JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 469308 10.1155/2014/469308 469308 Research Article Modified Block Pulse Functions for Numerical Solution of Stochastic Volterra Integral Equations Maleknejad K. Khodabin M. Hosseini Shekarabi F. Diethelm Kai Department of Mathematics Islamic Azad University Karaj Branch Karaj Iran iau.ac.ir 2014 1332014 2014 13 12 2013 22 01 2014 13 3 2014 2014 Copyright © 2014 K. Maleknejad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present a new technique for solving numerically stochastic Volterra integral equation based on modified block pulse functions. It declares that the rate of convergence of the presented method is faster than the method based on block pulse functions. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.

1. Introduction

The numerical study and simulation of stochastic Volterra integral equations (SVIEs) have been an active field of research for the past years . Most SVIEs do not have analytic solutions and hence it is of great importance to provide numerical schemes. Numerical schemes to stochastic differential equations (SDEs) have been well developed . However, there are still few papers discussing the numerical solutions for stochastic Volterra integral equations.

Study in economics, sociology, and various biological and medical models leads to the stochastic Volterra integral equations. These systems are dependent on a noise source, on a Gaussian white noise, so that modeling such phenomena naturally requires the use of various stochastic Volterra integral equations.

In this paper, we consider the linear stochastic Volterra integral equation: (1) u ( t ) = u 0 ( t ) + 0 t k 1 ( s , t ) u ( s ) d s + 0 t k 2 ( s , t ) u ( s ) d B ( s ) t [ 0 , T ] , where u ( t ) , u 0 ( 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 , , P ) with a filtration { t , t 0 } that is increasing and right continuous and 0 contains all P -null sets. u ( t ) is unknown random function and B ( t ) is a standard Brownian motion defined on the probability space and 0 t k 2 ( s , t ) u ( s ) d B ( s ) is the Itô integral. Numerous papers have been focusing on existence solution of (1) .

The paper  solves stochastic Volterra integral equations by block pulse functions (BPFs) and  applies this method for solving m -dimensional stochastic Itô Volterra integral equations. However, BPFs are very common in use; it seems that their convergence is weak. Maleknejad and Rahimi apply in  ε Modified Block Pulse Functions ( ε MBPFs) to solve Volterra integral equation of the first kind numerically. Here, we use this method for solving SVIEs.

This paper is organized as follows. In the rest of this section we describe some general concepts concerning the block pulse functions and epsilon modified block pulse functions and some concepts related to stochastic and Itô integral. Section 2 is devoted to stochastic integration operational matrix. In Section 3, the method is employed to solve stochastic integral equations. Section 4 discusses error analysis of this method. Section 5 gives numerical example. Finally, Section 6 provides the conclusion of this work.

1.1. Block Pulse Functions

BPFs have been variously studied  and applied for solving different problems. The goal of this section is to recall notations and definition of the BPFs that are used in the next sections.

The block pulse functions are defined on the time interval [ 0 , T ) by (2) ψ i ( t ) = { 1 ( i - 1 ) T m t < i T m , 0 elsewhere , where i = 1 , , m and for convenience we put h = T / m .

The block pulse functions on [ 0 , T ) have the following properties:

disjointness: for i , j = 1 , , m (3) ψ i ( t ) ψ j ( t ) = δ i j ψ i ( t ) ,

where δ i j is Kronecker delta;

orthogonality: (4) 0 T ψ i ( t ) ψ j ( t ) d t = δ i j h ;

completeness: for every f L 2 ( [ 0 , T ) ) when m approaches infinity, Parseval’s identity holds: (5) 0 T f 2 ( t ) d t = lim m i = 1 m ( f i ) 2 ψ i ( t ) 2 ,

where (6) f i = 1 h 0 T f ( t ) ψ i ( t ) d t .

