Stochastic volatility models with possible extremal clustering

Bernoulli

Volumn 19, Issue 5 A, 2013, Pages 1688-1713

  Mikosch, Thomas ^a   Rezapour, Mohsen ^b  

^a UNIVERSITY OF COPENHAGEN   (Denmark)

^b SHAHID BAHONAR UNIVERSITY OF KERMAN   (Iran)

Author keywords

EGARCH; Exponential AR(1); Extremal clustering; Extremal index; GARCH; Multivariate regular variation; Point process; Stationary sequence; Stochastic volatility process

Indexed keywords

EID: 84888332614     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/12-BEJ426     Document Type: Article

References (31)

1. Andersen, T.G., Davis, R.A., Kreiss, J.P. and Mikosch, T. (2009). Handbook of Financial Time Series. Berlin: Springer.
2. Basrak, B., Davis, R.A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908-920. MR1925445
3. Basrak, B., Davis, R.A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95-115. MR1894253
4. Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stochastic Process. Appl. 119 1055-1080. MR2508565
5. Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press. MR0898871
6. Boman, J. and Lindskog, F. (2009). Support theorems for the Radon transform and Cramér-Wold theorems. J. Theoret. Probab. 22 683-710. MR2530109
7. Breidt, F.J. and Davis, R.A. (1998). Extremes of stochastic volatility models. Ann. Appl. Probab. 8 664-675. MR1627756
8. Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323-331.
9. Davis, R.A. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179-195. MR0770636
10. Davis, R.A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879-917. MR1334176
11. Davis, R.A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049-2080. MR1673289
12. Davis, R.A. and Mikosch, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelation. J. Appl. Probab. 38A 93-104. MR1915537
13. Davis, R.A. and Mikosch, T. (2009). Extremes of stochastic volatility models. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.P. Kreiss and T. Mikosch, eds.) 355-364. Berlin: Springer.
14. Davis, R.A. and Mikosch, T. (2009). Fundamental properties of stochastic volatility models. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.P. Kreiss and T. Mikosch, eds.) 255-267. Berlin: Springer.
15. Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. New York: Springer. MR1312160
16. Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer. MR1458613
17. Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55-72. MR0652832
18. Goldie, C.M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166. MR1097468
19. Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stochastic Process. Appl. 115 249-274. MR2111194
20. Hult, H. and Lindskog, F. (2006). On Kesten's counterexample to the Cramér-Wold device for regular variation. Bernoulli 12 133-142. MR2202325
21. Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94) 121-140. MR2281910
22. Jessen, A.H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80(94) 171-192. MR2281913
23. Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248. MR0440724
24. Klüppelberg, C. and Pergamenchtchikov, S. (2007). Extremal behaviour of models with multivariate random recurrence representation. Stochastic Process. Appl. 117 432-456. MR2305380
25. Kulik, R. and Soulier, P. (2011). The tail empirical process for long memory stochastic volatility sequences. Stochastic Process. Appl. 121 109-134. MR2739008
26. Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. New York: Springer. MR0691492
27. Mikosch, T. and Stǎricǎ, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28 1427-1451. MR1805791
28. Mokkadem, A. (1990). Propriétés de mélange des processus autorégressifs polynomiaux. Ann. Inst. Henri Poincaré Probab. Stat. 26 219-260. MR1063750
29. Nelson, D.B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347-370. MR1097532
30. Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. New York: Springer. MR0900810
31. Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer. MR2271424