Numerical Solution of Two-Dimensional Nonlinear Stochastic Itô-Volterra Integral Equations by Applying Block Pulse Functions

This paper investigates the numerical solution of two-dimensional nonlinear stochastic Itô-Volterra integral equations based on block pulse functions. The nonlinear stochastic integral equation is transformed into a set of algebraic equations by operational matrix of block pulse functions. Then, we give error analysis and prove that the rate of convergence of this method is efficient. Lastly, a numerical example is given to confirm the method.


Introduction
Two-dimensional stochastic Itô-Volterra integral equations arise from many phenomena in physics and engineering fields [1].Some different orthogonal basis functions, polynomials and wavelets are used to approximate the solution of two-dimensional Volterra integral equations.For example, block pulse functions, triangular functions, modification of hat functions, Legender polynomials and Haar wavelet and the like (see [2] [3] [4] [5] [6]).
Especially, Fallahpour et al. [3] introduced the following two-dimensional linear stochastic Volterra integral equation by Haar wavelet where ( )

∫ ∫
is the double Itô integral.The authors transformed stochastic Volterra integral equations to algebra equations by Haar wavelet and gave the numerical solutions to the equations.Similarly, Fallahpour et al. [7] obtained a numerical method for two-dimensional linear stochastic Volterra integral equations by block pulse functions.
For nonlinear determinate Volterra integral equations, Maleknejad et al. [8] and Nemati et al. [6] used two-dimensional block pulse functions and Legendre polynomials to solve those respectively.Both Babolian et al. [2] and Maleknejad et al. [9] employed triangular functions to get the numerical solutions.Mirzaee et al. [5] [10] applied modified two-dimensional block pulse functions to approximate the following determinate equation where nonlinear term , x s s     is power function and ( ) , x s s is unknown, n is a positive integer.( ) , , , k t t s s  is determinate kernel function The authors revealed the accuracy and efficiency of the proposed method by some examples and gave the rate of convergence to the numerical solution.
However, as far as we known, there are hardly any papers about the numerical solution of two-dimensional nonlinear stochastic Itô-Volterra integral equations.
Inspired by the above literatures, we introduce an efficient numerical method for the following nonlinear stochastic integral equation based on block pulse functions. ( where ( ) ( ) ( ) , , , k t t s s  and ( ) are two independent Brownian motions.σ and g are analytical functions.
In Section 2, we recall the definition and properties of block pulse function.In Section 3 and 4, we show the integration operational matrix about two-dimensional block pulse functions.In Section 5, an efficient numerical method to nonlinear stochastic Itô-Volterra integral equation is obtained.In Section 6, the error and the rate of convergence of this method are given.It's important to emphasize that the error is analyzed by Gronwall's inequality and the interchangeability of integral and expectation.However, the norm was used in the literature [11], it is a pity that the interchangeability of norm and integral wasn't proved.In Section 7, we give a numerical example to illustrate the validity of the method.In the final Section 8, we make some conclusions and look ahead to further work.

Two-Dimensional Block Pulse Functions
One dimensional block pulse functions (BPFs) have been widely studied and applied to solve different problems.For example, the article [12] and their relative references give a detailed description.A 1 2 m m -set of two-dimensional block pulse functions (2D-BPFs) ( ) , m and n are arbitrary positive integers and 1, 2 i = .
Similar to the one-dimensional case [12].There are some elementary properties for 2D-BPFs as follows: 1) Disjointness: where , 1, 2, , 3) Completeness: for every ( ) ( ) , when 1 m and 2 m approach to the infinity, Parseval's identity holds: where ( ) ( ) where ( ) From the above representation and disjointness property, it follows that: ( ) ( ) where G is a ( ) m m -vector and the matrix . Moreover, it is easy to conclude that for every ( ) ( ) Similarly, a function of four variables ( ) m m m m × two-dimensional block pulse coefficient matrix in the following form ( ) ( ) and two-dimensional block pulse coefficients , ,  ,  , d d d d .
The more details can also reference to [7].
Then the vector ( ) , m m t t Φ can be showed as following The integration of the vector ( ) , m m t t Φ defined in ( 7) can be approximately obtained as following where , P is the ( ) ( ) m m m m × operational matrix of integration for 2D-BPFs and i P , ( ) are the operational matrix of one-dimensional BPFs [12] defined over [ ) 0,1 as following.

Stochastic Integration Operational Matrix
Similarly, we obtain the stochastic integration of the vector ( ) m m m m × stochastic operational matrix of integration for 2D-BPFs and i s P , ( ) are the stochastic operational matrix of one-dimensional BPFs [12] defined over [ ) 0,1 as following.

Numerical Method
In this section, we first provide a useful result for solving two-dimensional nonlinear stochastic Itô-Volterra integral Equation (3).
Lemma 1.Let ( ) be the analytic functions for positive integer where ( ) m m X are derived in (7) and ( 12), Proof.By virtue of the known conditions and the disjointness properties of 2D-BPFs defined in (4), we can get ( ) , , The proof is completed.□ Now we suppose ( ) , g x t t , ( ) , , , k t t s s  and ( ) ˆ, , , k t t s s can be approximated in terms of 2D-BPFs.
( ) ( ) ( ) ( )   15) and (18), we have ( ) m m m m × matrices.By (10), we have where Q and ˆs Q are ( ) m m -vectors with elements equal to the diagonal entries of matrices Q and s Q .Then There are various methods to solve the nonlinear system of Equation ( 27) of m m X .In this paper, we will use the int () function provided by Matlab 2015b [14] to solve it.According to the coefficient vector , we obtain that the approximation solution of Equation ( 3) , ,

Error Analysis
In this section, for convenience, we assume 1 2 m m m = = and prove that the approximation solution is convergent of order ( ) , v s s be an arbitrary bounded function on Proof.Similar to [15] [16].
, , , v t t s s be an arbitrary bounded function on D D × and ( ) ( ) ( )

t t s s v t t s s v t t s s = −
, which ( ) , , , is m 2 approximations of 2D-BPFs of ( ) , , , v t t s s , then , ,  d d d d .
, mm x t t is the approximation solution of ( ) , , , ( ) , , , , , x t t k t t s s  and ( ) ˆ, , , k t t s s , respectively.
Theorem 1.For analytic functions σ and g, there are constant numbers sa- tisfy the following conditions: 1) According to Itô isometry, Cauchy-Schwartz inequality and Lipschitz conditions, we can write )

∫ ∫
Then, for )   From these figures, we find the general trends of the solutions are similar for different m, and the absolute error of mean solution is very small.This method is efficient and the accuracy is credible.

Conclusion
For some stochastic Volterra integral equations, exact solutions cannot be expressed.But, the numerical solution can be conveniently obtained based on different stochastic numerical methods.As the complexity of the system, we use    BPFs as the basis function to solve the two-dimensional nonlinear stochastic Volterra integral equation.This numerical method is simple and effective.In the mm The front view and the top view of the approximation solutions of the Example 1 for m = 8 are given in Figure1.The front view and the top view of the mean solutions of the Example 1 for m = 8 are given in Figure2.The front view and the top view of the approximation solutions of the Example 1 for m = 16 are given in Figure3.The front view and the top view of the mean solutions of the Example 1 for m = 16 are given in Figure4.

Figure 1 .
Figure 1.The front view and top view of the approximation solutions for m = 8.

Figure 2 .
Figure 2. The front view and top view of the mean solutions for m = 8.

Figure 3 .
Figure 3.The front view and top view of the approximation solutions for m = 16.

Figure 4 .
Figure 4.The front view and top view of the mean solutions for m = 16. , , . ) . .