Simulation of the fractional Brownian process with given accuracy and reliability
DOI:
https://doi.org/10.17721/1812-5409.2024/1.27Keywords:
fractional Brownian motion, simulation, accuracy, reliability, Linear time-invariant system (LTI)Abstract
Background. Random process theory is being used more and more in various fields of science due to the high computing power of modern computers. However, it's often important to know how much we can rely on the models we use.
Methods. This paper examines the modelling of the fractional Brownian motion with given accuracy and reliability. The modelling is based on Dzhaparidze and van Zanten series representation of the fractional Brownian motion. We consider the fractional Brownian motion as an input process to a time-invariant linear system with a real-valued square-integrable impulse response function, which is defined on the finite domain.
Results. We prove the theorem that gives the conditions, specifically the value of the upper limit of the summing in the model, under which the obtained model approximates fractional Brownian motion with given accuracy and reliability taking into account the response of the system.
Conclusions. For the proof, we use the properties of square-Gaussian stochastic processes.
Pages of the article in the issue: 147 - 153
Language of the article: English
References
Dzhaparidze, K., Zanten, H. (2004). A series expansion of fractional Brownian motion. Probability Theory and Related Fields, 130, 39–55. https://doi.org/10.1007/s00440-003-0310-2
Kozachenko, Y., Pogorilyak, O., Rozora, I., Tegza, A. (2016). Simulation of Stochastic Processes with Given Accuracy and Reliability. Elsevier/ISTE Press.
Kozachenko, Y., Sottinen, T., Vasylyk, O. (2005). Simulation of Weakly Self-Similar Stationary Increment Subϕ(Ω)-processes: A Series Expansion Approach. Methodology and Computing in Applied Probability, 7, 379–400. https://doi.org/10.1007/s11009-005-4523-y
Rozora, I., Lyzhechko, M. (2018). On the modeling of linear system input stochastic processes with given accuracy and reliability. Monte Carlo Methods and Applications, 24(2), 129–137. https://doi.org/10.1515/mcma-2018-0011
Watson, G. N. (1944). A Treatise of the Theory of Bessel Functions. Cambridge UniversityPress.
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