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

Authors: Mengting Deng; Guo Jiang; Ting Ke
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## 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 parameterH∈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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## Body

## 1. Introduction

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. However, the vast majority of these equations do not have exact solutions, so it is meaningful to discuss numerical solutions of the equations [1, 2, 3, 4, 5, 6, 7].In the present work, we use BPFs to solve the following nonlinear SIVIEs driven by FBM:(1)xu=x0u+∫0ug1s,uνxsds+∫0ug2s,uκxsdBsH,u∈0,1,where g1s,u and g2s,u are kernel functions, for s∈0,u. xu is an unknown stochastic process, BsH is FBM with H∈1/2,1, and the stochastic integral is pathwise Riemann–Stieltjes (R-S) integral. They are all defined on the same probability spaceΩ,ℱ,P. ν and κ are bounded analytic functions and satisfy Lipschitz conditions.Although FBM is not semimartingale or Markov process, FBM can describe more phenomena than Brownian motion. This is also the main reason why FBM can be used in the Internet, traffic, weather, and so on. In the past two decades, many scholars have extensively researched the stochastic equations driven by FBM [8–16]. According to [15], we only discuss the case where H∈1/2,1, and the stochastic integral in equation (1) is the pathwise R-S integral. In [17], Zähle used the fractional integration by parts formula to express the pathwise R-S integral in terms of fractional derivative operators. In [18], Nualart and Rǎşcanu 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. Moreover, Mishura and Shevchenko studied the Euler approximate solutions of stochastic differential equations in regard to FBM in [10]. Hu et al. [20] also gave the Euler scheme of stochastic differential equation with respect to FBM.Since FBM is not martingale, the classical Itô formula is invalid. Hashemi and Khodabin [8] used Hat functions to solve nonlinear SIVIEs driven by FBM, but the error analysis is insufficient. On the contrary, 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. In this paper, we employ BPFs to solve equation (1) and a high-precision numerical solution is acquired. In [10], authors prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by FBM with H>1/2 can be estimated by Oδ2H−1 (δ is the diameter of partition). Compared with [10], the convergence rate of the approximate solution obtained by this method may be faster when H⟶1. In addition, the calculation also is simpler.This article’s structure is as follows. In Section2, the definitions and related properties of FBM and BPFs are given. In Section 3, the effective numerical method is used to solve equation (1). The error analysis is acquired in Section 4. In Section 5, the validity and accuracy of this approach are validated by two examples. Section 6 is the conclusion.


## 2. Preliminaries

FBM and BPFs have been extensively studied by many scholars. For more contents, please refer to [3, 4, 14, 16].


### 2.1. Fractional Brownian Motion

FBM is a stochastic process (BuH,u≥0) with H∈1/2,1 on Ω,ℱ,P, and BuH is a continuous Gaussian process that has zero mean and the following covariance function:(2)CovBsH,BuH=12s2H+u2H−u−s2H.It satisfies the following properties:(i)
B0H=0 and EBuH=0, u≥0(ii)
BuH has homogeneous increments(iii)
EBuH2=u2H, u≥0(iv)
BuH has continuous trajectoriesIt is obvious thatBuH is a standard Brownian motion when H=1/2 in [4, 6].In this paper, the stochastic integral of equation (1) is the stochastic pathwise R-S integral. For any α∈1−H,1/2, the stochastic process f=fu,u∈0,T,T<∞, belongs to the space W0α,10,T that is the space of measurable functions f on 0,T such that(3)fα,1≔∫0ufssαds+∫0u∫0sfs−fys−yα+1dyds<∞,and then the integral exists (for the details, see [18]). And, the pathwise integral satisfies the following:(4)∫0ufsdBsH≤Λ∫0ufssαds+∫0u∫0sfs−fys−yα+1dyds,where(5)Λ=1Γ1−αsup0<s<u<TDu−1−αBu−sis a random variable. It is well known that Λ has finite moments of all orders (see [19]).


