Maximum Quasi-likelihood Estimation in Fractional Levy Stochastic Volatility Model

Jaya Prakasah Narayan Bishwal

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DOI: 10.4236/jmf.2011.13008                

Abstract

Usually asset price process has jumps and volatility process has long memory. We study maximum quasi- likelihood estimators for the parameters of a fractionally integrated exponential GARCH, in short FIECO- GARCH process based on discrete observations. We deal with a compound Poisson FIECOGARCH process and study the asymptotic behavior of the maximum quasi-likelihood estimator. We show that the resulting estimators are consistent and asymptotically normal.

Keywords

Absolute Ruin, Debit Interest, Moment-Generating Function, Markovian Regime-Switching Risk Model, Dividend Barrier, Integro-Differential Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	[1] A. J. Wolfe, “On a Continuous Analogue of the Stochastic Difference Equation Xn = ρXn + Bn,” Stochastic Processes and their Applications, Vol. 12, No. 3, 1982, pp. 301- 312. doi:/10.1016/0304-4149(82)90050-3
[2]	O. E. Barndorff-Nielsen and N. Shephard, “Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses In Financial Economics (with Discussion),” Journal of the Royal Statistical Society, Series B, Vol. 63, No. 2, 2001, pp. 167-241. doi:/10.1111/1467-9868.00282
[3]	O. E. Barn-dorff-Neilsen and N. Shephard, “Normal modified stable processes,” Theory of Probability and Mathematical Sta-tistics, Vol. 65, 2002, pp. 7-20.
[4]	D. P. Gaver and P. A. W. Lewis, “First Order Autoregressive Gamma Se-quences and Point Processes,” Advances in Applied Probability, Vol. 12, No. 3, 1980, pp. 727-745. doi:/10.2307/1426429
[5]	R. A. Davis and W. P. McCormick, “Estimation for first order autoregressive processes with positive or bounded innovations,” Sto-chastic Processes and their Applications, Vol. 31, No. 2, 1989, pp. 237-250. doi:/10.1016/0304-4149(89)90090-2
[6]	B. Neilsen and N. Shephard, “Likelihood Analysis of a First Order Autoregressive Model with Exponential Innovations,” Journal of Time Series Analysis, Vol. 24, No. 3, 2003, pp. 337-344. doi:/10.1111/1467-9892.00310
[7]	P. J. Brockwell, R. A. Davis and Y. Yang, “Estimation for Non-Negative Levy driven Ornstein-Uhlenbeck Processes,” Journal of Applied Probability, Vol. 44, No. 4, 2007, pp. 977-989. doi:/10.1239/jap/1197908818
[8]	G. Jongbloed, F. H. van der Meulen and A. W. van der Vaart, “Nonparametric Inference for Levy Driven Ornstein-Uhlenbeck Processes,” Bernoulli, Vol. 11, No. 5, 2005, pp. 759-791. doi:/10.3150/bj/1130077593
[9]	J. P. N. Bishwal, “Parameter Estimation in Stochastic Dif-ferential Equations,” Springer-Verlag, Berlin, 2008. doi:/10.1007/978-3-540-74448-1
[10]	F. Comte and E. Renault, “Long Memory in Continuous- Time Stochastic Volatility Models,” Mathematical Finance, Vol. 8, No. 4, 1998, pp. 291-323. doi:/10.1111/1467-9965.00057
[11]	F. Comte, L. Coutin and E. Renault, “Affine Fractional Stochastic Volatility Models with application to Option Pricing,” Re-cherche,Vol. 13, No. 1993, 2010, 1-35.
[12]	T. Mar-quardt, “Fractional Levy processes with applica- tion to long memory moving average processes,” Ber- noulli, Vol. 12, No. 6, 2006, pp. 1009-1126. doi:/10.3150/bj/1165269152
[13]	T. Marquardt, “Multi-variate FICARMA Processes,” Journal of Multivariate Analysis, Vol. 98, No. 9, 2007, pp. 1705-1725. doi:/10.1016/j.jmva.2006.07.001
[14]	C. Bender and T. Marquardt, “Stochastic Calculus for Convoluted Levy Processes,” Bernoulli, Vol. 14, No. 2, 2007, pp. 499-518. doi:/10.3150/07-BEJ115
[15]	S. Haug and C. Czado, “An Exponential Continuous Time GARCH Process,” Journal of Applied Probability, Vol. 44, No. 4, 2007, pp. 960-976. doi:/10.1239/jap/1197908817
[16]	S. Haug and C. Czado, “Fractionally Integrated ECO- GARCH Process,” Sonder-forschungsbereich, Vol. 386, No. 484, 2006.
[17]	C. Czado and S. Haug, “An ACD-ECOGARCH(1,1) Mod-el,” Journal of Financial Econometrics, Vol. 8, No. 3, 2010, pp. 335-344. doi:/10.1093/jjfinec/nbp023
[18]	S. Haug, C. Kluppelberg, A. Lindner and M. Zapp, “Method of Moment Estimation in the COGARCH(1,1) Model,” Econometrics Journal, Vol. 10, No. 2, 2007, pp. 320-341. doi:/10.1111/j.1368-423X.2007.00210.x
[19]	D. Strau-mann and T. Mikosch, “Quasi-Maximum Likelihood Es-timation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach,” Annals of Statistics, Vol. 34, 2006, pp. 2449-2495. doi:/10.1214/009053606000000803
[20]	S. Haug and C. Czado, “Quasi Maximum Likelihood Estimation And Prediction in the Compound Poisson ECOGARCH(1,1) Model,” Sonderforschungsbereich, 386, No. 516, 2006.
[21]	D. Straumann, “Estimation in Conditionally Heterosce- dastic Time Series Models,” Lecture Notes in Statistics Vol. 181, Springer-Verlag, Berlin, 2005.