Abstract

In this paper, a Haar wavelet method is proposed for solving linear and nonlinear stochastic Itô–Volterra integral equation (SIVIE) driven by fractional Brownian motion (FBM) with Hurst parameter $H\in (\frac{1}{2},1).$ This approach reduces the solution of the problem under study to the solution of a linear and nonlinear system of algebraic equations, which is based on the operational matrix and stochastic operational matrix of the Haar wavelets basis (HWB). Moreover, the error estimate and convergence analysis of the method are established in detail. Several numerical examples are presented to test the applicability and efficiency of the proposed method.

摘要

在本文中，提出了一种 Haar 小波方法来求解由具有 Hurst 参数的分数布朗运动 (FBM) 驱动的线性和非线性随机 Itô-Volterra 积分方程 (SIVIE)。 $H\in (\frac{1}{2},1).$这种方法将所研究问题的解简化为代数方程的线性和非线性系统的解，该系统基于 Haar 小波基 (HWB) 的运算矩阵和随机运算矩阵。并详细建立了该方法的误差估计和收敛性分析。给出了几个数值例子来测试所提出方法的适用性和效率。