Abstract
In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.
Highlights
We consider the problem of simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion
Simulation of random processes and fields is used in many areas of natural and social sciences
A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc
Summary
У цьому роздiлi наведено необхiднi означення та властивостi з теорiї φ-субгауссових випадкових величин i процесiв [1, 3, 7]. Нехай φ = {φ(x), x ∈ R} — деяка N-функцiя Орлiча. Нехай (Ω, F, P) – стандартний iмовiрнiсний простiр. Нехай φ — N-функцiя Орлiча, для якої виконується умова Q. Випадкова величина ξ належить простору Subφ(Ω) (простору φ-субгауссових випадкових величин), якщо Eξ = 0, E exp{λξ} iснує для всiх λ ∈ R та iснує така стала a > 0, що для всiх λ ∈ R виконується нерiвнiсть. T , де сiм’я випадкових величин {ξk, k = 1, ∞} є строго φ-субгауссовою з визначальною сталою. Тодi випадковий процес X(t) є строго φ-субгауссовим випадковим процесом з визначальною сталою CX = Cξ. Теорема 1.1. [1] Простiр Subφ(Ω) є простором Банаха з нормою φ−1 (ln E exp (λξ))
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