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

ESAIM: Probability and Statistics (2013)

- Volume: 17, page 455-471
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topFuchs, Florian, and 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 17 (2013): 455-471. <http://eudml.org/doc/273624>.

@article{Fuchs2013,

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.},

author = {Fuchs, Florian, Stelzer, Robert},

journal = {ESAIM: Probability and Statistics},

keywords = {infinitely divisible process; mixing; mixed moving average process; supOU process; stochastic volatility model; codifference},

language = {eng},

pages = {455-471},

publisher = {EDP-Sciences},

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

url = {http://eudml.org/doc/273624},

volume = {17},

year = {2013},

}

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

PB - EDP-Sciences

VL - 17

SP - 455

EP - 471

AB - 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.

LA - eng

KW - infinitely divisible process; mixing; mixed moving average process; supOU process; stochastic volatility model; codifference

UR - http://eudml.org/doc/273624

ER -

## References

top- [1] R.B. Ash and M.F. Gardner, Topics in Stochastic Processes, Prob. Math. Stat., vol. 27. Academic Press, New York (1975). Zbl0317.60014MR448463
- [2] O.E. Barndorff-Nielsen, Superposition of Ornstein-Uhlenbeck type processes. Teor. Veroyatnost. i Primenen.45 (2000) 289–311. Zbl1003.60039MR1967758
- [3] O.E. Barndorff-Nielsen and N. Shephard, 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. Zbl0983.60028MR1841412
- [4] O.E. Barndorff-Nielsen and R. Stelzer, Multivariate supOU processes. Ann. Appl. Probab.21 (2011) 140–182. Zbl1235.60032MR2759198
- [5] O.E. Barndorff-Nielsen and R. Stelzer, The multivariate supOU stochastic volatility model. Math. Finance23 (2013) 275–296. Zbl1262.91139MR3034078
- [6] C. Bender, A. Lindner and M. Schicks, Finite variation of fractional Lévy processes. J. Theor. Probab.25 (2012) 595–612. Zbl1254.60040MR2914443
- [7] 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. Zbl1192.62194
- [8] S. Cambanis, K. Podgórski and A. Weron, Chaotic behavior of infinitely divisible processes. Stud. Math.115 (1995) 109–127. Zbl0835.60008MR1347436
- [9] R. Cont and P. Tankov, Financial Modelling with Jump Processes. CRC Financial Mathematics Series. Chapman & Hall, London (2004). Zbl1052.91043MR2042661
- [10] I.P. Cornfeld, S.V. Fomin and Y.G. Sinaǐ, Ergodic Theory, Grundlehren der mathematischen Wissenschaften, vol. 245. Springer-Verlag, New York (1982). Zbl0493.28007MR832433
- [11] 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. Zbl1136.60034MR2397794
- [12] D.M. Guillaume, M.M. Dacorogna, R.R. Davé, U.A. Müller, R.B. Olsen and O.V. Pictet, 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. Zbl0889.90021
- [13] L.P. Hansen, Large sample properties of generalized method of moments estimators. Econometrica50 (1982) 1029–1054. Zbl0502.62098MR666123
- [14] U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter, Berlin (1985). Zbl0575.28009MR797411
- [15] M. Magdziarz, A note on Maruyama’s mixing theorem. Theory Probab. Appl.54 (2010) 322–324. Zbl05820340MR2761568
- [16] T. Marquardt, Fractional Lévy processes with an application to long memory moving average processes. Bernoulli12 (2006) 1099–1126. Zbl1126.60038MR2274856
- [17] T. Marquardt and R. Stelzer, Multivariate CARMA processes. Stoc. Proc. Appl.117 (2007) 96–120. Zbl1115.62087MR2287105
- [18] G. Maruyama, Infinitely divisible processes. Theory Probab. Appl.15 (1970) 1–22. Zbl0268.60036MR285046
- [19] 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).
- [20] K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge, UK (1983). Zbl0507.28010MR833286
- [21] C. Pigorsch and R. Stelzer, A Multivariate Ornstein-Uhlenbeck Type Stochastic Volatility Model. Available from http://www.uni-ulm.de/mawi/finmath.html (2009). Zbl1221.60074
- [22] B.S. Rajput and J. Rosiński, Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields82 (1989) 451–487. Zbl0659.60078MR1001524
- [23] J. Rosiński and T. Żak, Simple conditions for mixing of infinitely divisible processes. Stoch. Proc. Appl.61 (1996) 277–288. Zbl0849.60031
- [24] J. Rosiński and T. Żak, The equivalence of ergodicity and weak mixing for infinitely divisible processes. J. Theor. Probab.10 (1997) 73–86. Zbl0870.60029
- [25] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge, UK (1999). Zbl0973.60001
- [26] D. Surgailis, J. Rosiński, V. Mandrekar and S. Cambanis, Stable mixed moving averages. Probab. Theory Relat. Fields97 (1993) 543–558. Zbl0794.60026MR1246979
- [27] T. Tosstorff and R. Stelzer, Moment based estimation of supOU processes and a related stochastic volatility model. In preparation (2011). Zbl1309.62145

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.