Bernoulli

  • Bernoulli
  • Volume 12, Number 6 (2006), 1099-1126.

Fractional Lévy processes with an application to long memory moving average processes

Tina Marquardt

Full-text: Open access

Abstract

Starting from the moving average (MA) integral representation of fractional Brownian motion (FBM), the class of fractional Lévy processes (FLPs) is introduced by replacing the Brownian motion by a general Lévy process with zero mean, finite variance and no Brownian component. We present different methods of constructing FLPs and study second-order and sample path properties. FLPs have the same second-order structure as FBM and, depending on the Lévy measure, they are not always semimartingales. We consider integrals with respect to FLPs and MA processes with the long memory property. In particular, we show that the Lévy-driven MA process with fractionally integrated kernel coincides with the MA process with the corresponding (not fractionally integrated) kernel and driven by the corresponding FLP.

Article information

Source
Bernoulli, Volume 12, Number 6 (2006), 1099-1126.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1165269152

Digital Object Identifier
doi:10.3150/bj/1165269152

Mathematical Reviews number (MathSciNet)
MR2274856

Zentralblatt MATH identifier
1126.60038

Keywords
CARMA process fractional integration fractional Lévy process long memory Lévy process stochastic integration

Citation

Marquardt, Tina. Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 (2006), no. 6, 1099--1126. doi:10.3150/bj/1165269152. https://projecteuclid.org/euclid.bj/1165269152


Export citation

References

  • [1] Barndorff-Nielsen, O.E. and Shephard, N. (2001) Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics (with discussion). J. Roy. Statist. Soc. Ser. B, 63, 167-241.
  • [2] Benassi, A., Cohen, S. and Istas, J. (2004) On roughness indices for fractional fields. Bernoulli, 10, 357-373.
  • [3] Bender, C. (2003) Integration with respect to a fractional Brownian motion and related market models. Doctoral thesis, University of Konstanz.
  • [4] Brockwell, P.J. (2001) Lévy-driven CARMA processes. Ann. Inst. Statist. Math., 52, 1-18.
  • [5] Brockwell, P.J. (2004) Representations of continuous-time ARMA processes. J. Appl. Probab., 41A, 375-382.
  • [6] Brockwell, P.J. and Marquardt, T. (2005) Lévy driven and fractionally integrated ARMA processes with continuous time parameter. Statist. Sinica, 15, 477-494.
  • [7] Cohen, S., Lacaux, C. and Ledoux, M. (2005) A general framework for simulation of fractional fields. Preprint, Université Paul Sabatier, Toulouse. http://www.lsp.ups-tlse.fr/Fp/Cohen/.
  • [8] Cont, R. and Tankov, P. (2004) Financial Modelling with Jump Processes. Boca Raton, FL: Chapmann & Hall/CRC.
  • [9] Decreusefond, L. and Savy, N. (2004) Anticipative calculus for filtered Poisson processes. Preprint. http://perso.enst.fr/~decreuse/recherche/fpp.pdf.
  • [10] Decreusefond, L. and Üstünel, A.S. (1999) Stochastic analysis of the fractional Brownian motion. Potential Anal., 10, 177-214.
  • [11] Doukhan, P., Oppenheim, G. and Taqqu, M.S. (2003) Theory and Applications of Long-Range Dependence, Boston: Birkhäuser.
  • [12] Duncan, T.E., Hu, Y. and Pasik-Duncan, B. (2000) Stochastic calculus for fractional Brownian motion I. Theory. SIAM J. Control. Optim., 28, 582-612.
  • [13] Eberlein, E. and Raible, S. (1999) Term structure models driven by general Lévy processes. Math. Finance, 9, 31-53.
  • [14] Fasen, V. (2004) Extremes of Lévy driven moving average processes with application in finance. Doctoral thesis, Munich University of Technology.
  • [15] Gripenberg, N. and Norros, I. (1996) On the prediction of fractional Brownian motion. J. Appl. Probab., 33, 400-410.
  • [16] Kallenberg, O. (1997) Foundations of Modern Probability. New York: Springer-Verlag.
  • [17] Loève, M. (1960) Probability Theory. Princeton, NJ: Van Nordstrand.
  • [18] Mandelbrot, B.B. and Van Ness, J.W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev., 10, 422-437.
  • [19] Marcus, M.B. and Rosinski, J. (2005) Continuity and boundedness of infinitely divisible processes: a Poisson point process approach. J. Theoret. Probab., 18, 109-160.
  • [20] Nualart, D. (2003) Stochastic calculus with respect to the fractional Brownian motion and applications. In J.M. González-Barrios, J.A. Leo´n and A. Meda (eds), Stochastic Models: Seventh Symposium on Probability and Stochastic Processes, Contemp. Math. 336, pp. 3-39. Providence, RI: American Mathematical Society.
  • [21] Pipiras, V. and Taqqu, M. (2000) Integration questions related to fractional Brownian motion. Probab. Theory Related Fields, 118, 251-291.
  • [22] Protter, P. (2004) Stochastic Integration and Differential Equations, 2 edn. New York: Springer-Verlag.
  • [23] Rajput, B.S. and Rosinski, J. (1989) Spectral representations of infinitely divisible processes. Probab. Theory Related Fields, 82, 451-487.
  • [24] Rosinski, J. (1989) On path properties of certain infinitely divisible processes, Stochastic Process. Appl., 33, 73-87.
  • [25] Rosinski, J. (1990) On series representations of infinitely divisible random vectors. Ann. Probab., 18, 405-430.
  • [26] Rosinski, J. (2002) Series representations of Lévy processes from the perspective of point processes. In O.E. Barndorff-Nielsen, T. Mikosch and S. Resnick (eds), Lévy Processes - Theory and Applications, pp. 401-415. Boston: Birkhäuser.
  • [27] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Lausanne: Gordon and Breach.
  • [28] Samorodnitsky, G. and Taqqu, M. (1994) Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman & Hall.
  • [29] Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press.
  • [30] Shiryaev, A.N. (1996) Probability. New York: Springer-Verlag.
  • [31] Zähle, M. (1998) Integration with respect to fractal functions and stochastic calculus. Probab. Theory Related Fields, 111, 333-374.