Simulation of a fractional Brownian motion in the space
Authors:
Yu. V. Kozachenko, A. O. Pashko and O. I. Vasylyk
Translated by:
N. N. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 97 (2017).
Journal:
Theor. Probability and Math. Statist. 97 (2018), 99-111
MSC (2010):
Primary 60G15, 60G22, 60G51, 68U20
DOI:
https://doi.org/10.1090/tpms/1051
Published electronically:
February 21, 2019
MathSciNet review:
3746002
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A model that approximates the fractional Brownian motion with parameter with a given reliability
,
, and accuracy
in the space
is constructed. An example of a simulation in the space
is given.
- [1] Francesca Biagini, Yaozhong Hu, Bernt Øksendal, and Tusheng Zhang, Stochastic calculus for fractional Brownian motion and applications, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008. MR 2387368
- [2] J.-F. Coeurjolly, Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study, Journal of Statistical Software 5 (2000), no. 7, 1-53.
- [3] A. B. Dieker and M. Mandjes, On spectral simulation of fractional Brownian motion, Probab. Engrg. Inform. Sci. 17 (2003), no. 3, 417–434. MR 1984656, https://doi.org/10.1017/S0269964803173081
- [4] A. B. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, Vrije Universiteit Amsterdam (2002, updated in 2004).
- [5] K. O. Dzhaparidze and J. H. van Zanten, A series expansion of fractional Brownian motion, CWI, Probability, Networks and Algorithms R 0216 (2002).
- [6] S. M. Ermakov and G. A. Mikhaĭlov, Statisticheskoe modelirovanie, 2nd ed., “Nauka”, Moscow, 1982 (Russian). MR 705787
- [7] Yu. V. Kozachenko and O. Ē. Kamenshchikova, Approximation of 𝑆𝑆𝑢𝑏ᵩ(Ω) random processes in the space 𝐿_{𝑝}(𝕋), Teor. Ĭmovīr. Mat. Stat. 79 (2008), 73–78 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 79 (2009), 83–88. MR 2494537, https://doi.org/10.1090/S0094-9000-09-00782-0
- [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. Kolmogoroff, The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.) 30 (1941), 301–305. MR 0004146
- [10] Yuriy Kozachenko, Andriy Olenko, and Olga Polosmak, Uniform convergence of wavelet expansions of Gaussian random processes, Stoch. Anal. Appl. 29 (2011), no. 2, 169–184. MR 2774235, https://doi.org/10.1080/07362994.2011.532034
- [11] Yu. V. Kozachenko and A. O. Pashko, The accuracy of modeling random processes in norms of Orlicz spaces. II, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 75–90 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 77–92 (2000). MR 1793766
- [12] Yu. V. Kozachenko and A. O. Pashko, The accuracy of modeling random processes in norms of Orlicz spaces. II, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 75–90 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 77–92 (2000). MR 1793766
- [13] Yu. V. Kozachenko and A. O. Pashko, On the modeling of random fields. I, Teor. Ĭmovīr. Mat. Stat. 61 (1999), 59–71 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 61 (2000), 61–74 (2001). MR 1866969
- [14] Yu. V. Kozachenko and A. O. Pashko, On the modeling of random fields. I, Teor. Ĭmovīr. Mat. Stat. 61 (1999), 59–71 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 61 (2000), 61–74 (2001). MR 1866969
- [15] Yu. Kozachenko and A. Pashko, Accuracy and Reliability of Simulation of Random Processes and Fields in Uniform Metrics, Kyiv, 2016. (Ukrainian)
- [16] Yu. Kozachenko, A. Pashko, and I. Rozora, Simulation of Random Processes and Fields, ``Zadruga'', Kyiv, 2007. (Ukrainian)
- [17] Yuriy Kozachenko, Oleksandr Pogorilyak, Iryna Rozora, and Antonina Tegza, Simulation of stochastic processes with given accuracy and reliability, Mathematics and Statistics Series, ISTE, London; Elsevier, Inc., Oxford, 2016. MR 3644192
- [18] Yu. V. Kozachenko and Ī. V. Rozora, Accuracy and reliability of modeling random processes in the space 𝑆𝑢𝑏ᵩ(Ω), Teor. Ĭmovīr. Mat. Stat. 71 (2004), 93–104 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 71 (2005), 105–117. MR 2144324, https://doi.org/10.1090/S0094-9000-05-00651-4