### 2.2. Block Pulse Functions

The definition of BPFs is as follows:(6)φiu=1,i−1h≤u<h,i=1,2,…,m,0,otherwise,where m is a positive integer and h=1/m and u∈0,1.The BPFs have the following basic properties:(i)
Disjointness:(7)φiuφju=δijφiu,i,j=1,2,…,m,whereδij is Kronecker delta.(ii)
Orthogonality:(8)∫01φiuφjudu=hδij,i,j=1,2,…,m.(iii)
Completeness: if the functionf is square integrable, then(9)∫01f2udu=limm⟶∞∑n=1mfn2φnu2,where fn=1/h∫01fuφnudu and the above equation is called Parseval’s identity.The vector form of BPFs is as follows:(10)Φu=φ1u,φ2u,…,φmuT,(11)ΦuΦTu=φ1u0⋯00φ2u⋯0⋮⋮⋱⋮00⋯φmum×m,ΦTuΦu=1,ΦuΦTuF=DFΦu,where F=f1,f2,…,fmT and DF denotes a diagonal matrix whose diagonal entries are the constant vector F.


### 2.3. Function Approximation

The square integrable functionxu on 0,1 can be expressed as(12)xu≃xmu=∑i=1mxiφu=XTΦu=ΦTuX,where(13)X=x1,x2,…,xmT.Letgs,u∈L20,1×0,1. It can also be approximated as(14)gs,u=ΦTsGΨu,where Φs and Ψs are m1 and m2 dimensional BPFs vectors, respectively, G=gijm1×m2,(15)gij=1h1h2∫01∫01gs,uφisψjudsdu,and h1=1/m1andh2=1/m2.


### 2.4. Related Lemmas

Lemma 1 (see [4]).
Suppose thatΦu is defined in (10); then, we have(16)∫0uΦsds≃PΦu,where(17)P=h2122⋯2012⋯2001⋯2⋮⋮⋮⋱⋮000⋯1m×m.Lemma 2 (see [12]).
The integral ofΦu with respect to FBM has the following approximation:(18)∫0uΦsdBsH≃PHΦu,where PH is the stochastic operational matrix with(19)PH=Bh/2HBhHBhH⋯BhH0B3h/2H−BhHB2hH−BhH⋯B2hH−BhH00B5h/2H−B2hH⋯B3hH−B2hH⋮⋮⋮⋱⋮000⋯B2m−1h/2H−Bm−1hHm×m.


## 2.1. Fractional Brownian Motion

FBM is a stochastic process (BuH,u≥0) with H∈1/2,1 on Ω,ℱ,P, and BuH is a continuous Gaussian process that has zero mean and the following covariance function:(2)CovBsH,BuH=12s2H+u2H−u−s2H.It satisfies the following properties:(i)
B0H=0 and EBuH=0, u≥0(ii)
BuH has homogeneous increments(iii)
EBuH2=u2H, u≥0(iv)
BuH has continuous trajectoriesIt is obvious thatBuH is a standard Brownian motion when H=1/2 in [4, 6].In this paper, the stochastic integral of equation (1) is the stochastic pathwise R-S integral. For any α∈1−H,1/2, the stochastic process f=fu,u∈0,T,T<∞, belongs to the space W0α,10,T that is the space of measurable functions f on 0,T such that(3)fα,1≔∫0ufssαds+∫0u∫0sfs−fys−yα+1dyds<∞,and then the integral exists (for the details, see [18]). And, the pathwise integral satisfies the following:(4)∫0ufsdBsH≤Λ∫0ufssαds+∫0u∫0sfs−fys−yα+1dyds,where(5)Λ=1Γ1−αsup0<s<u<TDu−1−αBu−sis a random variable. It is well known that Λ has finite moments of all orders (see [19]).


## 2.2. Block Pulse Functions

The definition of BPFs is as follows:(6)φiu=1,i−1h≤u<h,i=1,2,…,m,0,otherwise,where m is a positive integer and h=1/m and u∈0,1.The BPFs have the following basic properties:(i)
Disjointness:(7)φiuφju=δijφiu,i,j=1,2,…,m,whereδij is Kronecker delta.(ii)
Orthogonality:(8)∫01φiuφjudu=hδij,i,j=1,2,…,m.(iii)
Completeness: if the functionf is square integrable, then(9)∫01f2udu=limm⟶∞∑n=1mfn2φnu2,where fn=1/h∫01fuφnudu and the above equation is called Parseval’s identity.The vector form of BPFs is as follows:(10)Φu=φ1u,φ2u,…,φmuT,(11)ΦuΦTu=φ1u0⋯00φ2u⋯0⋮⋮⋱⋮00⋯φmum×m,ΦTuΦu=1,ΦuΦTuF=DFΦu,where F=f1,f2,…,fmT and DF denotes a diagonal matrix whose diagonal entries are the constant vector F.