Also the Fourier coefficients f i and the block pulse functions depend on m . The set of block pulse functions may be written as a vector Ψ ( t ) of dimension m : (7) Ψ ( t ) = [ ψ 1 ( t ) , , ψ m ( t ) ] T t [ 0 , T ) . From the above representation and disjointness property, it follows that (8) Ψ ( t ) Ψ T ( t ) = ( ψ 1 ( t ) 0 0 0 0 ψ 2 ( t ) 0 0 0 0 0 ψ m ( t ) ) m × m . Ψ T ( t ) Ψ ( t ) = 1 , Ψ ( t ) Ψ T ( t ) F = D F Ψ ( t ) , where F is an m -dimensional vector and D F = diag ( F ) . Let G be an m × m matrix so that (9) Ψ T ( t ) G Ψ ( t ) = G ^ T Ψ ( t ) , where G ^ is a vector with elements equal to the diagonal entries of G .

The expansion of a function f ( t ) over [ 0 , T ) with respect to ψ i ( t ) , i = 1 , , m , is given by (10) f ( t ) i = 1 m f i ψ i ( t ) = F T Ψ ( t ) = Ψ T ( t ) F , where F = [ f 1 , , f m ] T and f i is defined by (6).

Let k ( s , t ) L 2 ( [ 0 , T 1 ) × [ 0 , T 2 ) ) . It is expanded with respect to BPFs as (11) k ( s , t ) Ψ T ( s ) K Λ ( t ) , where Ψ ( s ) and Λ ( t ) are   m 1 - and m 2 -dimensional BPFs vectors, respectively, and K is the m 1 × m 2 block pulse coefficient matrix with the below k i j , i = 1 , , m 1 , j = 1 , , m 2 : (12) k i j = m 1 m 2 T 1 T 2 0 T 1 0 T 2 k ( s , t ) ψ i ( s ) λ j ( t ) d s d t . For convenience, we put m 1 = m 2 = m .

Now, integration operational matrix is considered and computed: (13) 0 t ψ i ( s ) d s = { 0 0 t ( i - 1 ) h , t - ( i - 1 ) h ( i - 1 ) h t i h , h i h t < 1 . Since t - i h is equal to h / 2 at midpoint of [ i h , ( i + 1 ) h ) , we can approximate t - ( i - 1 ) h , for ( i - 1 ) h t < i h by h / 2 . Therefore (14) 0 t ψ i ( s ) d s ( 0 , , 0 , h 2 , h , , h ) Ψ ( t ) , where the i th component is h / 2 . As a result (15) 0 t Ψ ( s ) d s Q Ψ ( t ) , where Q is operational matrix of integration that is given by (16) Q = h 2 ( 1 2 2 2 0 1 2 2 0 0 1 2 0 0 0 1 ) . So, (17) 0 t f ( s ) d s 0 t F T Ψ ( s ) d s F T Q Ψ ( t ) .

1.2. Epsilon Modified Block Pulse Functions (EMBPFs)

A set of epsilon modified block pulse functions θ i ( t ) , i = 0,1 , , m , on the interval [ 0 , T ) are defined as (18) θ 0 ( t ) = { 1 t [ 0 , T m - ε ) = I 0 , 0 otherwise , θ i ( t ) = { 1 t [ i T m - ε , ( i + 1 ) T m - ε ) = I i , 0 otherwise , for i = 1 , . . , m - 1 , and (19) θ m ( t ) = { 1 t [ T - ε , T ) = I m , 0 otherwise .

Similar to BPFs, the most important properties of EMBPFs are

disjointness: (20) θ i ( t ) θ j ( t ) = { θ i ( t ) i = j , 0 i j ,

where i , j = 0 , , m ;

orthogonality: if we put h = T / m , (21) 0 T θ i ( t ) θ j ( t ) d t = h δ i j , i , j = 1 , , m - 1 ;

completeness: (22) 0 T f 2 ( t ) d t = i = 0 f i 2 θ i ( t ) 2 ,

where (23) f i = 1 Δ ( I i ) 0 T f ( t ) θ i ( t ) d t

and Δ ( I i ) is length of interval I i .