- [19] Yu. V. Kozachenko, I. V. Rozora, and Ye. V. Turchyn, On an expansion of random processes in series, Random Oper. Stoch. Equ. 15 (2007), no. 1, 15–33. MR 2316186, https://doi.org/10.1515/ROSE.2007.002
- [20] Yurii V. Kozachenko and Oksana I. Vasilik, On the distribution of suprema of 𝑆𝑢𝑏ᵩ(Ω) random processes, Proceedings of the Donetsk Colloquium on Probability Theory and Mathematical Statistics (1998), 1998, pp. 147–160. MR 2026624
- [21] Yuriy Kozachenko, Tommi Sottinen, and Olga Vasylyk, Simulation of weakly self-similar stationary increment 𝑆𝑢𝑏ᵩ(Ω)-processes: a series expansion approach, Methodol. Comput. Appl. Probab. 7 (2005), no. 3, 379–400. MR 2210587, https://doi.org/10.1007/s11009-005-4523-y
- [22] Peter R. Kramer, Orazgeldi Kurbanmuradov, and Karl Sabelfeld, Comparative analysis of multiscale Gaussian random field simulation algorithms, J. Comput. Phys. 226 (2007), no. 1, 897–924. MR 2356863, https://doi.org/10.1016/j.jcp.2007.05.002
- [23] Benoit B. Mandelbrot and John W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437. MR 242239, https://doi.org/10.1137/1010093
- [24] Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- [25] F. J. Molz and 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 and Yu. Shusharin, Solving linear stochastic equations with random coefficients using Monte Carlo methods, Scientific Bulletin of Chernivtsi University, Series: Computer Systems and Components 5 (2014), no. 2, 21-27. (Ukrainian)
- [29] Jean Picard, Representation formulae for the fractional Brownian motion, Séminaire de Probabilités XLIII, Lecture Notes in Math., vol. 2006, Springer, Berlin, 2011, pp. 3–70. MR 2790367, https://doi.org/10.1007/978-3-642-15217-7_1
- [30] S. M. Prigarin, Numerical Modeling of Random Processes and Fields, Novosibirsk, 2005. (Russian)
- [31] S. M. Prigarin and P. V. Konstantinov, Spectral numerical models of fractional Brownian motion, Russian J. Numer. Anal. Math. Modelling 24 (2009), no. 3, 279–295. MR 2528923, https://doi.org/10.1515/RJNAMM.2009.017
- [32] Irving S. Reed, P. C. Lee, and 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. MR 1366329, https://doi.org/10.1109/18.412687
- [33] K. K. Sabel′fel′d, Metody Monte-Karlo v kraevykh zadachakh, “Nauka” Sibirsk. Otdel., Novosibirsk, 1989 (Russian). MR 1007305
- [34] Georgiy Shevchenko, Fractional Brownian motion in a nutshell, Analysis of fractional stochastic processes, Int. J. Modern Phys. Conf. Ser., vol. 36, World Sci. Publ., Hackensack, NJ, 2015, pp. 1560002, 16. MR 3642847
- [35] Tommi Petteri Sottinen, Fractional Brownian motion in finance and queueing, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Helsingin Yliopisto (Finland). MR 2715575
- [36]
O. Vasylyk, Yu. Kozachenko, and R. Yamnenko,
-sub-Gaussian Stochastic Processes, Kyivskyi Universytet, Kyiv, 2008. (Ukrainian)
- [37] A. M. Yaglom, Correlation theory of processes with random stationary 𝑛th increments, Mat. Sb. N.S. 37(79) (1955), 141–196 (Russian). MR 0071672
Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60G15, 60G22, 60G51, 68U20
Retrieve articles in all journals with MSC (2010): 60G15, 60G22, 60G51, 68U20
Additional Information
Yu. V. Kozachenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine; Vasyl’ Stus Donetsk National University, 600th Anniversary Street, 21, Vinnytsya 21021, Ukraine
Email:
ykoz@ukr.net
A. O. Pashko
Affiliation:
Faculty for Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
aap2011@ukr.net
O. I. Vasylyk
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
ovasylyk@univ.kiev.ua
DOI:
https://doi.org/10.1090/tpms/1051
Keywords:
Gaussian processes,
fractional Brownian motion,
simulation,
sub-Gaussian processes
Received by editor(s):
September 20, 2017
Published electronically:
February 21, 2019
Dedicated:
Dedicated to the memory of our teacher Mykhailo Yosypovych Yadrenko
Article copyright:
© Copyright 2019
American Mathematical Society