## 2.3. Function Approximation

The square integrable functionxu on 0,1 can be expressed as(12)xu≃xmu=∑i=1mxiφu=XTΦu=ΦTuX,where(13)X=x1,x2,…,xmT.Letgs,u∈L20,1×0,1. It can also be approximated as(14)gs,u=ΦTsGΨu,where Φs and Ψs are m1 and m2 dimensional BPFs vectors, respectively, G=gijm1×m2,(15)gij=1h1h2∫01∫01gs,uφisψjudsdu,and h1=1/m1andh2=1/m2.


## 2.4. Related Lemmas

Lemma 1 (see [4]).
Suppose thatΦu is defined in (10); then, we have(16)∫0uΦsds≃PΦu,where(17)P=h2122⋯2012⋯2001⋯2⋮⋮⋮⋱⋮000⋯1m×m.Lemma 2 (see [12]).
The integral ofΦu with respect to FBM has the following approximation:(18)∫0uΦsdBsH≃PHΦu,where PH is the stochastic operational matrix with(19)PH=Bh/2HBhHBhH⋯BhH0B3h/2H−BhHB2hH−BhH⋯B2hH−BhH00B5h/2H−B2hH⋯B3hH−B2hH⋮⋮⋮⋱⋮000⋯B2m−1h/2H−Bm−1hHm×m.


## 3. Numerical Method

We solve equation (1) by BPFs in this section. To solve this integral equation, a lemma is given to deal with nonlinear analytic functions in the first place.Lemma 3 (see [6, 7]).
Letνu=∑ajuj and κu=∑bjuj be the analytic functions and j∈ℤ+; then,(20)νxmu=νTXΦu,κxmu=κTXΦu,where(21)νTX=νx1,νx2,…,νxm,κTX=κx1,κx2,…,κxm.On the basis of (12) and (14), xu,x0u,g1s,u, and g2s,u have the following approximate forms, respectively:(22)xu≃XTΦu=ΦTuX,x0u≃CTΦu=ΦTuC,g1s,u≃ΦTsG1Ψu=ΨTuG1TΦs,g2s,u≃ΦTsG2Ψu=ΨTuG2TΦs,where X and C are BPFs coefficients vectors and G1 and G2 are BPFs coefficients matrices similar to (15). For convenience, we put m1=m2=m. Substituting above equations into equation (1), we obtain(23)XTΦu≃CTΦu+νTX∫0uΦsΦTsdsG1Φu+κTX∫0uΦsΦTsdBsHG2Φu.Applying the operational matrixesP and PH for BPFs derived in (16) and (18), equation (23) can be rewritten as(24)XTΦu≃CTΦu+νTXM1Φu+κTXM2Φu,where M1 and M2 are m×m matrixes similar to [12]; see appendix, for details,(25)M1=h2g1112g1212g131⋯2g1m10g2212g231⋯2g2m100g331⋯2g3m1⋮⋮⋮⋱⋮000⋯gmm1m×m,M2=g112Bh/2Hg122BhHg132BhH⋯g1m2BhH0g222B3h/2H−BhHg232B2hH−BhH⋯g2m2B2hH−BhH00g332B5h/2H−B2hH⋯g3m2B3hH−B2hH⋮⋮⋮⋱⋮000⋯gmm2B2m−1h/2H−Bm−1hHm×m,with gij1 and gij2 are the same as (15).Then,(26)XT≃CT+νTXM1+κTXM2.