With defining Θ m + 1 ( t ) = [ θ 0 ( t ) , , θ m ( t ) ] T , we have (24) Θ m + 1 ( t ) Θ m + 1 T ( t ) = ( θ 0 ( t ) 0 0 0 0 θ 1 ( t ) 0 0 0 0 0 θ m ( t ) ) m + 1 × m + 1 , Θ m + 1 T ( t ) Θ m + 1 ( t ) = 1 , Θ m + 1 ( t ) Θ m + 1 T ( t ) F = D F Θ m + 1 ( t ) , Θ m + 1 T ( t ) G Θ m + 1 ( t ) = G ^ T Θ m + 1 ( t ) . Similar to BPFs, (25) 0 t Θ m + 1 ( s ) d s P Θ m + 1 ( t ) , where the operational matrix P of EMBPFs is given by (26) P = ( h - ε 2 h - ε h - ε h - ε 0 h 2 h h 0 0 h 2 h 0 0 0 ε 2 ) m + 1 × m + 1 , and we have the following approximation: (27) 0 t f ( s ) d s 0 t F T Θ m + 1 ( s ) d s F T P Θ m + 1 ( s ) .

1.3. Stochastic Concepts of Itô Integral Definition 1 (Brownian motion process).

A real-valued stochastic process { B ( t ) , t 0 } is called Brownian motion, if it satisfies the following properties:

independence of increments: B ( t ) - B ( s ) , t > s , is independent of the past, that is, of B ( u ) , 0 u s , or of s , the σ -field generated by B ( u ) , u s ;

normal increments: B ( t ) - B ( s ) has normal distribution with mean 0 and variance t - s ;

continuity of paths: B ( t ) , t 0 , are continuous functions of t .

Definition 2.

Let { N ( t ) } t 0 be an increasing family of σ -algebras of subsets of Ω . A process g ( t , ω ) from [ 0 , ) × Ω to R n is called N ( t ) -adapted if for each t 0 the function ω g ( t , ω ) is N ( t ) -measurable .

Definition 3 (see [<xref ref-type="bibr" rid="B15">19</xref>]).

Let ν = ν ( S , T ) be the class of functions f ( t , ω ) : [ 0 , ) × Ω R such that

( t , ω ) f ( t , ω ) is B × -measurable, where B denotes the Borel σ -algebra on [ 0 , ) and is the σ -algebra on Ω ;

f ( t , ω ) is t -adapted, where t is the σ -algebra generated by the random variables B ( s ) , s t ;

E [ S T f 2 ( t , ω ) d t ] < .

Definition 4 (the Itô integral, [<xref ref-type="bibr" rid="B15">19</xref>]).

Let f ν ( S , T ) ; then the Itô integral of f (from S to T ) is defined by (28) S T f ( t , ω ) d B ( t ) ( ω ) = lim n S T ϕ n ( t , ω ) d B ( t ) ( ω ) , 000000000000000000000000000 ( limit in    L 2 ( P ) ) , where ϕ n is a sequence of elementary functions such that (29) E [ S T ( f ( t , ω ) - ϕ n ( t , ω ) ) 2 d t ] 0 , as    n .

Theorem 5 (the Itô isometry).

Let f ν ( S , T ) ; then (30) E [ ( S T f ( t , ω ) d B ( t ) ( w ) ) 2 ] = E [ S T f 2 ( t , ω ) d ( t ) ] .

Proof.

See .

Definition 6 (1-dimensional Itô processes, [<xref ref-type="bibr" rid="B15">19</xref>]).

Let B ( t ) be 1-dimensional Brownian motion on ( Ω , , P ) . A 1-dimensional Itô process (stochastic integral) is a stochastic process X ( t ) on ( Ω , , P ) of the form (31) X ( t ) = X ( 0 ) + 0 t u ( s , ω ) d s + 0 t v ( s , ω ) d B ( s ) , or (32) d X ( t ) = u d t + v d B ( t ) , where (33) P [ 0 t v 2 ( s , ω ) d s < , t 0 ] = 1 , P [ 0 t | u ( s , ω ) | d s < , t 0 ] = 1 .

