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.

中文翻译：

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分数布朗运动驱动的非线性随机Itô-Volterra积分方程的数值解

目的

本文旨在研究分数布朗运动及其在非线性随机积分方程中的应用。Bernstein多项式已用于获得非线性分数阶随机积分方程的数值结果。

设计/方法/方法

Bernstein多项式已用于获得非线性分数阶随机积分方程的数值解。基于伯恩斯坦多项式的分数随机操作矩阵已经被用来离散非线性分数随机积分方程。讨论了该方法的收敛性和误差分析。

发现

给出了两个示例，以证明所提出方法的效率和适用性。将获得的相应数值结果与精确解进行比较，以建立所提出方法的准确性和效率。

创意/价值

据作者所知，由分数布朗运动驱动的非线性随机Itô-Volterra积分方程首次使用伯恩斯坦多项式求解。数值结果很好地证明了该方法的准确性和有效性。