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Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation
An efficient method to determine a numerical solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (FBM) with Hurst parameter and independent one-dimensional standard Brownian motion (SBM) is proposed. The method is stated via a stochastic operational matrix based on the block pulse functions (BPFs). With using this approach, the SDE is reduced to a stochastic linear system of equations and unknowns. Then, the error analysis is demonstrated by some theorems and defnitions. Finally, the numerical examples demonstrate applicability and accuracy of this method.
In many fields of science and engineering, there are a large number of problems which are intrinsically involving stochastic excitations of a Gaussian white noise type. Having in mind a Gaussian white noise mathematically described as a formal derivative of a Brownian motion process, all such problems are mathematically modeled by stochastic differential equations. Most of them cannot be solved analytically, so it is important to provide their numerical solutions. There has been a growing interest in numerical solutions of stochastic differential equations for the last years [1–10].
In the presented work, we consider SDE as follows: or where denotes the FBM with Hurst parameter on probability space and is independent one-dimensional SBM defined on the same probability space. Also, and is the stochastic process of unknown on the probability space.
Investigations concerning the SDE driven by the FBM have been done by Zähle , Coutin , Decreusefond and Üstünel , Nualart [4, 14], Lisei and Soós , and other authors. Also, there exist several ways for solving it, pathwise and related techniques, Dirichlet forms, Euler approximations, Malliavin calculus, and Skorohod integral [1, 4, 15–17]; almost all methods have very poor numerical convergence.
It is important to find approximate solutions of the stochastic equations driven by the FBM, since these equations cannot be solved analytically in most cases and have many applications in models arising in physics, telecommunication networks, and finance . Also, we cannot use from the classical Ito theory for their stochastic calculus, since these processes are not Markovian and semimartingale. Hence, in this work, we implement the stochastic operational matrix based on the BPFs for solving (2). The benefits of this method are lower cost of setting up the system of equations; moreover, the computational cost of operations is low. Also, convergence of this method is faster than other methods. These advantages make the method easier to apply.
The rest of the paper is organized as follows. In Section 2, some essential definitions and the following assumptions on the coefficients of (2) are stated. Also, the necessary properties of the block pulse functions (BPFs) are introduced. In Section 3, first a theorem is proved; then (2) is reduced to a stochastic linear system by using the properties of the BPFs. In Section 4, the error analysis is demonstrated. Efficiency of this method and good reasonable degree of accuracy are confirmed by some numerical examples, in Section 5. Finally, in Section 6, a brief conclusion is given.
Definition 1. Let be the step function and denotes the characteristic function on , , and . Then, the wiener integral with respect to the FBM is defined as where and (see ).
Definition 2. Let denote the class of function on such that (1)the function is measurable;(2)the function is adapted to ;(3) and .
Let us consider the following assumptions on the coefficients.) is differentiable in and there exist constants and such that ()There exist constants such that ()There exist constants such that for all .
Theorem 3. Let , and hold in condition , , , and . Then, there exists a unique solution for (2).
Proof. See .
A function is approximated by using properties of the BPFs as where with where denotes the BPFs and with
A function is approximated as follows: where
In , it is proved that where
In , it is proved that where
3. Solving the SDE Driven by FBM and Independent One-Dimensional SBM
Theorem 4. Let denote the BPFs, , and , ; then where
Proof. First, we compute stochastic operational matrix driven by the FBM based on the BPFs as follows.(A1)If , then
(A2)If , the function is defined as
where denotes the characteristic function and , where if and if . Also,
Now, for computation (), we can write
Then by using Definition 1, we obtain
(A3)If , then
where , if , if , and
For computation (), we can write
so, we get
From (A1), (A2), and (A3), we get
Furthermore, we suppose that
so, we can write
Hence, by using the relation (33), we can write
Now, let be the th row of matrix , let be the ith row of the matrix , and let be the ith row of matrix . We have where is given by (21).
Let where and are the block pulse coefficients vector and , , and , are the block pulse coefficients matrix.
By substituting the relation (37) in (2), we get or Therefore, by using properties of the BPFs and Theorem 4, we can write where withwithwith Now, with replacing by , we have or where . Clearly, (46) is the stochastic linear system of equations and unknowns.
4. Error Analysis
In , it is stated that if and , then
Theorem 5. Let be an arbitrary bounded function on and such that is the BPFs of . Then,
Proof. See .
Theorem 6. Let be an arbitrary bounded function on and such that is the BPFs of . Then,
Proof. See .
Let where is the approximate solution of defined in (46) and , , , and are approximated by using properties of the BPFs.
Proof. Consider by using , we can write First, by using the relation (47), we can write Cleary, we have and consequently, Hence, or Now, by using the property of the Ito isometry for the SBM defined in  and , we get By using Theorems 5 and 6, we can write By substituting the relation (60) in (59), we get or where and . If , we get Now, by using Gronwall inequality, we have or
5. Numerical Examples
The SDE driven by the FBM is applied in modeling the price of a stock with various Hurst parameters (see ). Hence, we show applicability and accuracy of this method in two numerical examples.
Example 1. Let us consider a SDE with the exact solution . The numerical results have been shown in Tables 1, 2, and 3 (with various Hurst parameters), where and are error mean and standard deviation of error, respectively.
Example 2. Let us consider a SDE with the exact solution . The numerical results have been shown in Tables 4, 5, and 6 (with various Hurst parameters), where and are error mean and standard deviation of error, respectively.
This paper presents a numerical comparison between the approximation solution of the SDE driven by the FBM with Hurst parameter and independent one-dimensional SBM and the exact solution of it. Also, the method is applied with two examples to illustrate the accuracy and implementation of the method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Islamic Azad University for supporting this work. The authors are also grateful to the anonymous referee for his/her constructive comments and suggestions.
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