Theorem 7 (the 1-dimensional Itô formula).

Let X ( t ) be an Itô process given by (1) and g ( t , x ) C 2 ( [ 0 , ) × R ) ; then (34) Y ( t ) = g ( t , X ( t ) ) is again an Itô process, and (35) d Y ( t ) = g t ( t , X ( t ) ) d t + g x ( t , X ( t ) ) d X ( t ) + 1 2 2 g x 2 ( t , X ( t ) ) ( d X ( t ) ) 2 , where ( d X ( t ) ) 2 = ( d X ( t ) ) ( d X ( t ) ) is computed according to the rules (36) d t · d t = d t · d B ( t ) = d B ( t ) · d t = 0 , d B ( t ) · d B ( t ) = d t .

Proof.

See .

Moreover, · is notation of (37) f ( t ) 2 = 0 1 | f ( t ) | 2 d t .

Lemma 8 (the Gronwall inequality).

Let α , β [ t 0 , T ] R be integral with (38) 0 α ( t ) β ( t ) + L t 0 t α ( s ) d s for t [ t 0 , T ] , where L > 0 . Then (39) α ( t ) β ( t ) + L t 0 t e L ( t - s ) β ( s ) d s , t [ t 0 , T ] .

For more details see [19, 20].

2. Stochastic Integral Operational Matrix for EMBPFs

In this section stochastic integral operational matrix for EMBPFs is considered. For finding vector form of 0 t θ i ( s ) d B ( s ) , with EMBPFs, the Itô integral of each single EMBPF θ i ( t ) can be computed as follows. It is clear that the integrals are stochastic and nondeterministic: (40) 0 t θ 0 ( s ) d B ( s ) = { B ( t ) - B ( 0 ) 0 t < h - ε , B ( h - ε ) - B ( 0 ) h - ε t < T . 0 t θ i ( s ) d B ( s ) = { 0 0 t < i h - ε , B ( t ) - B ( i h - ε ) i h - ε t < ( i + 1 ) h - ε , B ( ( i + 1 ) h - ε ) - B ( i h - ε ) ( i + 1 ) h - ε t < T , for i = 1 , , m , and (41) 0 t θ m ( s ) d B ( s ) = { 0 0 t < T - ε , B ( t ) - B ( T - ε ) T - ε t < T . We approximate

B ( t ) - B ( i h - ε ) , by B ( ( i + 0.5 ) h - ε ) - B ( i h - ε ) , at midpoint of [ i h - ε , ( i + 1 ) h - ε ) ;

B ( t ) - B ( 0 ) by B ( ( h - ε ) / 2 ) in θ 0 ( t ) at midpoint of [ 0 , h - ε ) ;

B ( t ) - B ( T - ε ) by B ( T - ( ε / 2 ) ) - B ( T - ε ) in θ m ( t ) , at midpoint of [ T - ε , T ) .

As a result, vector form of 0 t θ i ( s ) d B ( s ) , with EMBPFs, is given by (42) 0 t θ 0 ( s ) d B ( s ) ( B ( h - ε 2 ) , B ( h - ε ) , , B ( h - ε ) ) Θ ( t ) , 0 t θ i ( s ) d B ( s ) ( 0,0 , , 0 , B ( ( i + 0.5 ) h - ε ) - B ( i h - ε ) , B ( ( i + 1 ) h - ε ) - B ( i h - ε ) , , B ( ( i + 1 ) h - ε ) - B ( i h - ε ) ) Θ ( t ) , in which the ( i + 1 ) th component is B ( ( i + 0.5 ) h - ε ) - B ( i h - ε ) , (43) 0 t θ m ( s ) d B ( s ) ( 0,0 , , B ( T - ε 2 ) - B ( T - ε ) ) Θ ( t ) . Therefore (44) 0 t Θ ( s ) d B ( s ) P S Θ ( t ) , where stochastic operational matrix of integration is given by (45) P S = ( B ( h - ε 2 ) B ( h - ε ) B ( h - ε ) B ( h ) 0 B ( 3 h 2 - ε ) - B ( h - ε ) B ( 2 h - ε ) - B ( h - ε ) B ( 2 h - ε ) - B ( h - ε ) 0 0 B ( 5 h 2 - ε ) - B ( 2 h - ε ) B ( 3 h - ε ) - B ( 2 h - ε ) 0 0 0 B ( ( 2 m - 1 ) h 2 - ε ) - B ( ( m - 1 ) h - ε ) 0 0 0 B ( T - ε 2 ) - B ( T - ε ) ) m + 1 × m + 1 . So, the Itô integral of every function f ( t ) can be approximated as follows: (46) 0 t f ( s ) d B ( s ) 0 t F T Θ ( s ) d B ( s ) F T P S Θ ( t ) .

