A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations

In this paper, the stochastic operational matrix of Itô-integration for the Chebyshev wavelets is applied for solving stochastic Volterra-Fredholm integral equations. The main characteristic of the presented method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Chebyshev wavelets basis is considered. The efficiency and accuracy of the proposed method was demonstrated by some non-trivial examples and comparison with the other existing methods.


Introduction
Stochastic analysis has been an interesting research area in mathematics, fluid mechanics, geophysics, biology, chemistry, epidemiology, microelectronics, theoretical physics, economics, and finance. The behavior of dynamical systems in these fields are often dependent on a noise source and a Gaussian white noise, governed by certain probability laws. This noise might be either due to thermal fluctuations, noise in somecontrol parameter, coarse-graining of a high-dimensional deterministic system with random initial conditions or the stochastic parameterization of small scales. The dynamical systems subject to noise can be modeled accurately using stochastic differential equations, stochastic integral equations, stochastic integro-differential equations or in more complicated cases stochastic partial differential equation [1,2,3,4,5,6,7,8].
Since in many cases it is difficult to derive an explicit form of the solution of stochastic functional equations numerical approximation becomes a practical way to face this problem. Recently, many studies have been appeared which describe numerical solution of stochastic differential and integral equations [1,4,7,8,9,3,10,11,12,13].
As a powerful tool, wavelets have been extensively used in signal processing, numerical analysis, and many other areas. Wavelets permit the accurate representation of a variety of functions and operators [14,15,17,18,19,20,21,16]. In this paper, an stochastic operational matrix for the Chebyshev wavelets is derived. Then application of this stochastic operational matrix in solving stochastic Volterra-Fredholm integral equations is investigated. Some non-trivial examples are included to demonstrate the efficiency and accuracy of the proposed method. Also to verify the proposed method, numerical results are compared with the block pulse functions (BPFs) methed presented in [8].
This paper is organized as follows: In section 2 some basic definition and preliminaries about stochastic process and Itô integral are presented. The Chebyshev wavelets and their properties are introduced in section 3. In section 4 stochastic operational matrix of the Chebyshev wavelets is derived. In section 5 application of this stochastic operational matrix in solving stochastic Voltera-Fredholm integral equations are described. In section 6 the efficiency of the proposed method is demonstrated by some non-trivial examples. Finally, a conclusion is given in section 7.

Preliminaries
In this section we review some basic definition of the stochastic calculus and the block pulse functions (BPFs).

Stochastic calculus
Definition 2.1. (Brownian motion process) A real-valued stochastic process B(t), t ∈ [0, T ] is called Brownian motion, if it satisfies the following properties (i) The process has independent increments for 0 has Normal distribution with mean 0 and variance h, (iii) The function t → B(t) is continuous functions of t.
where B denotes the Borel algebra on [0, ∞) and F is the σ -algebra on Ω.
(ii) f is adapted to F t , where F t is the σ -algebra generated by the random variables B(s), s ≤ t. where, ϕ n is a sequence of elementary functions such that For more details about stochastic calculus and integration please see [2].

Block pulse functions
BPFs have been studied by many authors and applied for solving different problems. In this section we recall definition and some properties of the block pulse functions [7,8,24]. The m-set of BPFs are defined as in which t ∈ [0, T ), i = 1, 2, ..., m and h = T m . The set of BPFs are disjointed with each other in the interval [0, T ) and where δ ij is the Kronecker delta. The set of BPFs defined in the interval [0, T ) are orthogonal with each other, that is If m → ∞ the set of BPFs is a complete basis for L 2 [0, T ), so an arbitrary real bounded function f (t), which is square integrable in the interval [0, T ), can be expanded into a block pulse series as where Rewritting Eq. (4) in the vector form we have in which Morever, any two dimensional function k(s, t) ∈ L 2 ([0, can be expanded with respect to BPFs such as where Φ(t) is the m-dimensional BPFs vectors respectively, and Π is the m × m BPFs coefficient matrix with and h 1 = T1 m and h 2 = T2 m . Let Φ(t) be the BPFs vector, then we have and For an m-vector F we have whereF is an m × m matrix, andF = diag(F ). Also, it is easy to show that for an m × m matrix A

Chebyshev wavelets
Wavelets constitute a family of functions constructed from dilation and translation of a single function ψ called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets The Chebyshev wavelets ψ nm (x) = ψ(k, n, m, x) are defined on the interval [0, 1) by wherẽ  [19,20]. By using the orthonormality of the Chebyshev wavelets, any function f (t) over [0, 1); square-integrable with respect to the measure w(t)dt; with w(t) = w nk (t); for n 2 k ≤ t ≤ n+1 2 k ; and w nk (t) = w(2 k+1 t − 2n + 1); being w(t) = 1 √ 1−t 2 can be expanded in terms of the Chebyshev wavelets as where c mn = (f (t), ψ mn (t)) w nk and (., .) w nk denotes the inner product on L 2 w nk [0, 1]. If the infinite series in (16) is truncated, then it can be written as where C and Ψ(t) arem = 2 k M column vectors given by By changing indices in the vectors Ψ(t) and C the series (4) can be rewritten as where where K = [k ij ] is anm ×m matrix and k ij = ψ i (s), (k(s, t), ψ j (t)) w nk w nk .

