Extremes

, Volume 12, Issue 3, pp 265–296

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

Article
  • 65 Downloads

Abstract

We investigate the extremal behavior of stationary mixed MA processes \(Y(t)=\int_{\mathbb{R}_+\times\mathbb{R}}f(r,t-s)\,d\,\Lambda(r,s)\) for t ≥ 0, where f is a deterministic function and Λ is an infinitely divisible and independently scattered random measure. Particular examples of mixed MA processes are superpositions of Ornstein-Uhlenbeck processes, applied as stochastic volatility models in Barndorff-Nielsen and Shephard (2001a). We assume that the finite dimensional distributions of Λ are in the class of convolution equivalent tails and in the maximum domain of attraction of the Gumbel distribution. On the one hand, we compute the tail behavior of Y(0) and supt ∈ [0,1]Y(t). On the other hand, we study the extremes of Y by means of marked point processes based on maxima of Y in random intervals. A complementary result guarantees the convergence of the running maxima of Y to the Gumbel distribution.

Keywords

Convolution equivalent distribution Extreme value theory MA process Marked point process Mixed MA process Point process Random measure Shot noise process Subexponential distribution SupOU process 

AMS 2000 Subject Classifications

Primary—60G70; Secondary—60F05, 60G10, 60G55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barndorff-Nielsen, O.E.: Superposition of Ornstein–Uhlenbeck type processes. Theory Probab. Appl. 45, 175–194 (2001)CrossRefMathSciNetGoogle Scholar
  2. Barndorff-Nielsen, O.E., Shephard, N.: Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Lévy Processes: Theory and Applications, pp. 283–318. Birkhäuser, Boston (2001a)Google Scholar
  3. Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B 63, 167–241 (2001b)MATHCrossRefMathSciNetGoogle Scholar
  4. Billingsley, P.: Convergence of Probability and Measures, 2nd edn. Wiley, New York (1999)Google Scholar
  5. Braverman, M., Samorodnitsky, G.: Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207–231 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. Cline, D.B.H.: Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72, 529–557 (1986)MATHCrossRefMathSciNetGoogle Scholar
  7. Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes, vol. I: Elementary Theory and Methods, 2nd edn. Springer, New York (2003)MATHGoogle Scholar
  8. Davis, R., Resnick, S.: Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Process. Appl. 30, 41–68 (1988)MATHCrossRefMathSciNetGoogle Scholar
  9. Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Process. Appl. 13, 263–278 (1982)MATHCrossRefMathSciNetGoogle Scholar
  10. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)MATHGoogle Scholar
  11. Fasen, V.: Extremes of Lévy Driven MA Processes with Applications in Finance. Ph.D. thesis, Munich University of Technology (2004)Google Scholar
  12. Fasen, V.: Extremes of regularly varying mixed moving average processes. Adv. Appl. Probab. 37, 993–1014 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. Fasen, V.: Extremes of subexponential Lévy driven moving average processes. Stoch. Process. Appl. 116, 1066–1087 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. Fasen, V., Klüppelberg, C.: Extremes of SupOU processes. In: Benth, F.E., Di Nunno, G., Lindstrom, T., Oksendal, B., Zhang, T. (eds.) Stochastic Analysis and Applications: The Abel Symposium 2005, pp. 340–359. Springer, New York (2007)Google Scholar
  15. Fasen, V., Klüppelberg, C., Lindner, A.: Extremal behavior of stochastic volatility models. In: Shiryaev, A.N., Grossinho, M.d.R., Oliviera, P.E., Esquivel, M.L. (eds.) Stochastic Finance. Springer, New York (2006)Google Scholar
  16. Goldie, C.M., Resnick, S.: Subexponential distribution tails and point processes. Commun. Stat. Stoch. Models 4, 361–372 (1988)MATHCrossRefMathSciNetGoogle Scholar
  17. Hsing, T., Teugels, J.L.: Extremal properties of shot noise processes. Adv. Appl. Probab. 21, 513–525 (1989)MATHCrossRefMathSciNetGoogle Scholar
  18. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)MATHGoogle Scholar
  19. Kingman, J.F.C.: Poisson Processes. Oxford University Press, Oxford (1993)MATHGoogle Scholar
  20. Kwapień, S., Woyczyzński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992)MATHGoogle Scholar
  21. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)MATHGoogle Scholar
  22. Pakes, A.G.: Convolution equivalence and infinite divisibility. J. Appl. Probab. 41, 407–424 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82, 453–487 (1989)CrossRefMathSciNetGoogle Scholar
  24. Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)MATHGoogle Scholar
  25. Rootzén, H.: Extreme value theory for moving average processes. Ann. Probab. 14, 612–652 (1986)MATHCrossRefMathSciNetGoogle Scholar
  26. Rosinski, J., Samorodnitsky, G.: Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21, 996–1014 (1993)MATHCrossRefMathSciNetGoogle Scholar
  27. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)MATHGoogle Scholar
  28. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  29. Schoutens, W.: Lévy Processes in Finance. Wiley, Chichester (2003)CrossRefGoogle Scholar
  30. Urbanik, K., Woyczyński, W.A.: Random integrals and Orlicz spaces. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 15, 161–169 (1967)MATHGoogle Scholar
  31. Watanabe, T.: Convolution equivalence and distributions of random sums. Probab. Theory Relat. Fields 142, 367–397 (2008)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnische Universität MünchenGarchingGermany

Personalised recommendations