## 4. Error Analysis

The error analysis of the presented method is provided in this section. Firstly, two important lemmas are shown.Lemma 4 (see [3]).
Letes=fs−fms, where fms is approximations of bounded function fs by BPFs on 0,1. Then,(27)esL20,12=∫01e2sds≤Oh2.Lemma 5 (see [3]).
Letes,u=gs,u−gms,u, where gms,u is approximations of bounded function gs,u by BPFs on c=0,1×0,1. Then,(28)es,uL2c2=∫01∫01e2s,udsdu≤Oh2.Secondly, considereu=xu−xmu. We have(29)eu=xu−xmu=x0u−x0mu+∫0ug1s,uνxs−g1ms,uνxmsds+∫0ug2s,uκxs−g2ms,uκxmsdBsH,where xmu, x0mu, g1ms,u, and g2ms,u are approximation of xu, x0u, g1s,u, and g2s,u by BPFs, respectively.Then, for eachR>1, we define the following stopping time τR:(30)τR≔infu≥0:Λ≥R∧T.Finally, we give the main theorem and prove it.Theorem 1.
Assume that bounded analytic functionsνxu and κxu satisfy Lipschitz conditions:(I)
νq−νp≤l1q−pandκq−κp≤l3q−p(II)
νq≤l2andκq≤l4(III)
g1s,u≤l5andg2s,u≤l6
In (I)–(III),q,p∈ℝ, li>0,i=1,…,6 are constants. Then,(31)∫0TEeu∧τR2du=∫0TExu∧τR−xmu∧τR2du≤CROh2−4α,T∈0,1,where CR is a constant related to R.Proof.
According to (29), we obtain(32)Eeu∧τR2≤3Ex0u−x0mu2+E∫0u∧τRg1s,u∧τRνxs−g1ms,u∧τRνxmsds2+E∫0u∧τRg2s,u∧τRκxs−g2ms,u∧τRκxmsdBsH2≔3J1+3J2+3J3.
For the second item, using Lipschitz conditions and Cauchy–Schwarz inequality, we have(33)J2≤∫0uE|g1s,u∧τRνxs−νxms+νxmsg1s,u∧τR−g1ms,u∧τR|2ds≤2l12l52∫0uEes2ds+2l22∫0ug1s,u∧τR−g1ms,u∧τR2ds.
For the last item, by (4), we obtain(34)J3≤2EΛ∫0u|g2s,u∧τRκxs−g2ms,u∧τRκxmss−αds2+2EΛ∫0u∫0s|g2s,u∧τRκxs−g2mxsκxms−g2y,u∧τRκxy+g2my,u∧τRκxmys−y−α−1dyds2≔J31+J32.
Then,(35)J31≤2R2E∫0u|g2s,u∧τRκxs−κxms+κxmsg2s,u∧τR−g2ms,u∧τRs−αds2≤4R2l32l62∫0uEes2ds+4R2l42×∫0ug2s,u∧τR−g2ms,u∧τR2ds∫0us−2αds≤4R2l32l62C0∫0uEes2ds+4R2l42C0∫0ug2s,u∧τR−g2ms,u∧τR2ds,(36)J32≤2R2E∫0u∫0s|g2s,u∧τRκxs−κxms+μxmsg2s,u∧τR−g2ms,u∧τR−g2y,u∧τRκxy−κxmy−μxmyg2y,u∧τR−g2my,u∧τRs−y−α−1dyds2≤8R2l32l62E∫0u∫0usess−y−α−1dyds+∫0u∫ussess−y−α−1dyds2+8R2l42∫0u∫0usg2s,u∧τR−g2ms,u∧τRs−y−α−1dyds+∫0u∫ussg2s,u∧τR−g2ms,u∧τRs−y−α−1dyds2+8R2l32l62E∫0u∫uyueys−y−α−1dyds+∫0u∫yuyeys−y−α−1dyds2+8R2l42∫0u∫uyug2y,u∧τR−g2my,u∧τRs−y−α−1dyds+∫0u∫yuyg2y,u∧τR−g2my,u∧τRs−y−α−1dyds2≤8R2l32l62α−1+12∫0uEes2ds∫0us−us−2αds+8R2l42α−1+12∫0ug2s,u∧τR−g2ms,u∧τR2ds∫0us−us−2αds+8R2l32l62α−1+12E∫0ueyuy−y−α−u−y−αdy2+8R2l42α−1+12∫0ug2y,u∧τR−g2my,u∧τRuy−y−α−u−y−αdy2≤8R2l32l62C1h−2α∫0uEes2ds+8R2l42C1h−2α∫0ug2s,u∧τR−g2ms,u∧τR2ds+8R2l32l62C1h−2α∫0uEey2dy+8R2l42C1h−2α∫0ug2y,u∧τR−g2my,u∧τR2dy,where ςn=n/m=nh,n=0,…,m, us=maxςn:ςn≤s, uy=minςn:ςn≥y, and C0=1−2α−1 and C1=C0α−1+12.
Finally, by combining (33)–(36), we obtain(37)Eeu∧τR2≤3x0u−x0mu2+2l12l52∫0uEes2ds+2l22∫0ug1s,u∧τR−g1ms,u∧τR2ds+4R2l32l62C0∫0uEes2ds+4R2l42C0×∫0ug2s,u∧τR−g2ms,u∧τR2ds+8R2l32l62C1h−2α∫0uEes2ds+8R2l42C1h−2α×∫0ug2s,u∧τR−g2ms,u∧τR2ds+8R2l32l62C1h−2α∫0uEey2dy+8R2l42C1h−2α×∫0ug2y,u∧τR−g2my,u∧τR2dy.
Then,(38)Eeu∧τR2≤3x0u−x0mu2+2l22∫0ug1s,u∧τR−g1ms,u∧τR2ds+4R2l42C0+16R2l42C1h−2α×∫0ug2s,u∧τR−g2ms,u∧τR2ds+6l12l52+12R2l32l62C0+48R2l32l62C1h−2α×∫0uEes2ds,or(39)Eeu∧τR2≤σu+λ∫0uEes2ds,where(40)σu=3x0u−x0mu2+2l22∫0ug1s,u∧τR−g1ms,u∧τR2ds+4R2l42C0+16R2l42C1h−2α×∫0ug2s,u∧τR−g2ms,u∧τR2ds,λ=6l12l52+12R2l32l62C0+48R2l32l62C1h−2α.
LetVu∧τR=Eeu∧τR2; we obtain(41)Vu∧τR≤σu+λ∫0uVθdθ,θ∈0,u.
According to Gronwall’s inequality, we have(42)Vu∧τR≤σu+λ∫0ueλu∧τR−θσθdθ,u∈0,T.
Then,(43)∫0TVu∧τRdu=∫0TEeu∧τR2du≤∫0Tσu+λ∫0ueλu∧τR−θσθdθdu=∫0Tσudu+λ∫0T∫0ueλu∧τR−θσθdθdu≤∫0Tσudu+λeλT∫0T∫0uσθdθdu=3∫0Tx0u−x0mu2du+6l22∫0T∫0ug1s,u∧τR−g1ms,u∧τR2dsdu+12R2l42C0+48R2l42C1h−2α∫0T∫0ug2s,u∧τR−g2ms,u∧τR2dsdu+λeλT3∫0T∫0ux0θ−x0mθ2dθdu+6l22∫0T∫0u∫0θg1s,θ∧τR−g1ms,θ∧τR2dsdθdu+12R2l42C0+48R2l42C1h−2α∫0T∫0u∫0θg2s,θ∧τR−g2ms,θ∧τR2dsdθdu≔3N1+6l22N2+12R2l42C0+48R2l42C1h−2αN3+λeλT3N4+6l22N5+12R2l42C0+48R2l42C1h−2αN6.
By (27) and (28), we have(44)Ni≤wih2,i=1,…,6.
So,(45)∫0TEeu∧τR2du≤3w1+6l22w2+12R2l42C0+48R2l42C1h−2αw3+λeλT3w4+6l22w5+12R2l42C0w6+48R2l42C1h−2αw6h2≤CROh2−4α,where wi>0,i=1,…,6, are constants.
The proof is accomplished.


