Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-13T08:00:29.949Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail behavior of multivariate lévy-driven mixed moving average processes and supOU Stochastic Volatility Models

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

Martin Moser  and
Robert Stelzer
Martin Moser*
Affiliation:
Technische Universität München
Robert Stelzer*
Affiliation:
Ulm University
*
∗ Postal address: TUM Institute for Advanced Study and Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching bei München, Germany. Email address: moser@ma.tum.de
∗∗ Postal address: Institute of Mathematical Finance, Ulm University, Helmholtzstrasse 18, 89081 Ulm, Germany. Email address: robert.stelzer@uni-ulm.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Multivariate Lévy-driven mixed moving average (MMA) processes of the type Xt = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

Keywords

Lévy basismixed moving average processmultivariate regular variationsupOU processstochastic volatility modeltail behavior

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G70: Extreme value theory; extremal processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 1109 - 1135
Copyright
Copyright © Applied Probability Trust 2011 

References

Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein-Uhlenbeck type processes. Theory Prob. Appl. 45, 175–194.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Statist. Soc. B 63, 167–241.Google Scholar
Barndorff-Nielsen, O. E. and Stelzer, R. (2011). Multivariate supOU processes. Ann. Appl. Prob. 21, 140–182.Google Scholar
Barndorff-Nielsen, O. E. and Stelzer, R. (2011). The multivariate supOU stochastic volatility model. To appear in Math. Finance.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
Bender, C., Lindner, A. and Schicks, M. (2011). Finite variation of fractional Lévy processes. To appear in J. Theoret. Prob. Google Scholar
Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113–124.Google Scholar
Brockwell, P. J. (2004). Representations of continuous-time ARMA processes. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), eds Gani, J. and Seneta, E., Applied Probability Trust, Sheffield, pp. 375–382.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Springer, Berlin.Google Scholar
Fasen, V. (2005). Extremes of regularly varying Lévy-driven mixed moving average processes. Adv. Appl. Prob. 37, 993–1014.Google Scholar
Fasen, V. (2009). Extremes of Lévy driven mixed MA processes with convolution equivalent distributions. Extremes 12, 265–296.Google Scholar
Fasen, V. and Klüppelberg, C. (2007). Extremes of supOU processes. In Stochastic Analysis and Applications (Abel Symp. 2), eds Benth, F. et al., Springer, Berlin, pp. 339–359.Google Scholar
Griffin, J. E. and Steel, M. F. J. (2010). Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes. Comput. Statist. Data Anal. 54, 2594–2608.Google Scholar
Horn, R. A. and Johnson, C. R. (1991). Topics in Matrix Analysis. Cambridge University Press.Google Scholar
Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274.Google Scholar
Hult, H. and Lindskog, F. (2006). On regular variation for infinitely divisible random vectors and additive processes. Adv. Appl. Prob. 38, 134–148.Google Scholar
Jacobsen, M., Mikosch, T., Rosiński, J. and Samorodnitsky, G. (2009). Inverse problems for regular variation of linear filters, a cancellation property for σ-finite measures and identification of stable laws. Ann. Appl. Prob. 19, 210–242.Google Scholar
Lindskog, F. (2004). Multivariate extremes and regular variation for stochastic processes. , ETH Zurich.Google Scholar
Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099–1126.Google Scholar
Marquardt, T. (2007). Multivariate fractionally integrated CARMA processes. J. Multivariate Anal. 98, 1705–1725.Google Scholar
Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA processes. Stoch. Process. Appl. 117, 96–120.Google Scholar
Pedersen, J. (2003). The Lévy-Ito decomposition of an independently scattered random measure. MaPhySto Res. Rep. 2, MaPhySto, University of Aarhus. Available at http://www.maphysto.dk/.Google Scholar
Pigorsch, C. and Stelzer, R. (2009). A multivariate Ornstein-Uhlenbeck type stochastic volatility model. Submitted.Google Scholar
Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.Google Scholar
Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer, New York.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Sato, K.-I. (2002). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1993). Stable mixed moving averages. Prob. Theory Relat. Fields 97, 543–558.Google Scholar