Numerical simulation of the Hurst index of solutions of fractional stochastic dynamical systems driven by fractional Brownian motion

2021 ◽  
Vol 386 ◽  
pp. 113210
Author(s):  
A. Shahnazi-Pour ◽  
B. Parsa Moghaddam ◽  
A. Babaei
Keyword(s):  
Numerical Simulation ◽  
Brownian Motion ◽  
Dynamical Systems ◽  
Fractional Brownian Motion ◽  
Hurst Index ◽  
Stochastic Dynamical Systems

scholarly journals Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Anas Dheyab Khalaf ◽  
Mahmoud Abouagwa ◽  
Xiangjun Wang
Keyword(s):  
Brownian Motion ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Lipschitz Condition ◽  
Averaging Method ◽  
Averaging Principle ◽  
Mean Square ◽  
Stochastic Dynamical Systems ◽  
Theoretical Results

AbstractThis paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

scholarly journals Second-order limit laws for occupation times of fractional Brownian motion

2017 ◽  
Vol 54 (2) ◽  
pp. 444-461 ◽  
Author(s):  
Fangjun Xu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Method Of Moments ◽  
Second Order ◽  
Hurst Index ◽  
Occupation Times ◽  
Limit Laws ◽  
Additive Functionals ◽  
Finite Dimensional ◽  
Functional Version

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

scholarly journals Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

2022 ◽  
Vol 9 ◽  
Author(s):  
Han Gao ◽  
Rui Guo ◽  
Yang Jin ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Hurst Index ◽  
Interacting Diffusion ◽  
Two Parameters

Let SH be a sub-fractional Brownian motion with index 12<H<1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ &lt; 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

scholarly journals Optimal Sequential Change Detection for Fractional Diffusion-Type Processes

2013 ◽  
Vol 50 (1) ◽  
pp. 29-41
Author(s):  
Alexandra Chronopoulou ◽  
Georgios Fellouris
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Abrupt Change ◽  
Fractional Diffusion ◽  
Hurst Index ◽  
Continuous Path ◽  
Cusum Test ◽  
Random Drift ◽  
Diffusion Type ◽  
Drift Term

The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H+1/2.

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