Chebyshev wavelets and BPFs
In this section we will review the relation between the Chebyshev wavelets and BPFs. It is worth mention that here we set T = 1 in definition of BPFs.
Theorem 3.1. Let Ψ(t) and Φ(t) be them-dimensional Chebyshev wavelets and BPFs vector respectively, the vector Ψ(t) can be expanded by BPFs vector Φ(t) as where Q is anm ×m block matrix and Proof. Let φ i (t), i = 1, 2, ...,m be the i-th element of Chebyshev wavelets vector. Expanding φ i (t) into anm-term vector of BPFs, we have where Q i is the i-th row and Q ij is the (i, j)-th element of matrix Q. By using the orthogonality of BPFs we have by using mean value theorem for integrals in the last equation we can write now by choosing η j = 2j−1 2m we have and this prove the desired result.
The following Remark is the consequence of relations (12), (13) and Theorem 3.1.

Remark 3.2. For anm-vector F we have
in whichF is anm ×m matrix as whereF = diag Q T F . Moreover, it can be easy to show that for anm ×m matrix A whereÂ T = U Q −1 and U = diag(Q T AQ) is am-vector.

Convergence analysis
Here we investigate the convergence and error analysis of the Chebyshev wavelets basis. ∞ m=0 c mn ψ mn (x) be its infinite Chebyshev wavelets expansion, then this means the Chebyshev wavelets series converges uniformly to f (x) and Proof. See [23]. where Proof. We have now by considering the relation (29) the desired result is achieved.

Stochastic operational matrix of Chebyshev wavelets
In this section we derive an stochastic operational matrix for Chebyshev wavelets. For this purpose we first remind some useful results for BPFs [7,8].
Lemma 4.1. [7] Let Φ(t) be them-dimensional BPFs vector defined in (7), then integration of this vector can be derived as where P is called the operational matrix of integration for BPFs and is given by  (7), the Itô integral of this vector can be derived as where P s is called the stochastic operational matrix of BPFs and is given by Now we are ready to derive a new operational matrix of stochastic integration for the Chebyshev wavelets basis. For this end we use BPFs and the matrix Q introduced in (20). (17), the integral of this vector can be derived as

Theorem 4.3. Suppose Ψ(t) be them-dimensional Chebyshev wavelets vector defined in
where Q is introduced in (20) and P is the operational matrix of integration for BPFs derived in (33).
and this complete the proof.
Theorem 4.4. Suppose Ψ(t) be them-dimensional Chebyshev wavelets vector defined in (17), the Itô integral of this vector can be derived as where Λ s is called stochastic operational matrix for Chebyshev wavelets, Q is introduced in (20) and P s is the stochastic operational matrix of integration for BPFs derived in (35).
Proof. Let Ψ(t) be the Chebyshev wavelets vector, by using Theorem 3.1 and Lemma 4.2 we have now Theorem 3.1 result and this complete the proof.

Numerical solution of stochastic Voltera-Fredholm integral equation
In this section, we use the stochastic operational matrix of Chebyshev wavelets for solving stochastic Voltera-Fredholm integral equations. In this way, consider the following stochastic Voltera-Fredholm integral equation where X(t), f (t) and k i (s, t), i = 1, 2, 3 are the stochastic processes defined on the same probability space (Ω, F, P ), and X(t) is unknown. Also B(t) is a Brownian motion process and t 0 k 3 (s, t)X(s)dB(s) are the Itô integral. For sake of simplicity and without loss of generality we set (α, β) = (0, 1). Now, we approximate X(t), f (t) and k i (s, t), i = 1, 2, 3 in term ofm-dimensional Chebyshev wavelets as follows where X and F are Chebyshev wavelets coefficients vector, and K i , i = 1, 2, 3 are Chebyshev wavelets coefficient matrices defined in Eq. (17) and Eq. (19). Substituting above approximations in Eq. (42), we have using relation 1 0 Ψ(s)Ψ T (s)ds = Im ×m and Remark 3.2 we get whereX is anm ×m matrix. Now applying the operational matrices Λ and Λ s for Haar wavelets derived in Eqs.
(36) and (39) we have by setting Y 2 = K T 2X Λ, Y 3 = K T 3X Λ s and using Remark 3.2 we derive in whichŶ 2 andŶ 3 arem×m matrix and they are linear function of vector X. This equation is hold for all t ∈ [0, 1), so we can write SinceŶ 2 andŶ 3 are linear function of X, Eq. (48) is a linear system for unknown vector X. Solving this linear system and determining X, we can approximate solution of stochastic Voltera-Fredholm integral equation (42) by substituting obtained vector X in Eq. (44).

Numerical examples
Here we demonstrate the efficiency and accuracy of the Chebyshev wavelets method (CWM) by some non-trivial examples. All algorithms are performed by Maple 17 with 20 digits precision.
and X(t) is an unknown stochastic process defined on the probability space (Ω, , P ) and B(t) is a Brownian motion process. The proposed method in Section 5 are used for solving this stochastic Volterra-Fredholm integral equation. Fig. 1 presents the approximate solution computed by CWM form = 64. The numerical results derived by the CWM and BPFs method [8] are shown in Table 1.   and X(t) is an unknown stochastic process defined on the probability space (Ω, , P ) and B(t) is a Brownian motion process. The stochastic operational matrix of Chebyshev wavelets is employed for deriving numerical solution of this Volterra-Fredholm integral equation. Fig. 2 presents the approximate solution computed by the CWM form = 64. Table 2 shows the numerical results given by the CWM and BPFs method [8]. The numerical results reveal the efficiency of the proposed method.

Conclusion
A computational method based on the Chebyshev wavelets and their Itô-integration operational matrix is proposed for solving stochastic Volterra-Fredholm integral equations. The main advantage of the proposed method is that it transforms stochastic Volterra-Fredholm integral equations into linear systems of algebraic equations which can be simply solved. Convergence and error analysis of the Chebyshev wavelets is investigated. The efficiency and accuracy of this method is shown by comparison with other existing methods on some non-trivial examples.