A Numerical Method for Linear Stochastic Ito-Volterra Integral Equation Driven by Fractional Brownian Motion

2019 ◽  
Author(s):  
Xiaoxia Wen ◽  
Jin Huang
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Numerical Method ◽  
Fractional Brownian Motion ◽  
Volterra Integral Equation

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 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 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.

NON-LIPSCHITZ BACKWARD STOCHASTIC VOLTERRA TYPE EQUATIONS WITH JUMPS

2007 ◽  
Vol 07 (04) ◽  
pp. 479-496 ◽  
Author(s):  
ZHIDONG WANG ◽  
XICHENG ZHANG
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Jump Process ◽  
Uniqueness Of Solution ◽  
Volterra Type

In this paper, we prove the existence and uniqueness of solution for the backward stochastic Volterra integral equation with non-Lipschitz coefficients and driven by Brownian motion and jump process. Moreover, when the equation is driven only by Brownian motion, we also study the continuity of the solution with respect to the time.

scholarly journals Orthonormal Bernstein polynomials for Volterra integral equations of the second kind

2020 ◽  
Vol 9 (1) ◽  
pp. 9
Author(s):  
Jafar Biazar ◽  
Hamed Ebrahimi
Keyword(s):  
Mathematical Modeling ◽  
Integral Equation ◽  
Numerical Method ◽  
Integral Equations ◽  
Method Of Moments ◽  
Volterra Integral Equation ◽  
Bernstein Polynomials ◽  
Volterra Integral Equations ◽  
Linear Volterra Integral Equations ◽  
Relative Errors

The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the method of moments (Galerkin- Ritz) using orthonormal Bernstein polynomials. To solve a Volterra integral equation, the ap-proximation for a solution is considered as an expansion in terms of Bernstein orthonormal polynomials. Ultimately, the usefulness and extraordinary accuracy of the proposed approach will be verified by a few examples where the results are plotted in diagrams, Also the re-sults and relative errors are presented in some Tables.  

On the prediction of fractional Brownian motion

1996 ◽  
Vol 33 (02) ◽  
pp. 400-410 ◽  
Author(s):  
Gustaf Gripenberg ◽  
Ilkka Norros
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Singular Integral Equation ◽  
Weight Function ◽  
Fractional Brownian Motion ◽  
Singular Integral ◽  
Hurst Parameter ◽  
Weakly Singular

Integration with respect to the fractional Brownian motionZwith Hurst parameteris discussed. The predictoris represented as an integral with respect toZ,solving a weakly singular integral equation for the prediction weight function.

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).

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