Extremes of regularly varying Lévy-driven mixed moving average processes

Advances in Applied Probability

Volumn 37, Issue 4, 2005, Pages 993-1014

  Fasen, Vicky ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Continuous time moving average process; Extreme value theory; Independently scattered random measure; Long range dependence; Marked point process; Point process; Regular variation; Shot noise process; SupOU process

Indexed keywords

ALGEBRA; FUNCTIONS; POISSON DISTRIBUTION; PROBABILITY; THEOREM PROVING;

CONTINUOUS-TIME MOVING AVERAGE PROCESS; EXTREME-VALUE THEORY; INDEPENDENTLY SCATTERED RANDOM MEASURE; LONG-RANGE DEPENDENCE; MARKED POINT PROCESS; REGULAR VARIATION; SHOT NOISE PROCESS;

RANDOM PROCESSES;

EID: 31944450365     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1134587750     Document Type: Article

References (35)

1. Albin, J. M. P. (2000). Extremes and upcrossing intensities for P-differentiable stationary processes. Stoch. Process. Appl. 87, 199-234.
2. Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein-Uhlenbeck type processes. Theory Prob. Appl. 45, 175-194.
3. Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, eds O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, Birkhäuser, Boston, MA, pp. 283-318.
4. Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908-920.
5. Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95-115.
6. Bingham, N. H., Goldie, C. M. And Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
7. Braverman, M. and Samorodnitsky, G. (1995). Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207-231.
8. Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323-331.
9. Blockwell, P. J. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statist. Sinica 15, 477-494.
10. Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods, 2nd edn. Springer, New York.
11. Davis, R. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879-917.
12. Davis, R. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049-2080.
13. Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179-195.
14. Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
15. Fasen, V. (2004). Extremes of Lévy driven MA processes with applications in finance. Doctoral Thesis, Munich University of Technology. Available at http://tumbl.biblio.tu-muenchen.de/publ/diss/ma/ 2004/fasen.html.
16. Fasen, V. (2005). Extremes of subexponential Lévy driven moving average processes. Submitted.
17. Gushchin, A. A. and Küchler, U. (2000). On stationary solutions of delay differential equations driven by a Lévy process. Stoch. Process. Appl. 88, 195-211.
18. Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Prob. Theory Relat. Fields 95, 331-356.
19. Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513-525.
20. Hult, H. And Lindskøg, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249-274.
21. Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
22. Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
23. Leadbetter, M. R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Prob. 16, 431-478.
24. Lebedev, A. V. (2000). Extremes of subexponential shot noise. Math. Notes 71, 206-210.
25. Lindskøg, F. (2004). Multivariate extremes and regular variation for stochastic processes. Doctoral Thesis, ETH Zürich. Available at http://e-colection.ethbib.ethz.ch/cgi-bin/ show.pl?type=diss&nr=15319.
26. McCormick, W. P. (1997). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643-656.
27. Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 1025-1064.
28. Pedersen, J. (2003). The Lévy-Ito decomposition of an independently scattered random measure. Tech. Rep. 2003-2, Centre for Mathematical Physics and Stochastics, University of Aarhus. Available at http://www.maphysto.dk/oldpages/publications/ Publications2003_static.html.
29. Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 453-487.
30. Renick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
31. Rootzén, H. (1978). Extremes of moving averages of stable processes. Ann. Prob. 6, 847-869.
32. Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612-652.
33. Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996-1014.
34. Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
35. Urbanic, K. and Woyczyński, W. A. (1968). Random measures and harmonizable sequences. Studia Math. 31, 61-88.