Abstract
We present a new technique for solving numerically stochastic Volterra integral equation based on modified block pulse functions. It declares that the rate of convergence of the presented method is faster than the method based on block pulse functions. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.
1. Introduction
The numerical study and simulation of stochastic Volterra integral equations (SVIEs) have been an active field of research for the past years [1–7]. Most SVIEs do not have analytic solutions and hence it is of great importance to provide numerical schemes. Numerical schemes to stochastic differential equations (SDEs) have been well developed [8–12]. However, there are still few papers discussing the numerical solutions for stochastic Volterra integral equations.
Study in economics, sociology, and various biological and medical models leads to the stochastic Volterra integral equations. These systems are dependent on a noise source, on a Gaussian white noise, so that modeling such phenomena naturally requires the use of various stochastic Volterra integral equations.
In this paper, we consider the linear stochastic Volterra integral equation: where , , , and , for , are the stochastic processes defined on the same probability space with a filtration that is increasing and right continuous and contains all -null sets. is unknown random function and is a standard Brownian motion defined on the probability space and is the Itô integral. Numerous papers have been focusing on existence solution of (1) [13–15].
The paper [3] solves stochastic Volterra integral equations by block pulse functions (BPFs) and [4] applies this method for solving -dimensional stochastic Itô Volterra integral equations. However, BPFs are very common in use; it seems that their convergence is weak. Maleknejad and Rahimi apply in [16] Modified Block Pulse Functions (MBPFs) to solve Volterra integral equation of the first kind numerically. Here, we use this method for solving SVIEs.
This paper is organized as follows. In the rest of this section we describe some general concepts concerning the block pulse functions and epsilon modified block pulse functions and some concepts related to stochastic and Itô integral. Section 2 is devoted to stochastic integration operational matrix. In Section 3, the method is employed to solve stochastic integral equations. Section 4 discusses error analysis of this method. Section 5 gives numerical example. Finally, Section 6 provides the conclusion of this work.
1.1. Block Pulse Functions
BPFs have been variously studied [16–18] and applied for solving different problems. The goal of this section is to recall notations and definition of the BPFs that are used in the next sections.
The block pulse functions are defined on the time interval by where and for convenience we put .
The block pulse functions on have the following properties:(1)disjointness: for where is Kronecker delta;(2)orthogonality: (3)completeness: for every when approaches infinity, Parseval’s identity holds: where Also the Fourier coefficients and the block pulse functions depend on . The set of block pulse functions may be written as a vector of dimension : From the above representation and disjointness property, it follows that where is an -dimensional vector and . Let be an matrix so that where is a vector with elements equal to the diagonal entries of .
The expansion of a function over with respect to , , is given by where and is defined by (6).
Let . It is expanded with respect to BPFs as where and are - and -dimensional BPFs vectors, respectively, and is the block pulse coefficient matrix with the below : For convenience, we put .
Now, integration operational matrix is considered and computed: Since is equal to at midpoint of , we can approximate , for by . Therefore where the th component is . As a result where is operational matrix of integration that is given by So,
1.2. Epsilon Modified Block Pulse Functions (EMBPFs)
A set of epsilon modified block pulse functions , , on the interval are defined as for , and
Similar to BPFs, the most important properties of EMBPFs are(1)disjointness: where ;(2)orthogonality: if we put , (3)completeness: where and is length of interval .
With defining , we have Similar to BPFs, where the operational matrix of EMBPFs is given by and we have the following approximation:
1.3. Stochastic Concepts of Itô Integral
Definition 1 (Brownian motion process). A real-valued stochastic process is called Brownian motion, if it satisfies the following properties:(i)independence of increments: , , is independent of the past, that is, of , or of , the -field generated by ;(ii)normal increments: has normal distribution with mean 0 and variance ;(iii)continuity of paths: , are continuous functions of .
Definition 2. Let be an increasing family of -algebras of subsets of . A process from to is called -adapted if for each the function is -measurable [19].
Definition 3 (see [19]). Let be the class of functions such that(i) is -measurable, where denotes the Borel -algebra on and is the -algebra on ;(ii) is -adapted, where is the -algebra generated by the random variables , ;(iii).
