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 . 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.
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:where and are kernel functions, for . is an unknown stochastic process, is FBM with , and the stochastic integral is pathwise Riemann–Stieltjes (R-S) integral. They are all defined on the same probability space. and 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 , 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 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 can be estimated by ( is the diameter of partition). Compared with [10], the convergence rate of the approximate solution obtained by this method may be faster when . In addition, the calculation also is simpler.
This article’s structure is as follows. In Section 2, 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 () with on , and is a continuous Gaussian process that has zero mean and the following covariance function:
It satisfies the following properties:(i) and , (ii) has homogeneous increments(iii), (iv) has continuous trajectories
It is obvious that is a standard Brownian motion when in [4, 6].
In this paper, the stochastic integral of equation (1) is the stochastic pathwise R-S integral. For any , the stochastic process , belongs to the space that is the space of measurable functions on such thatand then the integral exists (for the details, see [18]). And, the pathwise integral satisfies the following:whereis a random variable. It is well known that has finite moments of all orders (see [19]).
2.2. Block Pulse Functions
The definition of BPFs is as follows:where is a positive integer and and .
The BPFs have the following basic properties:(i)Disjointness: where is Kronecker delta.(ii)Orthogonality:(iii)Completeness: if the function is square integrable, thenwhere and the above equation is called Parseval’s identity.
The vector form of BPFs is as follows:where and denotes a diagonal matrix whose diagonal entries are the constant vector .
2.3. Function Approximation
The square integrable function on can be expressed aswhere
Let . It can also be approximated aswhere and are and dimensional BPFs vectors, respectively, ,and .
2.4. Related Lemmas
Lemma 1 (see [4]). Suppose that is defined in (10); then, we havewhere
Lemma 2 (see [12]). The integral of with respect to FBM has the following approximation:where is the stochastic operational matrix with
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]). Let and be the analytic functions and ; then,where
On the basis of (12) and (14), , and have the following approximate forms, respectively:where and are BPFs coefficients vectors and and are BPFs coefficients matrices similar to (15). For convenience, we put . Substituting above equations into equation (1), we obtain
Applying the operational matrixes and for BPFs derived in (16) and (18), equation (23) can be rewritten aswhere and are matrixes similar to [12]; see appendix, for details,with and are the same as (15).
Then,
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]). Let , where is approximations of bounded function by BPFs on . Then,
Lemma 5 (see [3]). Let , where is approximations of bounded function by BPFs on . Then,
Secondly, consider . We havewhere , , , and are approximation of , , , and by BPFs, respectively.
Then, for each , we define the following stopping time :
Finally, we give the main theorem and prove it.
Theorem 1. Assume that bounded analytic functions and satisfy Lipschitz conditions:(I)(II)(III)In (I)–(III), , are constants. Then,where is a constant related to .
Proof. According to (29), we obtainFor the second item, using Lipschitz conditions and Cauchy–Schwarz inequality, we haveFor the last item, by (4), we obtainThen,where , , , and and .
Finally, by combining (33)–(36), we obtainThen,orwhereLet ; we obtainAccording to Gronwall’s inequality, we haveThen,By (27) and (28), we haveSo,where , 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]:where is the exact solution.
Let and . For different values of , the error means , error standard deviations , and confidence intervals for error means of Example 1 with are given in Tables 1 and 2 , where and are obtained by trajectories. Also, when and , the simulation result of the approximate and precise solutions of this example are shown in Figure 1, and for and , the results are exhibited in Figure 2.
Example 2. The following nonlinear SIVIE is discussed [8]:where is the exact solution.
Let and . Similarly, for different values of , , and and confidence intervals for error means of Example 2 with are given in Tables 3 and 4 , where and are obtained by trajectories. Also, when and , simulation result of the approximate and precise solutions of this example is shown in Figure 3, and for and , the results are shown in Figure 4.
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 Section 4, error analysis is given, and the convergence rate of the approximate solution is faster than the method of [21] when .
Appendix
In this section, we give detailed proofs of and .
Let and be the ith row of the matrix and , respectively, and be the ith row of the integral operation matrix and the stochastic integral matrix , respectively, and and be a diagonal matrix with and as the diagonal elements, respectively. By (16) and (18), we obtain
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors state that there are no conflicts of interest in the publication of this paper.
Acknowledgments
This article was funded by the NSF (Grant 2016CFB526) of Hubei Province.