Numerical Approximation of Stochastic Volterra-Fredholm Integral Equation using Walsh Function

In this paper, a computational method is developed to find an approximate solution of the stochastic Volterra-Fredholm integral equation using the Walsh function approximation and its operational matrix. Moreover, convergence and error analysis of the method is carried out to strengthen the validity of the method. Furthermore, the method is numerically compared to the block pulse function method and the Haar wavelet method for some non-trivial examples.


Introduction
Stochastic differential equations (SDE) are widely used in a variety of fields, including the physical sciences, biological sciences, agricultural sciences, and financial mathematics, which includes option pricing, where stochastic Volterra-Fredholm integral equation (SVFIE) plays a crucial role [2][3][4].As with other differential equations, it is practically impossible to find the solution to many SDE, and the problem becomes more complex in the case of SVFIE.Therefore, numerical approximation method becomes crucial for finding the solutions to the problems.Numerous SVFIEs are determined approximately using a variety of numerical techniques.In recent decades, orthogonal functions such as block pulse function (BPF), Haar wavelet, Legendre polynomials, Laguerre polynomials, and Chebyshev's polynomials have been utilised to approximate the solution of SVFIE.
The Walsh functions form an orthonormal system that only accepts the values 1 and −1.As a result, a number of mathematicians consider the Walsh system to be an artificial orthonormal system, which was introduced in 1923 citeWalsh and has numerous applications in digital technology.Walsh functions have a significant advantage over traditional trigonometric functions because a computer can determine the exact value of any Walsh function at any given time with high accuracy.Chen and Hsiao used the Walsh function to solve the variational problem in 1997, is cited in [8].They used the same concept to solve the integral equation [9] in 1979.The technique's key property is that it converts the problem into a system of algebraic equations, which are then solved to yield an approximation of the solution.In this paper, we use the Walsh function to approximate the solution x(t) of the following linear SVFIE where x(t), f (t), k(s, t), k 1 (s, t) and k 2 (s, t) for s, t ∈ [0, T ), represent the stochastic processes primarily based on the identical probability space (Ω, F, P ) and x(t) is unknown.In addition, B(t) represents Brownian motion [2,3], and t 0 k 2 (s, t)x(s)dB(s) represents the Itô integral.In the majority of previous works, the evaluation is predicated on the assumption that the derivativesf ′ (t), ∂ 2 k ∂s∂t , ∂ 2 k i ∂s∂t for i = 1, 2, exists and bounded.By converting BPF approximation to Walsh function approximation in this paper, we expect only Lipschitz continuity of the functions f (t), k(s, t), k 1 (s, t) and k 2 (s, t) to have the same rate of convergence, which allows us to consider the general form of SVFIE to be integrated.In the final portion, the method is compared to similar techniques [13,14] that approximate the solution of the SVFIE using block pulse function and Haar wavelet.
The first m Walsh functions for m ∈ N can be written as an m-vector by T , t ∈ [0, 1).The Walsh functions satisfy the following properties.

Orthonormality
The set of Walsh functions is orthonormal.i.e., w i (t)w j (t)dt = 1 i=j, 0 otherwise.

Walsh Function Approximation
Any real-valued function f (t) ∈ L 2 ([0, 1)) can be approximated as where, The matrix form is given by where Here, T W = [w i (η j )] is called as the Walsh operational matrix where η j ∈ [jh, (j + 1)h).
The BPFs in vector form satisfy T is the m-vector with elements equal to the diagonal entries of A. The integration of BPF vector Φ(t), t ∈ [0, 1) can be performed by [1] Hence, the integral of every function f (t) ∈ L 2 [0, 1) can be approximated as The Itô integral of BPF vector Φ(t), t ∈ [0, 1) can be performed by [12] t 0 Hence, the Itô integral of every function f (t) ∈ L 2 [0, 1) can be approximated as The following theorem describes a relationship between the Walsh function and the block pulse function.Theorem 3.2.[7] Let the m-set of Walsh function and BPF vectors be W (t) and Φ(t) respectively.Then the BPF vectors Φ(t) can be used to approximate W (t) as W (t) = T W Φ(t), m = 2 k , and k = 0, 1, . .., where T W = c ij m×m , c ij = w i (η j ), for some η j = j m , j+1 m and i, j = 0, 1, 2, . . .m − 1.
One can see that [10] T W T T W = mI and

Numerical Solution of Stochastic Volterra-Fredholm Integral Equation
We consider following linear Stochastic Volterra-Fredholm Integral equation(LSVFIE) where x(t), f (t), k(s, t), k 1 (s, t) and k 2 (s, t) for s, t ∈ [0, T ), are the stochastic processes defined on the same probability space (Ω, F, P ) and x(t) is unknown.Also B(t) is Brownian motion and t 0 k 2 (s, t)x(s)dB(s) is the Ito Integral.Using equation ( 2) and( 3) in (6) we have Now Similarly, Substituting ( 8) and ( 9) in ( 7) and using the condition of orthonormality, we get where and Ĥi is the m-vector with elements equal to the diagonal elements of can be solved to obtain a non trivial solution of the given Stochastic Volterra-Fredholm integral equation ( 6).

Error Analysis
In this section, we analyse the error between the approximate solution and the exact solution of the stochastic Volterra-Fredholm integral equation.Before we start the analysis let us define, Theorem 5.3.Suppose x m (t) be the approximate solution of the linear SFVIE (6).If Proof.Let (6) be the given SVFIE and x m (t) be the approximation to the solution using the Walsh function.Then We know that, (a Now for i = 1, 2, we have s)| ≤ σ and using Theorem 5.2, we get which gives, By Cauchy-Schwarz inequality, for t > 0 and f ∈ L 2 [0, 1) Therefore, Similarly, for |k(s, t)| ≤ ρ and using Theorem 5.2, we get Now, Hence, Using Theorem 5.1, equation ( 14), ( 15) and ( 16) in ( 12), we get where, and By using Gronwall's inequality, we have which implies that,

Numerical Examples
To illustrate the method given in the above section,we consider following examples and compute the approximate solution.The computations are done using Matlab 2013a.

Conclusion
Since it is challenging to find the exact solution for a majority of the SVFIEs, the numerical technique is crucial in solving these issues.Several numerical solutions have also been developed earlier to determine the approximate solution of SVFIEs.This article also proposes a numerical method to find an approximate solution to SVFIE.It also includes numerical estimates for some SVFIEs.The important part is that error analysis of the approach has been undergone by considering the functions satisfing Lipschitz condition to confirm the validity of the methodology which gives an upper hand to consider more general SVFIEs than the previous method .This method can further be developed to address the non linear stochastic integral equations.

Table 1 :
Numerical result for m=32 and m=64 in Example 6.1

Table 2 :
Numerical result for m=32 and m=64 in Example 6.2