.M., Davé, R.R., Müller, U.A., Olsen, R.B. and Pictet, O.V., 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. Google Scholar
Hansen, L.P., Large sample properties of generalized method of moments estimators. Econometrica 50 (1982) 1029–1054. Google Scholar
U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter, Berlin (1985).
Magdziarz, M., A note on Maruyama’s mixing theorem. Theory Probab. Appl. 54 (2010) 322–324. Google Scholar
Marquardt, T., Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 (2006) 1099–1126. Google Scholar
Marquardt, T. and Stelzer, R., Multivariate CARMA processes. Stoc. Proc. Appl. 117 (2007) 96–120. Google Scholar
Maruyama, G., Infinitely divisible processes. Theory Probab. Appl. 15 (1970) 1–22. Google Scholar
J. Pedersen, 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).
K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge, UK (1983).
C. Pigorsch and R. Stelzer, A Multivariate Ornstein-Uhlenbeck Type Stochastic Volatility Model. Available from http://www.uni-ulm.de/mawi/finmath.html (2009).
Rajput, B.S. and Rosiński, J., Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82 (1989) 451–487. Google Scholar
Rosiński, J. and Żak, T., Simple conditions for mixing of infinitely divisible processes. Stoch. Proc. Appl. 61 (1996) 277–288. Google Scholar
Rosiński, J. and Żak, T., The equivalence of ergodicity and weak mixing for infinitely divisible processes. J. Theor. Probab. 10 (1997) 73–86. Google Scholar
K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge, UK (1999).
Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S., Stable mixed moving averages. Probab. Theory Relat. Fields 97 (1993) 543–558. Google Scholar
T. Tosstorff and R. Stelzer, Moment based estimation of supOU processes and a related stochastic volatility model. In preparation (2011).