3. Numerical Solution of SVIEs by EMBPFs

Here, we modify the method that has been used in  by EMBPFs. In the below equation: (47) u ( t ) = u 0 ( t ) + 0 t k 1 ( s , t ) u ( s ) d s + 0 t k 2 ( s , t ) u ( s ) d B ( s )       t [ 0 , T ] , we approximate functions u ( t ) , u 0 ( t ) , k 1 ( s , t ) , and k 2 ( s , t ) by EMBPFs: (48) u ( t ) U T Θ ( t ) = Θ T ( t ) U , u 0 ( t ) U 0 T Θ ( t ) = Θ T ( t ) U 0 , k 1 ( s , t ) Θ T ( s ) K 1 Θ ( t ) = Θ T ( t ) K 1 T Θ ( s ) , k 2 ( s , t ) Θ T ( s ) K 2 Θ ( t ) = Θ T ( t ) K 2 T Θ ( s ) , where the vectors U , U 0 and matrices K 1 , K 2 are EMBPFs coefficient of u , u 0 , k 1 , and k 2 , respectively.

Substituting (48) into (47) and using previous relations, (49) U T Θ ( t ) U 0 T Θ ( t ) + U T ( 0 t Θ ( s ) Θ T ( s ) d s ) K 1 Θ ( t ) + U T ( 0 t Θ ( s ) Θ T ( s ) d B ( s ) ) K 2 Θ ( t ) . Finally (50) U T Θ ( t ) U 0 T Θ ( t ) + U T B Θ ( t ) + U T B s Θ ( t ) , where (51) B = ( k 00 1 ( h - ε 2 ) k 01 1 ( h - ε ) k 02 1 ( h - ε ) k 0 m 1 ( h - ε ) 0 k 11 1 ( h 2 ) 2 k 12 1 h 2 k 1 m 1 h 0 0 k 33 1 h 2 2 k 3 m 1 h 0 0 0 k m m 1 ε 2 ) m + 1 × m + 1 , B s = ( k 00 2 B ( h - ε 2 ) k 01 2 B ( h - ε ) k 02 2 B ( h - ε ) k 0 m 2 B ( h - ε ) 0 k 11 2 ( B ( 3 h 2 - ε ) - B ( h - ε ) ) k 12 2 ( B ( 2 h - ε ) - B ( h - ε ) ) k 1 m 2 ( B ( 2 h - ε ) - B ( h - ε ) ) 0 0 k 22 2 ( B ( 5 h 2 - ε ) - B ( 2 h - ε ) ) k 2 m 2 ( B ( 3 h - ε ) - B ( 2 h - ε ) ) 0 0 0 k m - 1 , m 2 ( B ( ( 2 m - 1 ) h 2 - ε ) - B ( ( m - 1 ) h - ε ) ) 0 0 0 k m m 2 ( B ( T - ε 2 ) - B ( T - ε ) ) ) . Then (52) U T ( I - B - B s ) U 0 . With replacing by = , we have a linear system of equations.

Now if ε j = j h / k , j = 0,1 , , k - 1 , there will be k numerical answers f ^ j h / k . Solution is approximated by (53) f ¯ ( t ) = 1 k i = 0 k - 1 f ^ i h / k ( x ) .

