Weyl Multifractional Ornstein–Uhlenbeck Processes Mixed with A Gamma Distribution

• Autorzy: Es-Sebaiy Khalifa , Farah Fatima-Ezzahra , Hilbert Astrid

Identyfikatory

DOI

10.37190/0208-4147.40.2.5

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Weyl multifractional Ornstein–Uhlenbeck process; gamma distribution; aggregated process; multifractional Brownian motion

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2020
• Tom: Vol. 40, Fasc. 2
• Strony: 269--295
• Opis fizyczny: Bibliogr.19 poz.

Twórcy

autor

Es-Sebaiy Khalifa

• Department of Mathematics, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

autor

Farah Fatima-Ezzahra

• Department of Mathematics, National School of Applied Sciences, Marrakesh, Cadi Ayyad University, BP Guéliz Boulevard Abdelkrim Al Khattabi, Marrakesh, Morocco

autor

Hilbert Astrid

• Department of Mathematics, Faculty of Technology, Linnaeus University, P G Vejdes väg, 351 95 Växjö, Sweden

Bibliografia

• [1] G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge Univ. Press, 1999.
• [2] A. Ayache, S. Jaffard and M. S. Taqqu, Wavelet construction of generalized multifractional processes, Rev. Mat. Iberoamer. 23 (2007), 327-370.
• [3] A. Ayache and M. S. Taqqu, Multifractional processes with random exponent, Publ. Mat. 49 (2005), 459-486.
• [4] A. Benassi, S. Jaffard and 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 and M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003), no. 3, 14 pp.
• [9] S. Eryilmaz, δ-shock model based on Polya process and its optimal replacement policy, Eur. J. Oper. Res. 263 (2017), 690-697.
• [10] K. Es-Sebaiy and 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 and 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 and G. Terdik, Long-range dependence through Gamma-mixed Ornstein-Uhlenbeck process, Electron. J. Probab. 4 (1999), no. 16, 33 pp.
• [14] S. C. Lim and C. H. Eab, Riemann-Liouville and Weyl fractional oscillator processes, Phys. Lett. A 355 (2006), 87-93.
• [15] S. C. Lim and 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 and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, Research Report RR-2645, INRIA, 1995.
• [18] S. Samko, A. A. Kilbas and D. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993.
• [19] S. Stoev and M. S. Taqqu, How rich is the class of multifractional Brownian motions?, Stoch. Process. Appl. 116 (2006), 200-221.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).