Chapter 10. Simulation of Stochastic Processes with Given Reliability and Accuracy

$39.50

Iryna Rozora, Tetiana Ianevychy, Anatoliy Pashkoz and Dmytro Zatulax
Taras Shevchenko National University of Kyiv, Ukraine

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications

Abstract

In this chapter the authors deal with the problem of simulation of the random sequences and stochastic processes that are supposed to be an input on a system taking into account the system response with given accuracy and reliability. Some examples of simulation of AR, ARMA sequences are given. The simulation method for stochastic processes as input for the system of time lagging is described as well.

Keywords: simulation, given accuracy and reliability, stochastic process, random sequence, time series, spectral density, input and output process


References


[1] Sabelfeld K., Monte Carlo methods in boundary value problems, Scientific Computation, Berlin, Heidelberg: Springer-Verlag, 283 p., 1991.
[2] Sabelfeld K. and Simonov N., Random walks on boundary for solving PDEs, Utrecht:
VSP, 133 p., 1994.
[3] Prigarin S., Hahn K. and Winkler G., Comparative analysis of two numerical methods
to measure Hausdorff dimension of the fractional Brownian motion, Siberian J. Num.
Math., vol. 11, no.2, pp. 201–218, 2008.
[4] Prigarin S., Models of Random Processes and Fields in Monte Carlo Methods, Palmarium Academic Publishing, 2014.
[5] Belotserkovskii O. M. and Khlopkov Y. I., Monte Carlo methods in applied mathematics and computational aerodynamics. Comput. Math. and Math. Phys., vol. 46, pp.
1418–1441, 2006.
[6] Ermakov S. and Mikhailov G., Statistical modeling, Moskva: “Nauka”, 296 p., 1982.
(In Russian)
[7] Ermakov S. V., Jacobson S. C. and Ramsey J. M., Computer simulations of electrokinetic transport in microfabricated channel structures, Analytical Chemistry, vol. 70,
no. 21, pp. 4494–4504, 1998.
[8] Kozachenko Yu. V. and Rozora I. V., Construction of the Karhunen–Loeve model for
an input Gaussian process in a linear system by using the output process, Theory of
Probability and Mathematical Statistics, vol. 99, pp. 101–112, 2019.
[9] Pashko A. O., Lukovych O. V., Rozora I. V., Oleshko T. A. and Vasylyk O. I., Analysis
of simulation methods for fractional Brownian motion in the problems of intelligent
systems design, IEEE International Conference on Advanced Trends in Information
Theory, ATIT 2019 – Proceedings, pp. 373–378, 2019.
[10] Pashko A. O., and Rozora I. V., Accuracy of simulation for the network traffic in
the form of fractional Brownian motion, 14th International Conference on Advanced
Trends in Radioelectronics, Telecommunications and Computer Engineering, TCSET
2018 – Proceedings, pp. 840–845, 2018.
[11] Kramer P., Kurbanmuradov O. and Sabelfeld K., Comparative analysis of multiscale
Gaussian random field simulation algorithms, Journal of Computational Physics, vol.
226, no. 1, pp. 897-924, 2007.
[12] Paroka D., Ohkura Y. and Umeda N., Analytical prediction of capsizing probability
of a ship in beam wind and waves, Journal of Ship Research, vol. 50, pp. 187-195,
2006.
[13] Shinozuka M. and Jan C. M., Digital simulation of random processes and its applications, Journal of Sound and Vibration, vol. 25, pp. 111–128, 1972.
[14] Shinozuka M. and Deodatis G., Simulation of stochastic processes by spectral representation, Applied Mechanics Review, vol. 44, pp. 191–204, 1991.
[15] Dunne A., Time series simulation, Journal of the Royal Statistical Society. Series D
(The Statistician), vol. 41, no. 1, pp. 3–8, 1992.
[16] Nielsen A., Practical time series analysis: Prediction with statistics and machine
learning. O’Reilly Media, 504 p., 2019.
[17] Cai G. Q., and Zhu W. Q., Elements of stochastic dynamics, World Scientific Publisher, Singapore, 552 p., 2016.
[18] Kozachenko Yu. V.and Petranova M. Yu., Simulation of Gaussian stationary Ornstein-Uhlenbeck process with given reliability and accuracy in space C([0, T]). Monte
Carlo Methods and Applications, vol. 23, no. 4, pp. 277–286, 2017.
[19] Kozachenko Yu. V., Pogorilyak O. O., Rozora I. V. and Tegza A. M., Simulation of
stochastic processes with given accuracy and reliability. ISTE Press – Elsevier, 331
p., 2016.
[20] Kozachenko Yu. V., Rozora I. V. and Turchyn Ye. V., On an expansion of random
processes in series, Random Operators and Stochastic Equ., vol.15, pp. 15–33, 2007.
[21] Kozachenko Yu. V., Rozora I. V. and Turchyn Ye. V., Properties of some random