Definition 4 (the Itô integral, [19]). Let ; then the Itô integral of (from to ) is defined by where is a sequence of elementary functions such that
Theorem 5 (the Itô isometry). Let ; then
Proof. See [19].
Definition 6 (1-dimensional Itô processes, [19]). Let be 1-dimensional Brownian motion on . A 1-dimensional Itô process (stochastic integral) is a stochastic process on of the form or where
Theorem 7 (the 1-dimensional Itô formula). Let be an Itô process given by (1) and ; then is again an Itô process, and where is computed according to the rules
Proof. See [19].
Moreover, is notation of
Lemma 8 (the Gronwall inequality). Let be integral with for , where . Then
For more details see [19, 20].
2. Stochastic Integral Operational Matrix for EMBPFs
In this section stochastic integral operational matrix for EMBPFs is considered. For finding vector form of , with EMBPFs, the Itô integral of each single EMBPF can be computed as follows. It is clear that the integrals are stochastic and nondeterministic: for , and We approximate(1), by , at midpoint of ;(2) by in at midpoint of ;(3) by in , at midpoint of .
As a result, vector form of , with EMBPFs, is given by in which the th component is , Therefore where stochastic operational matrix of integration is given bySo, the Itô integral of every function can be approximated as follows:
3. Numerical Solution of SVIEs by EMBPFs
Here, we modify the method that has been used in [16] by EMBPFs. In the below equation: we approximate functions , , , and by EMBPFs: where the vectors , and matrices , are EMBPFs coefficient of , , , and , respectively.
Substituting (48) into (47) and using previous relations, Finally whereThen With replacing by , we have a linear system of equations.
Now if , there will be numerical answers . Solution is approximated by
4. Error Analysis
In this section, error analysis is studied. In the following theorems, for simplicity, we assume and .
Theorem 9. If and , then(1) achieves its minimum value;(2) approach pointwise;(3).
Proof. See [16].
Theorem 10. Assume the following.(1) is continuous and differentiable in , with bounded derivative; that is, .(2), are correspondingly BPFs. MBPFs, MBPFs expansions of base on EMBPFs over interval .(3).
Then
Proof. Trapezoidal rule for integral is
where is error of integration. Suppose and . The representation error when is represented by a series of BPFs over every subinterval , is
where
From (55),
It is obvious that if , then .
So, this error is computed for in interval , .
For this function , so
Then this error with BPFs is .
Similarly, the error when is represented in a series of EMBPFs over every subinterval is
So, the error with EMBPFs is .
For in we have
So, the error is also for .
Now,
We define the representation error between and its 2D-EMBPFs expansion, , over every subregion , is defined as
where
With Taylor’s expansion and similarity to the above discussion,
Theorem 11. Assume that(1),(2).
Then
Proof. Consider
So,
by the Cauchy-Schwartz inequality, Itô isometry formula, and the linearity of Itô integrals in their integrands.
The first term is satisfied by last theorem:
Now,
Furthermore,
Hence
Then by Gronwall’s inequality, we get
5. Numerical Example
In this section, we present an example for showing the features of the EMBPFs method in this paper. Let denote the EMBP coefficient of exact solution of the given example and let be the EMBP coefficient of computed solution by the presented method. In this example error is defined as
Example 1 (see [3]). Consider the following linear stochastic Volterra integral equation:
with the exact solution , for .
The numerical results are shown in Tables 1 and 2. In the tables, is the number of iterations, is error mean, and is standard deviation of error.
Table 3 is from [3] for comparison.
In some examples by applying BPFs when increases, accuracy decreases, but in EMBPFs we achieve good accuracy by increasing .
6. Conclusion
As some SVIEs cannot be solved analytically, in this paper we present a new technique for solving SVIEs numerically. Here, we consider a modification of the block pulse functions. Some theorems show that if EMBPFs are used for achieving numerical expansions with times more precision, there is no need to increase the number of BPFs, times, which leads to solving a system of equations with times more equations and unknowns. But the results of BPFs solution can be combined with solutions of systems of equations with one more unknown and nearly achieve times more precision. Parallel programming is so useful for this method. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.