## 5. Numerical Example

We verify the effectiveness and precision of the above method by two numerical examples in this section.Example 1.
The following nonlinear SIVIE is discussed [8]:(46)xu=x0−2Hβ2∫0us2H−1xs1−x2sds+β∫0u1−x2sdBsH,u∈0,1,where xu=tanhβBuH+arctanhx0 is the exact solution.
Letβ=1/30 and x0=1/10. For different values of H, the error means XE, error standard deviations XS, and confidence intervals for error means of Example 1 with m=16 are given in Tables 1 and 2 , where XE and XS are obtained by 102 trajectories. Also, when m=16 and H=0.6, the simulation result of the approximate and precise solutions of this example are shown in Figure 1, and for m=32 and H=0.6, the results are exhibited in Figure 2.Table 1
Whenm=16 and H=0.6, the table exhibits numerical results of Example 1.
uXEXS95% confidence interval for error meanLowerUpper1/161.23683212E − 056.18416058E − 062.47366423E − 072.44892759E − 052/162.15162737E − 051.07581368E − 054.30325473E − 074.26022219E − 053/162.89145059E − 051.44572530E − 055.78290119E − 075.72507217E − 054/163.48716777E − 051.74358389E − 056.97433554E − 076.90459219E − 055/164.00955361E − 052.00477681E − 058.01910722E − 077.93891615E − 05Table 2
Whenm=16 and H=0.8, the table exhibits numerical results of Example 1.
uXEXS95% confidence interval for error meanLowerUpper1/163.87689012E − 061.93844506E − 067.75378025E − 087.67624244E − 062/168.79318940E − 064.39659470E − 061.75863788E − 071.74105150E − 053/161.29658037E − 056.48290186E − 062.59316074E − 072.56722914E − 054/161.86048116E − 059.30240580E − 063.72096232E − 073.68375270E − 055/162.14168523E − 051.07084262E − 054.28337047E − 074.24053677E − 05Figure 1
Simulation result of Example1 for H=0.6 and m=16.Figure 2
Simulation result of Example1 for H=0.6 and m=32.Example 2.
The following nonlinear SIVIE is discussed [8]:(47)xu=x0−Hβ2∫0utanhxss2H−1sech2xsds+β∫0tsechxsdBsH,u∈0,1,where xu=arcsinhβBuH+sinhx0 is the exact solution.
Letβ=1/30 and x0=1/10. Similarly, for different values of H, XE, and XS and confidence intervals for error means of Example 2 with m=16 are given in Tables 3 and 4 , where XE and XS are obtained by 102 trajectories. Also, when m=16 and H=0.8, simulation result of the approximate and precise solutions of this example is shown in Figure 3, and for m=32 and H=0.8, the results are shown in Figure 4.Table 3
Whenm=16 and H=0.6, the table exhibits numerical results of Example 2.
uXEXS95% confidence interval for error meanLowerUpper1/164.82207134E − 062.41103567E − 069.64414269E − 089.54770126E − 052/169.11303123E − 064.55651561E − 061.82260625E − 071.80438018E − 053/161.31576729E − 056.57883646E − 062.63153458E − 072.60521924E − 054/161.60910855E − 058.04554277E − 063.21821711E − 073.18603494E − 055/162.03276041E − 051.01638020E − 054.06552081E − 074.02486560E − 05Table 4
Whenm=16 and H=0.8, the table exhibits numerical results of Example 2.
uXEXS95% confidence interval for error meanLowerUpper1/163.08317271E − 061.54158636E − 066.16634543E − 086.10468197E − 062/166.35447091E − 063.17723546E − 061.27089418E − 071.25818524E − 053/167.04978090E − 063.52489045E − 061.40995618E − 071.39585662E − 054/168.48079420E − 064.24039710E − 061.69615884E − 071.67919725E − 055/161.14691035E − 055.73455173E − 062.29382069E − 072.27088249E − 05Figure 3
Simulation result of Example2 for H=0.8 and m=16.Figure 4
Simulation result of Example2 for H=0.8 and m=32.


## 6. Conclusion

For nonlinear SIVIEs driven by FBM, this paper presents a valid method to solve them by BPFs. The simulation results show that the approximate solution of this method is very close to the exact solution. In Section4, error analysis is given, and the convergence rate of the approximate solution is faster than the method of [21] when H⟶1.



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