A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

In this paper, a linear combination of quadratic modified hat functions is proposed to solve stochastic Itô–Volterra integral equation with multi-stochastic terms. All known and unknown functions are expanded in terms of modified hat functions and replaced in the original equation. The operational matrices are calculated and embedded in the equation to achieve a linear system of equations which gives the expansion coefficients of the solution. Also, under some conditions the error of the method is O(h3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^3)$$\end{document}. The accuracy and reliability of the method are studied and compared with those of block pulse functions and generalized hat functions in some examples.

The paper is organized as follows: In "MHFs and their properties" section, the MHFs and their properties are described. In "Operational matrices" section, the operational matrices are found. In "Solving stochastic Itô-Volterra integral equation with multi-stochastic terms by the MHFs" section, the sets and operational matrices are applied in the above equation and the approximate solution is found. In "Error analysis" section, the error analysis of the present method is discussed. In the "Numerical examples" section, some numerical examples are solved by using this method. And finally, the last section concludes the paper.

MHFs and their properties
In this section, we recall the definition and properties of modified hat functions [33]. Let m ≥ 2 be an even integer and h = T m . Also assume that the interval [0, T) is divided into m 2 equal subintervals [ih, (i + 2)h], i = 0, 2, … , (m − 2) and let X m be the set of all continuous functions that are quadratic polynomials when restricted to each of the above subintervals. Because each element of X m is completely determined by its values at the (m + 1) nodes ih, i = 0, 1, … , m , the dimension of X m is (m + 1) . Considering that f ∈ = C 3 (D) can be approximated by its expansion with respect to the following set functions, (m + 1) set of MHFs are defined over D as

Properties of the MHFs
By considering the above definition, the following properties come as a result.

Function approximation
An arbitrary real function f on D can be expanded by these functions as [34] where = [f 0 , f 1 , … , f m ] T and (t) is defined in relation (2) and the coefficients in (3) are given by Similarly, an arbitrary real function of two variables g(s, t) on D × D can be expanded by these basic functions as where

Operational matrices
In this section, we present both operational matrix of integrating the vector (t) , denoted by , and stochastic operational matrix of Itô integrating the vector (t) , denoted by s . Therefore, by integrating the vector (t) defined in (2), we have [34,35] where is the following (m + 1) × (m + 1) operational matrix of integration of MHFs (2), the Itô integral of (t) can be expressed as

Theorem 1 Let (t) be the vector defined in
and so we obtain If i is odd and and Putting the obtained components in the matrix form ends the proof. □

Solving stochastic Itô-Volterra integral equation with multi-stochastic terms by the MHFs
Our problem is to define the MHFs coefficients of X(t) in the following linear stochastic Itô-Volterra integral equation with several independent white noise sources, where X, f , and j , j = 1, 2, … , n for s, t ∈ D , are stochastic processes defined on the same probability space(Ω, F, P) . Also B 1 (t), B 2 (t), … , B n (t) are Brownian motion processes, and ∫ t 0 j (s, t) dB j (s) , j = 1, 2, … , n are the Itô integrals.
We replace X(t), f (t), (s, t) and j (s, t) , j = 1, 2, … , n by their approximations which are obtained by MHFs: where and are stochastic MHFs coefficient vectors and and j , j = 1, 2, … , n are stochastic MHFs coefficient matrices. Substituting (9)-(12) in relation (8), we obtain Using the 6-th property in relation (13), we get Utilizing operational matrices defined in relations (5) and (6) in (14), we have Let = T diag( ) and j = T diag( ) , j = 1, 2, … , n. Applying property (7) in relation (15) yields therefore, by using the third property and replacing ≃ by = , we have which is a linear system of equations that gives the approximation of X with the help of MHFs.

Error analysis
In this section, the error analysis is studied. We propose some conditions to show that the rate of convergence for this method is O(h 3 ).   (8) and X m be the MHFs series approximate solution of (8) , and also assume that

Theorem 4 Let X be the exact solution of
By substituting (17) and (18)

Numerical examples
In this section, we use our algorithm to solve stochastic Itô-Volterra integral equation with multi-stochastic terms stated in "Solving stochastic Itô-Volterra integral equation with multi-stochastic terms by the MHFs" section. In order to compare it with the method proposed in [22,23], we consider some examples. The computations associated with the examples were performed using Matlab 7 and [36].
Example 1 Consider the following linear stochastic Itô-Volterra integral equation with multi-stochastic terms [22] ‖X Nodes t i Errors of BPFs in [22] Errors of GHFs in [23] Errors of present method Errors of BPFs in [22] Errors of GHFs in [23] Errors  a r e shown in Table 1. Also curves in Figs. 1 and 2 show the exact and approximate solutions computed by this method for m = 10 and m = 40 . Figures 3 and 4 represent the errors of the present method.

Example 2 Let [22]
be a linear stochastic Itô-Volterra integral equation with multi-stochastic terms with the exact solution  for 0 ≤ t < 1, where X is the unknown stochastic process defined on the probability space (Ω, F, P) a n d B 1 (t), B 2 (t), … , B n (t) a r e t h e Brownian motion processes. The numerical results for X 0 = 1 12 , r = 1 30 , 1 = 1 10 , 2 (s) = s 2 , 3 (s) = sin(s) 3 are inserted in Table 2

Conclusion
Finding an analytical exact solution for stochastic equations usually seems impossible. Therefore, it is convenient to use stochastic numerical methods to find some approximate solutions. The MHFs, as a simple and suitable basis, adopt to solve stochastic Itô-Volterra integral equations with multistochastic terms. With this choice, the vector and matrix coefficients are found easily. This method results in a linear system of equations that can be solved simply. Numerical results of the examples show that the MHFs tend to more accurate solutions than the BPFs and GHFs do.
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