Abstract
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.
Highlights
Stochastic equations have been widely used in engineering, economic management, biological sciences, finance, etc
A lot of problems in these areas are modeled by stochastic Volterra integral equations
We use block pulse functions (BPFs) to solve the following nonlinear stochastic Ito–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM): u u x(u) x0(u) + g1(s, u)](x(s))ds + g2(s, u)κ(x(s))dBHs, u ∈ [0, 1), (1)
Summary
Stochastic equations have been widely used in engineering, economic management, biological sciences, finance, etc. In [18], Nualart and Rǎscanu used the pathwise R-S integral to study stochastic differential equations in regard to FBM. Pei and Xu [19] solved the stochastic equations about FBM with H > (1/2) and standard Brownian motion by pathwise approach and stochastic average. Hashemi and Khodabin [8] used Hat functions to solve nonlinear SIVIEs driven by FBM, but the error analysis is insufficient. Maleknejad et al [3, 4], Sang et al [6], and Ezzati et al [12] studied numerical solution of stochastic equation using BPFs and gave the error analysis.
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