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

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).

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

Lietuvos matematikos rinkinys ◽

10.15388/lmr.2008.18148 ◽

2008 ◽

Vol 48 ◽

Cited By ~ 6

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.

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

Journal of Applied Probability ◽

10.1017/jpr.2017.10 ◽

2017 ◽

Vol 54 (2) ◽

pp. 444-461 ◽

Cited By ~ 2

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.

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

Engineering Computations ◽

10.1108/ec-01-2020-0039 ◽

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.

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700034535 ◽

2004 ◽

Vol 70 (2) ◽

pp. 321-328 ◽

Cited By ~ 5

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.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2021/4934658 ◽

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.

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

Journal of Applied Probability ◽

10.1239/jap/1395771410 ◽

2014 ◽

Vol 51 (1) ◽

pp. 1-18 ◽

Cited By ~ 1

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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Fuzzy stochastic differential equations driven by fractional Brownian motion

Advances in Difference Equations ◽

10.1186/s13662-020-03181-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hossein Jafari ◽

Marek T. Malinowski ◽

M. J. Ebadi

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Fractional Brownian Motion ◽

Approximation Method ◽

Existence And Uniqueness ◽

Financial Models ◽

Stochastic Integral ◽

Long Range Dependence ◽

World Systems

AbstractIn this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

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