4. Error Analysis

In this section, error analysis is studied. In the following theorems, for simplicity, we assume T = 1 and h = 1 / m .

Theorem 9.

If f ^ m ( x ) = i = 0 m f i θ i ( x ) and f i = ( 1 / Δ ( I i ) ) 0 1 f ( x ) θ i ( t ) d t , i = 0 , , m , then

δ = 0 1 ( f ( x ) - i = 0 m f i ϕ i ( x ) ) 2 d x achieves its minimum value;

f ^ m ( x ) approach f ( x ) pointwise;

0 1 f 2 ( x ) d x = i = 0 f i 2 ϕ i 2 .

Proof.

See .

Theorem 10.

Assume the following.

f ( x ) is continuous and differentiable in [ - h , 1 + h ] , with bounded derivative; that is, | f ( x ) | < M .

f ^ i h / k ( x ) , i = 0,1 , , k - 1 , are correspondingly BPFs. h / k MBPFs, , ( k - 1 ) h / k MBPFs expansions of f ( x ) base on m + 1 EMBPFs over interval [ 0,1 ) .

f ¯ ( t ) = ( 1 / k ) i = 0 k - 1 f ^ i h / k ( x ) .

Then (54) f ( x ) - f ^ i h / k ( x ) = O ( h ) , f ( x ) - f ¯ ( x ) = O ( h k ) i n [ h , 1 - h ] .

Proof.

Trapezoidal rule for integral is (55) a b f ( x ) d x = b - a 2 ( f ( a ) + f ( b ) ) - ( b - a ) 3 f ′′ ( η ) 12 = b - a 2 ( f ( a ) + f ( b ) ) + E , η [ a , b ] , where E is error of integration. Suppose t i = i / m = i h and I i = [ t i - 1 , t i ] . The representation error when f ( x ) is represented by a series of BPFs over every subinterval [ t i , t i + h / k ] , i = 0 , , m - 1 , is (56) e i ( x ) = f ( x ) - f i ϕ i ( x ) = f ( x ) - f i , where (57) f i = 1 h i h ( i + 1 ) h f ( x ) d x . From (55), (58) f i = 1 2 ( f ( t i ) + f ( t i + h ) ) + E .

It is obvious that if f ( x ) = C ( c o n s t a n t ) , then e i ( x ) = 0 .

So, this error is computed for f ( x ) = x in interval [ t i , t i + h / k ] , i = 1 , , m - 1 .

For this function E = 0 , so (59) e i ( x ) [ t i , t i + h / k ] = | x - f i | = | x - t i + t i + 1 2 | = | x - ( t i + h 2 ) | h 2 . Then this error with BPFs is ( h / 2 ) M .

Similarly, the error when f ( x ) is represented in a series of EMBPFs over every subinterval [ t i , t i + h / k ] is (60) e i ( x ) [ t i , t i + h / k ] = | x - ( j = 0 k - 1 ( t i - ( j h / k ) + t i + 1 - j h / k ) 2 k )    | = | x - ( j = 0 k - 1 ( t i - j h / k + t i + h - j h / k ) 2 k ) | = | x - ( t i + h 2 ) - ( k - 1 ) h 2 k | h 2 k . So, the error with EMBPFs is ( h / 2 k ) M .

For I 0 in [ 0 , h / k ] we have (61) e i ( x ) = | x - j = 0 k - 1 h - j h / k 2 k | = | x - ( h 2 - ( k - 1 ) h 4 k ) | = | x - ( h 4 + h 4 k ) | = O ( h 4 ) . So, the error is O ( h / 4 ) also for I n .

Now, (62) e i ( x ) 2 = t i t i + h / k | e i ( x ) | 2 d x = t i t i + h / k h 2 4 k 2 M 2 d x = h 3 4 k 3 M 2 , e 2 = 0 1 e 2 ( x ) d x = 0 1 ( i = 1 m j = 0 k - 1 e i ( x ) ) 2 d x = i = 1 m j = 0 k - 1 0 1 e i 2 ( x ) d x = i = 1 m j = 0 k - 1 e i ( x ) 2 = 1 h · k · h 3 4 k 3 M 2 = h 2 4 k 2 M 2 .

