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 sup t ∈ [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.
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Barndorff-Nielsen, O.E.: Superposition of Ornstein–Uhlenbeck type processes. Theory Probab. Appl. 45, 175–194 (2001)
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)
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)
Billingsley, P.: Convergence of Probability and Measures, 2nd edn. Wiley, New York (1999)
Braverman, M., Samorodnitsky, G.: Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207–231 (1995)
Cline, D.B.H.: Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72, 529–557 (1986)
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)
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)
Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Process. Appl. 13, 263–278 (1982)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)
Fasen, V.: Extremes of Lévy Driven MA Processes with Applications in Finance. Ph.D. thesis, Munich University of Technology (2004)
Fasen, V.: Extremes of regularly varying mixed moving average processes. Adv. Appl. Probab. 37, 993–1014 (2005)
Fasen, V.: Extremes of subexponential Lévy driven moving average processes. Stoch. Process. Appl. 116, 1066–1087 (2006)
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)
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)
Goldie, C.M., Resnick, S.: Subexponential distribution tails and point processes. Commun. Stat. Stoch. Models 4, 361–372 (1988)
Hsing, T., Teugels, J.L.: Extremal properties of shot noise processes. Adv. Appl. Probab. 21, 513–525 (1989)
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)
Kingman, J.F.C.: Poisson Processes. Oxford University Press, Oxford (1993)
Kwapień, S., Woyczyzński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992)
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)
Pakes, A.G.: Convolution equivalence and infinite divisibility. J. Appl. Probab. 41, 407–424 (2004)
Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82, 453–487 (1989)
Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)
Rootzén, H.: Extreme value theory for moving average processes. Ann. Probab. 14, 612–652 (1986)
Rosinski, J., Samorodnitsky, G.: Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21, 996–1014 (1993)
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Schoutens, W.: Lévy Processes in Finance. Wiley, Chichester (2003)
Urbanik, K., Woyczyński, W.A.: Random integrals and Orlicz spaces. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 15, 161–169 (1967)
Watanabe, T.: Convolution equivalence and distributions of random sums. Probab. Theory Relat. Fields 142, 367–397 (2008)
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Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.
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Fasen, V. Extremes of Lévy driven mixed MA processes with convolution equivalent distributions. Extremes 12, 265–296 (2009). https://doi.org/10.1007/s10687-008-0079-x
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DOI: https://doi.org/10.1007/s10687-008-0079-x
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