A Journal "Theory of Probability and Mathematical Statistics"
2024
2023
2022
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process   Special Issues   History  

Theory of Probability and Mathematical Statistics



Simulation of fractional Brownian motion in the space Lp([0,T])

Yu. V. Kozachenko, A. O. Pashko, O. I. Vasylyk

Download PDF

Abstract: In this paper, we construct the model of a fractional Brownian motion with parameter a, which approximates such process with given reliability and accuracy in the space Lp([0,T]). Example of simulation in L2([0,1]) is given.

Keywords: Gaussian processes, fractional Brownian motion, simulation, sub-Gaussian processes.

Bibliography:
1. F. Biagini, Y. Hu, B. Oksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Probability and its Applications (New York), Springer-Verlag, London, 2008.
2. J.-F. Coeurjolly, Simulation and identication of the fractional Brownian motion: a bibliographical and comparative study, Journal of Statistical Software, 5 (2000), no. 7, 153.
3. A. B. Dieker, M. Mandjes, On spectral simulation of fractional Brownian motion, Probab.Engrg. Inform. Sci., 17 (2003), 417-434.
4. A. B. Dieker, Simulation of fractional Brownian motion, Master's thesis, Vrije Universiteit Amsterdam (2002, updated in 2004).
5. K. O. Dzhaparidze, J. H. van Zanten, A series expansion of fractional Brownian motion, CWI,Probability, Networks and Algorithms, R 0216 (2002).
6. S. M. Ermakov, G. A. Mikhailov, Statistical simulation, Nauka, Moscow, 1982. (Russian)
7. Yu. Kozachenko, O. Kamenshchikova, Approximation of Sub stochastic processes in the space Lp(T), Theory Probab. Math. Statist., 79 (2009), 83-88.
8. A. N. Kolmogorov, Wiener spirals and some other interesting curves in a Hilbert space, Dokl.Akad. Nauk SSSR, 26 (1940), no. 2, 115-118. (Russian)
9. A. N. Kolmogorov, The local structure of turbulence in an incompressible uid at very high Reynolds numbers, Dokl. Akad. Nauk SSSR, 30 (1941), 299-303. (Russian)
10. Yu. Kozachenko, A. Olenko, O. Polosmak, Uniform convergence of wavelet expansions of Gaussian random processes, Stoch. Anal. Appl., 29 (2011), no. 2, 169-184.
11. Yu. Kozachenko, A. Pashko, Accuracy of simulation of stochastic processes in norms of Orlicz spaces. I, Theory Probab. Math. Statist., 58 (1999), 51-66.
12. Yu. Kozachenko, A. Pashko, Accuracy of simulation of stochastic processes in norms of Orlicz spaces. II, Theory Probab. Math. Statist., 59 (1999), 77-92.
13. Yu. Kozachenko, A. Pashko, On the simulation of random elds. I, Theory Probab. Math. Statist., 61 (2000), 61-74.
14. Yu. Kozachenko, A. Pashko, On the simulation of random elds. II, Theory Probab. Math. Statist., 62 (2001), 51-63.
15. Yu. Kozachenko, A. Pashko, Accuracy and reliability of simulation of random processes and elds in uniform metrics, Kyiv, 2016. (Ukrainian)
16. Yu. Kozachenko, A. Pashko, I. Rozora, Simulation of random processes and elds, Zadruga, Kyiv, 2007. (Ukrainian)
17. Yu. Kozachenko, O. Pogorilyak, I. Rozora, A. M. Tegza, Simulation of stochastic processes with given accuracy and reliability, ISTE Press Elsevier, 2016.
18. Yu. Kozachenko, I. Rozora, Accuracy and reliability of models of stochastic processes of the space Sub?(?), Theory Probab. Math. Statist., 71 (2005), 105-117.
19. Yu. Kozachenko, I. Rozora, Ye. Turchyn, On an expansion of random processes in series, Random Oper. Stoch. Equ., 15 (2007), no. 1, 15-33.
20. Yu. Kozachenko, O. Vasilik, On the distribution of suprema of Sub?(?) random processes, Theory Stoch. Process., 4(20) (1998), no. 1-2, 147-160.
21. Yu. Kozachenko, T. Sottinen, O. Vasylyk, Simulation of weakly self-similar stationary increment Sub?(?)-processes: a series expansion approach, Methodol. Comput. Appl. Probab., 7 (2005), 379-400.
22. P. Kramer, O. Kurbanmuradov, K. Sabelfeld, Comparative Analysis of Multiscale Gaussian Random Field Simulation Algorithms, J. Comput. Phys., September (2007).
23. B. B. Mandelbrot, J.W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), no. 4, 422-437.
24. Yu. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Math., vol. 1929, Springer, Berlin, 2008.
25. F. J. Molz, H. H. Liu, Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions, Water Resources Research, 33 (1997), no. 10 , 2273-2286.
26. A. Pashko, Statistical simulation of a generalized Wiener process, Bulletin of Taras Shevchenko National University of Kyiv, Series: Physics & Mathematics, 2 (2014), 180-183. (Ukrainian)
27. A. Pashko, Estimation of accuracy of simulation of a generalized Wiener process, Bulletin of Uzhgorod University, Series: Mathematics and Informatics, 25 (2014), no. 1, 109-116. (Ukrainian)
28. A. Pashko, Yu. Shusharin, Solving linear stochastic equations with random coecients using Monte Carlo methods, Scientic Bulletin of Chernivtsi University, Series: Computer Systems and Components, 5 (2014), no. 2, 2127. (Ukrainian)
29. J. Picard, Representation formulae for the fractional Brownian motion, Seminaire de Probabilites, Springer-Verlag, XLIII (2011), 370.
30. S. M. Prigarin, Numerical modelling of random processes and elds, Novosibirsk, 2005. (Russian)
31. S. M. Prigarin, P. V. Konstantinov, Spectral numerical models of fractional Brownian motion, Russ. J. Numer. Anal. Math. Modelling, 24 (2009), no. 3, 279-295.
32. I. S. Reed, P. C. Lee, T. K. Truong, Spectral representation of fractional Brownian motion in n dimensions and its properties, IEEE Trans. Inform. Theory, 41 (1995), no. 5, 1439-1451.
33. K. Sabelfeld, Monte Carlo Methods in Boundary Problems, Nauka, Novosibirsk, 1989. (Russian)
34. G. Shevchenko, Fractional Brownian motion in a nutshell, Int. J. Modern Phys. Conf. Ser., 36 (2015), id. 1560002.
35. T. Sottinen, Fractional Brownian motion in nance and queueing, Academic Dissertation, University of Helsinki, 2003.
36. O. Vasylyk, Yu. Kozachenko, R. Yamnenko, ?-sub-Gaussian random processes, Kyivskyi Universytet, Kyiv, 2008. (Ukrainian)
37. A. M. Yaglom, Correlation theory of stationary and related random processes with stationary n-th increments, Mat. Sbornik, 37 (1955), no. 1, 141-196.