scholarly journals On estimation of the Hurst index of solutions of stochastic integral equations

2008 ◽  
Vol 48 ◽  
Author(s):  
Kęstutis Kubilius ◽  
Dmitrij Melichov
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Hurst Index ◽  
Stochastic Integral Equation ◽  
Stochastic Integral Equations ◽  
Convergence Almost Surely ◽  
Strongly Consistent

Let X be a solution of a stochasti Let X be a solution of a stochastic integral equation driven by a fractional Brownian motion BH and let Vn(X, 2) = \sumn k=1(\DeltakX)2, where \DeltakX = X( k+1/n ) - X(k/n ). We study the ditions n2H-1Vn(X, 2) convergence almost surely as n → ∞ holds. This fact is used to obtain a strongly consistent estimator of the Hurst index H, 1/2 < H < 1.  

scholarly journals Estimating the Hurst index of the solution of a stochastic integral equation

2009 ◽  
Vol 50 ◽  
Author(s):  
Kęstutis Kubilius ◽  
Dmitrij Melichov
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Integral ◽  
Second Order ◽  
Quadratic Variation ◽  
Consistent Estimator ◽  
Hurst Index ◽  
Stochastic Integral Equation ◽  
Strongly Consistent

Let X(t) be a solution of a stochastic integral equation driven by fractional Brownian motion BH and let V2n (X, 2) = \sumn-1 k=1(\delta k2X)2 be the second order quadratic variation, where \delta k2X = X (k+1/N) − 2X (k/ n) +X (k−1/n). Conditions under which n2H−1Vn2(X, 2) converges almost surely as n → ∞ was obtained. This fact is used to get a strongly consistent estimator of the Hurst index H, 1/2 < H < 1. Also we show that this estimator retains its properties if we replace Vn2(X, 2) with Vn2(Y, 2), where Y (t) is the Milstein approximation of X(t).

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

2020 ◽  
Vol 37 (9) ◽  
pp. 3243-3268
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Stochastic Integral ◽  
Bernstein Polynomials ◽  
Numerical Results ◽  
Content Type ◽  
Stochastic Integral Equations

Purpose This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

scholarly journals Numerical Solution of Nonlinear Stochastic Itô–Volterra Integral Equations Driven by Fractional Brownian Motion Using Block Pulse Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mengting Deng ◽  
Guo Jiang ◽  
Ting Ke
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Hurst Parameter ◽  
Stochastic Integral Equation ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix

This paper presents a valid numerical method to solve nonlinear stochastic Itô–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ 1 / 2 , 1 . On the basis of FBM and block pulse functions (BPFs), a new stochastic operational matrix is proposed. The nonlinear stochastic integral equation is converted into a nonlinear algebraic equation by this method. Furthermore, error analysis is given by the pathwise approach. Finally, two numerical examples exhibit the validity and accuracy of the approach.

scholarly journals Prediction of fractional Brownian motion with Hurst index less than 1/2

2004 ◽  
Vol 70 (2) ◽  
pp. 321-328 ◽  
Author(s):  
V. V. Anh ◽  
A. Inoue
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Index ◽  
Prediction Formula

We give a proof based on an integral equation for an explicit prediction formula for fractional Brownian motion with Hurst index less than 1/2.

scholarly journals Linear filtering with fractional Brownian motion in the signal and observation processes

1999 ◽  
Vol 12 (1) ◽  
pp. 85-90 ◽  
Author(s):  
M. L. Kleptsyna ◽  
P. E. Kloeden ◽  
V. V. Anh
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Wiener Process ◽  
Integral Equations ◽  
Hurst Index ◽  
Mean Square ◽  
Linear Filtering ◽  
Observation Process ◽  
The Mean ◽  
Filtering Problem

Integral equations for the mean-square estimate are obtained for the linear filtering problem, in which the noise generating the signal is a fractional Brownian motion with Hurst index h∈(3/4,1) and the noise in the observation process includes a fractional Brownian motion as well as a Wiener process.

scholarly journals STOCHASTIC VOLTERRA EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER H > 1/2

2012 ◽  
Vol 12 (04) ◽  
pp. 1250004 ◽  
Author(s):  
MIREIA BESALÚ ◽  
CARLES ROVIRA
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Uniqueness Result ◽  
Volterra Equations ◽  
Hurst Parameter ◽  
Stieltjes Integral ◽  
Finite Moments

In this note we prove an existence and uniqueness result of solution for stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2, showing also that the solution has finite moments. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.

scholarly journals Theoretical Error Analysis of Solution for Two-Dimensional Stochastic Volterra Integral Equations by Haar Wavelet

2019 ◽  
Vol 5 (6) ◽  
Author(s):  
M. Fallahpour ◽  
M. Khodabin ◽  
K. Maleknejad
Keyword(s):  
Integral Equation ◽  
Error Analysis ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Stochastic Integral ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Two Dimensional ◽  
Essential Requirement ◽  
Stochastic Integral Equations

Abstract The finding an efficient way to the approximate solutions of the stochastic integral equations is an essential requirement. In this paper we discuss the convergence analysis of the two-dimensional Haar wavelet functions (2D-HWFs) method for solve 2D linear stochastic Volterra integral equation. The illustrative examples are included to demonstrate the validity and applicability of this numerical method.

Stochastic asymptotic exponential stability of stochastic integral equations

1972 ◽  
Vol 9 (01) ◽  
pp. 169-177
Author(s):  
Chris P. Tsokos ◽  
M. A. Hamdan
Keyword(s):  
Integral Equation ◽  
Exponential Stability ◽  
Stochastic Integral ◽  
Stability Theorem ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Classical Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β &gt; 0 and a γ &gt; 0 such that for t∈ R +. The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

Stochastic asymptotic exponential stability of stochastic integral equations

1972 ◽  
Vol 9 (1) ◽  
pp. 169-177 ◽  
Author(s):  
Chris P. Tsokos ◽  
M. A. Hamdan
Keyword(s):  
Integral Equation ◽  
Exponential Stability ◽  
Stochastic Integral ◽  
Stability Theorem ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Classical Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R+.The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

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.

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