Simulation of a fractional Brownian motion in the space $L_p([0,T])$
Authors:
Yu. V. Kozachenko, A. O. Pashko and O. I. Vasylyk
Translated by:
N. N. Semenov
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 |
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Additional Information
Abstract: A model that approximates the fractional Brownian motion with parameter $\alpha \in (0,2)$ with a given reliability $1- \delta$, $0<\delta <1$, and accuracy $\varepsilon > 0$ in the space $L_p([0,T])$ is constructed. An example of a simulation in the space $L_2([0,1])$ is given.
References
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- 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)
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- 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)
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References
- F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Probability and its Applications, Springer-Verlag, London, 2008. MR 2387368
- 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.
- A. B. Dieker and M. Mandjes, On spectral simulation of fractional Brownian motion, Probab. Engin. Inform. Sci. 17 (2003), 417–434. MR 1984656
- A. B. Dieker, Simulation of Fractional Brownian Motion, Master’s thesis, Vrije Universiteit Amsterdam (2002, updated in 2004).
- K. O. Dzhaparidze and J. H. van Zanten, A series expansion of fractional Brownian motion, CWI, Probability, Networks and Algorithms R 0216 (2002).
- S. M. Ermakov and G. A. Mikhailov, Statistical Simulation, “Nauka”, Moscow, 1982. (Russian) MR 705787
- Yu. Kozachenko and O. Kamenshchikova, Approximation of $SSub_\varphi (\Omega )$ stochastic processes in the space $L_p(T)$, Theory Probab. Math. Statist. 79 (2009), 83–88. MR 2494537
- 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)
- A. N. Kolmogorov, The local structure of turbulence in an incompressible fluid at very high Reynolds numbers, Dokl. Akad. Nauk SSSR 30 (1941), 299–303. (Russian) MR 0004146
- Yu. Kozachenko, A. Olenko, and O. Polosmak, Uniform convergence of wavelet expansions of Gaussian stochastic processes, Stoch. Anal. Appl. 29 (2011), no. 2, 169–184. MR 2774235
- Yu. Kozachenko and A. Pashko, Accuracy of simulation of stochastic processes in norms of Orlicz spaces. I, Theory Probab. Math. Statist. 58 (1999), 51–66. MR 1793766
- Yu. Kozachenko and A. Pashko, Accuracy of simulation of stochastic processes in norms of Orlicz spaces. II, Theory Probab. Math. Statist. 59 (1999), 77–92. MR 1793766
- Yu. Kozachenko and A. Pashko, On the simulation of random fields. I, Theory Probab. Math. Statist. 61 (2000), 61–74. MR 1866969
- Yu. Kozachenko and A. Pashko, On the simulation of random fields. II, Theory Probab. Math. Statist. 62 (2001), 51–63. MR 1866969
- Yu. Kozachenko and A. Pashko, Accuracy and Reliability of Simulation of Random Processes and Fields in Uniform Metrics, Kyiv, 2016. (Ukrainian)
- Yu. Kozachenko, A. Pashko, and I. Rozora, Simulation of Random Processes and Fields, “Zadruga”, Kyiv, 2007. (Ukrainian)
- Yu. Kozachenko, O. Pogorilyak, I. Rozora, and A. M. Tegza, Simulation of Stochastic Processes with Given Accuracy and Reliability, ISTE Press–Elsevier, 2016. MR 3644192
- Yu. Kozachenko and I. Rozora, Accuracy and reliability of models of stochastic processes of the space $\mathrm {Sub}_\varphi (\Omega )$, Theory Probab. Math. Statist. 71 (2005), 105–117. MR 2144324
- Yu. Kozachenko, I. Rozora, and Ye. Turchyn, On an expansion of stochastic processes in series, Random Oper. Stoch. Equ. 15 (2007), no. 1, 15–33. MR 2316186
- Yu. Kozachenko and O. Vasilik, On the distribution of suprema of $Sub_\varphi (\Omega )$ stochastic processes, Theory Stoch. Process. 4(20) (1998), no. 1–2, 147–160. MR 2026624
- Yu. Kozachenko, T. Sottinen, and O. Vasylyk, Simulation of weakly self-similar stationary increment $Sub_\varphi (\Omega )$-processes: a series expansion approach, Methodol. Comput. Appl. Probab. 7 (2005), 379–400. MR 2210587
- P. Kramer, O. Kurbanmuradov, and K. Sabelfeld, Comparative analysis of multiscale Gaussian random field simulation algorithms, J. Comput. Phys., September (2007). MR 2356863
- B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review 10 (1968), no. 4, 422–437. MR 0242239
- Yu. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Math., vol. 1929, Springer, Berlin, 2008. MR 2378138
- 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.
- 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)
- 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)
- 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)
- J. Picard, Representation formulae for the fractional Brownian motion, Séminaire de Probabilités, Springer-Verlag, XLIII (2011), 3–70. MR 2790367
- S. M. Prigarin, Numerical Modeling of Random Processes and Fields, Novosibirsk, 2005. (Russian)
- S. M. Prigarin and P. V. Konstantinov, Spectral numerical models of fractional Brownian motion, Russ. J. Numer. Anal. Math. Modeling 24 (2009), no. 3, 279–295. MR 2528923
- I. 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
- K. Sabelfeld, Monte Carlo Methods in Boundary Problems, “Nauka”, Novosibirsk, 1989. (Russian) MR 1007305
- G. Shevchenko, Fractional Brownian motion in a nutshell, Int. J. Modern Phys. Conf. Ser. 36 (2015), id. 1560002. MR 3642847
- T. Sottinen, Fractional Brownian Motion in Finance and Queuing, Academic Dissertation, University of Helsinki, 2003. MR 2715575
- O. Vasylyk, Yu. Kozachenko, and R. Yamnenko, $\varphi$-sub-Gaussian Stochastic Processes, Kyivskyi Universytet, Kyiv, 2008. (Ukrainian)
- A. M. Yaglom, Correlation theory of stationary and related stochastic processes with stationary $n$-th increments, Mat. Sbornik 37 (1955), no. 1, 141–196. MR 0071672
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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
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