We define the representation error between f ( x , y ) and its 2D-EMBPFs expansion, f i , j , over every subregion D i j , is defined as (63) e i j ( x , y ) = f ( x , y ) - f i j , where (64) D i j : = { ( x , y ) t i x t i + h k , t j x t j + h k } .

With Taylor’s expansion and similarity to the above discussion, (65) e ( x , y ) = h 2 k M .

Theorem 11.

Assume that

P ( w Ω : u ( ω , t ) < C ) = 1 ,

k i < C       i = 1,2 .

Then (66) sup 0 t T ( E ( ( u - u ¯ ) ) 2 ) 1 / 2 = O ( h k ) , t [ h , 1 - h ] .

Proof.

Consider (67) u ( t ) - u ¯ ( t ) = u 0 ( t ) - u 0 ¯ ( t ) + 0 t k 1 ( s , t ) u ( s ) - k 1 ¯ ( s , t ) u ( s ) ¯ d s + 0 t k 2 ( s , t ) u ( s ) - k 2 ¯ ( s , t ) u ( s ) ¯ d B ( s ) . So, (68) E ( u - u ¯ 2 ) 3 [ ( 0 t ( k 2 u - k 2 ¯ u ¯ ) d B ( s ) 2 ) E ( ( u 0 - u 0 ¯ ) 2 ) + E ( 0 t ( k 1 u - k 1 ¯ u ¯ ) d s 2 ) + E ( 0 t ( k 2 u - k 2 ¯ u ¯ ) d B ( s ) 2 ) ] 3 [ 0 t E ( k 2 u - k 2 ¯ u ¯ ) 2 ) E ( ( u 0 - u 0 ¯ ) 2 ) + ( 0 t E ( k 1 u - k 1 ¯ u ¯ 2 ) d s ) + 0 t E ( k 2 u - k 2 ¯ u ¯ 2 ) d s ] , by the Cauchy-Schwartz inequality, Itô isometry formula, and the linearity of Itô integrals in their integrands.

The first term is satisfied by last theorem: (69) E ( u 0 - u 0 ¯ 2 ) E ( C 2 h 2 k 2 ) = O ( h 2 k 2 ) . Now, (70) ( k i ( s , t ) u ( t ) - k i ¯ ( s , t ) u ¯ ( t ) 2 2 ( k i - k i ¯ ) u 2 + 2 k i ¯ ( u ¯ - u ) 2 C · ( k i - k i ¯ 2 ) + C · ( ( u ¯ - u ) 2 ) . Furthermore, (71) k i - k i ¯ 2 = O ( h 2 k 2 ) , i = 1,2 . Hence (72) E ( u - u ¯ 2 ) 3 [ 0 t E ( ( u 0 - u 0 ¯ ) 2 ) + 0 t E ( ( k 1 u - k 1 ¯ u ¯ ) 2 d s ) + 0 t E ( ( k 2 u - k 2 ¯ u ¯ ) 2 ) d s ] C 0 E ( u 0 - u 0 ¯ 2 ) + C 1 0 t E ( k 1 - k 1 ¯ 2 ) d s + C 2 0 t E ( k 2 - k 2 ¯ 2 ) d s + C 3 0 t E ( ( u - u ¯ ) 2 ) d s . Then by Gronwall’s inequality, we get (73) E ( ( u - u ¯ ) 2 ) C h 2 k 2 .

5. Numerical Example

In this section, we present an example for showing the features of the EMBPFs method in this paper. Let X i denote the EMBP coefficient of exact solution of the given example and let Y i be the EMBP coefficient of computed solution by the presented method. In this example error is defined as (74) E = max 1 i m | X i - Y i | .

Example 1 (see [<xref ref-type="bibr" rid="B10">3</xref>]).

