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Contents of PMS, Vol. 40, Fasc. 2,
pages 269 - 295
DOI: 10.37190/0208-4147.40.2.5
Published online 9.7.2020
 

Weyl multifractional Ornstein--Uhlenbeck processes mixed with a Gamma distribution

Khalifa Es-Sebaiy
Fatima-Ezzahra Farah
Astrid Hilbert

Abstract: The aim of this paper is to study the asymptotic behavior of aggregated Weyl multifractional Ornstein–Uhlenbeck processes mixed with Gamma random variables. This allows us to introduce a new class of processes, Gamma-mixed Weyl multifractional Ornstein–Uhlenbeck processes (GWmOU), and study their elementary properties such as Hausdorff dimension, local self-similarity and short-range dependence. We also prove that these processes approach the multifractional Brownian motion.

2010 AMS Mathematics Subject Classification: Primary 60G22; Secondary 60G17.

Keywords and phrases: Weyl multifractional Ornstein--Uhlenbeck process, Gamma distribution, aggregated process, multifractional Brownian motion.

References


[1] G. Andrews, R. Askey R. Roy, Special Functions, Cambridge Univ. Press, 1999.

[2] A. Ayache, S. Jaffard M. S. Taqqu, Wavelet construction of generalized multifractional processes, Rev. Mat. Iberoamer. 23 (2007), 327-370.

[3] A. Ayache M. S. Taqqu, Multifractional processes with random exponent, Publ. Mat. 49 (2005), 459-486.

[4] A. Benassi, S. Jaffard D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamer. 13 (1997), 19-90.

[5] S. Bianchi, Pathwise identification of the memory function of multifractional Brownian motion with application to finance, Int. J. Theoret. Appl. Finance 8 (2005), 255-281.

[6] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.

[7] P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley, New York, 1999.

[8] P. Cheridito, H. Kawaguchi M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003), no. 3, 14 pp.

[9] S. Eryilmaz, Delta-shock model based on Polya process and its optimal replacement policy, Eur. J. Oper. Res. 263 (2017), 690-697.

[10] K. Es-Sebaiy C. A. Tudor, Fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution, Fractals 23 (2015), no. 3, art. 1550032, 10 pp.

[11] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, 2nd ed., Wiley, 2003.

[12] P. Flandrin, P. Borgnat P. O. Amblard, From stationarity to self-similarity, and back: Variations on the Lamperti transformation, in: Processes with Long-Range Correlations, Lecture Notes in Phys. 621, Springer, New York, 2003, 88–117.

[13] E. Iglói G. Terdik, Long-range dependence through Gamma-mixed Ornstein–Uhlenbeck process, Electron. J. Probab. 4 (1999), no. 16, 33 pp.

[14] S. C. Lim C. H. Eab, Riemann–Liouville and Weyl fractional oscillator processes, Phys. Lett. A 355 (2006), 87-93.

[15] S. C. Lim L. P. Teo, Weyl and Riemann–Liouville multifractional Ornstein-Uhlenbeck processes, J. Phys. A 40 (2007), 6035-6060.

[16] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

[17] R. F. Peltier J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, Research Report RR-2645, INRIA, 1995.

[18] S. Samko, A. A. Kilbas D. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993.

[19] S. Stoev M. S. Taqqu, How rich is the class of multifractional Brownian motions?, Stoch. Process. Appl. 116 (2006), 200-221.

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