series, Communications in Statistics – Theory and Methods, vol. 40, no.19-20, pp.
3672–3683, 2011.
[22] Kozachenko Yu. V., Sottinen T. and Vasylyk O. V., Simulation of weakly self-similar
stationary increment Subϕ(Ω)-processes: a series expansion approach, Methodology
and computing in applied probability, vol.7, pp. 379–400, 2005.
[23] Rozora I. V., and Lyzhechko M. V., On the modeling of linear system input stochastic
processes with given accuracy and reliability, Monte Carlo Methods and Applications,
vol.24, no. 2, pp. 129–137, 2018.
[24] Box G. E. P., Jenkins G. M., Reinsel G. C. and Ljung G. M., Time series analysis.
Forecasting and control. 5th ed., Hoboken, NJ: John Wiley & Sons, 712 p., 2016.
[25] Gikhman I. I. and Skorokhod A. V., The theory of stochastic processes. I., Berlin:
Springer, 574 p., 2004.
[26] Yaglom A. M., Correlation theory of stationary and related random functions. Vol. 1:
Basic results; Vol. 2: Supplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.
[27] Priestley M., Spectral analysis and time series, Volumes I and II, Academic Press, 890
p., 1982.
[28] Koopmans L. H., The spectral analysis of time series. Academic Press, 366 p., 1995.
[29] Kozachenko Yu., Pashko A. and Rozora I., Simulation of random processes and fields,
Kyiv: “Zadruga”, 2007. (In Ukrainian)
[30] Ianevych T.O., Rozora I.V.and Pashko A. O., On one way of modeling a stochastic
process with given accuracy and reliability, Monte Carlo Methods and Applications,
vol. 28(2), pp. 135–147, 2022.
[31] Kozachenko Yu. V. and Rozora I. V., Simulation of Gaussian stochastic processes.
Random Oper. and Stochastic Equ., vol. 11, no.3, pp. 275–296, 2003.
[32] Kozachenko Yu. V. and and Rozora I. V., Accuracy and Reliability of models of
stochastic processes of the space Subϕ(Ω), Theory of Probability and Mathematical
Statistics, vol. 71, pp. 105–117, 2005.
[33] Pashko A.O. and Ianevych T.O., Methods for modeling the Ornstein-Uhlenbeck process, Bulletin of Taras Shevchenko National University of Kyiv Series: Physics and
Mathematics, vol. 3, pp.24–29, 2019. https://doi.org/10.17721/1812-5409.2019/3.3
[34] Rozora I.V., On simulation accuracy and reliability in the space Lp([0; T]) for the
input Gaussian process served by the linear system taking into account the output,
Bulletin of Taras Shevchenko National University of Kyiv Series: Physics and Mathematics, vol. 2, pp.75–80, 2018.
[35] Vasylyk O.I., Rozora I.V., Ianevych T.O. and Lovytska I.I., On some method onmodel
construction for strictly ϕ-sub-Gaussian generalized fractional Brownian motion, Bulletin of Taras Shevchenko National University of Kyiv Series: Physics and Mathematics, vol. 2, pp.18–25, 2021. https://doi.org/10.17721/1812-5409.2021/2.3
[36] Kolmogorov A. N., Stationary sequences in Hilbert’s space, Bull. Mosk. Gos. Univ.
Mat., vol.2, no.6, pp. 1–40, 1941.
[37] Kolmogorov A. N., Selected works by Kolmogorov A. N., Vol. II: Probability theory
and mathematical statistics. Ed. by Shiryaev. mathematics and Applications. Soviev
Series. 26, Kluwer Academic Publishers, 597 p., 1992.
[38] Buldygin V. V., and Kozachenko Yu. V., Metric characterization of random variables
and random processes, American Mathematical Society, Providence RI, 261 p., 2000.
[39] Vasylyk O. I., Kozachenko Yu. V. and Yamnenko R. Ye., ϕ-Sub-Gaussian stochastic
processes. Kyiv, VPTs “Ky¨ıvsky˘ıuniversytet”, 232 p., 2008. (in Ukrainian)
[40] Kozachenko Yu. V., Sottinen T. and Vasylyk O. I., Lipschitz conditions for Subϕ(Ω)-
processes with application to weakly self-similar stationary increment processes.
Preprint 483, University of Helsinki, 19p., 2008.
[41] Kozachenko Yu. V., and Moklyachuk O. M., Large deviation probabilities for square-Gaussian stochastic processes, Extremes, Vol.2, No. 3, p.269–293, 1999.
[42] Kozachenko Yu. V., and Moklyachuk O. M., Square-Gaussian stochastic processes,
Theory of Stoch. Processes, vol. 6(22), no.3-4, pp. 98–121, 2000.
[43] Dudley R. M., Sample functions of Gaussian process, The Annals of Probability, Vol.
1, No. 1, p. 66–103, 1973.
[44] Kozachenko Yu. V. and Rozora I. V., Conditions of the sample continuity with probability one for square-Gaussian stochastic processes, Theory of Probability and Mathematical Statistics, vol. 101, p. 153–166, 2020.
[45] Lo`eve, Probability theory. Third edition. Dover Publications, 705 p., 2017

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