Consider the following linear stochastic Volterra integral equation: (75) u ( t ) = 1 12 + 0 t cos ( s ) u ( s ) d s + 0 t sin ( s ) u ( s ) d B ( s ) s , t [ 0,0.5 ) ,

with the exact solution u ( t ) = ( 1 / 12 ) e - t / 4 + sin ( t ) + sin ( 2 t ) / 8 + 0 t sin ( s ) d B ( s ) , for 0 t < 0.5 .

The numerical results are shown in Tables 1 and 2. In the tables, n is the number of iterations, x ¯ E is error mean, and s E is standard deviation of error.

Mean, standard deviation, and confidence interval for error mean in Example 1 with m = 4 , k = 4 .

n x - E s E %95 confidence interval for mean of E
Lower Upper
30 3.5678 × 10 - 3 4.5802 × 10 - 3 1.9287 × 10 - 4 5.2068 × 10 - 3
50 5.0234 × 10 - 3 7.5849 × 10 - 3 2.9209 × 10 - 3 7.1258 × 10 - 3
100 3.3467 × 10 - 3 4.7983 × 10 - 3 2.4062 × 10 - 3 4.2871 × 10 - 3
125 4.3526 × 10 - 3 6.3657 × 10 - 3 3.2367 × 10 - 3 5.4685 × 10 - 3

Mean, standard deviation, and confidence interval for error mean in Example 1 with m = 8 , k = 4 .

n x - E s E %95 confidence interval for mean of E
Lower Upper
30 3 . 0924 × 10 - 3 5 . 1132 × 10 - 3 2 . 6266 × 10 - 3 4 . 9221 × 10 - 3
50 2 . 0598 × 10 - 3 6 . 1477 × 10 - 3 3 . 5574 × 10 - 4 3 . 7635 × 10 - 3
100 1 . 9728 × 10 - 3 2 . 2587 × 10 - 3 1 . 5300 × 10 - 3 2 . 4155 × 10 - 3
125 1 . 7054 × 10 - 3 2 . 6547 × 10 - 3 1 . 2400 × 10 - 3 2 . 1707 × 10 - 3

Table 3 is from  for comparison.

Mean, standard deviation, and confidence interval for error mean in Example 1 with m = 32 .

n x - E s E %95 confidence interval for mean of E
Lower Upper
30 0.02308947 0.00442835 0.02150480 0.02467413
50 0.02341165 0.00511389 0.02199415 0.02482915
100 0.02364843 0.00524000 0.02262139 0.02467548
125 0.02345691 0.00477156 0.02262042 0.02429340

In some examples by applying BPFs when m increases, accuracy decreases, but in EMBPFs we achieve good accuracy by increasing k .

6. Conclusion

As some SVIEs cannot be solved analytically, in this paper we present a new technique for solving SVIEs numerically. Here, we consider a modification of the block pulse functions. Some theorems show that if EMBPFs are used for achieving numerical expansions with k times more precision, there is no need to increase the number of BPFs, k times, which leads to solving a system of equations with k times more equations and unknowns. But the results of BPFs solution can be combined with solutions of k - 1 systems of equations with one more unknown and nearly achieve k times more precision. Parallel programming is so useful for this method. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Appley J. A. D. Devin S. Reynolds D. W. Almost sure convergence of solutions of linear stochastic Volterra equations to nonequilibrium limit Journal of Integral Equations and Applications 2007 19 4 405 437 10.1216/jiea/1192628616 MR2374162 ZBL1154.45012 Berger M. A. Mizel V. J. Volterra equations with Itô integrals. I Journal of Integral Equations 1980 2 3 187 245 MR581430 ZBL0442.60064 Maleknejad K. Khodabin M. Rostami M. Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions Mathematical and Computer Modelling 2012 55 3-4 791 800 2-s2.0-84855204134 10.1016/j.mcm.2011.08.053 MR2887420 ZBL1255.65247 Maleknejad K. Khodabin M. Rostami M. A numerical method for solving m-dimensional stochastic Itô-Volterra integral equations by stochastic operational matrix Computers & Mathematics with Applications 2012 63 1 133 143 2-s2.0-83655163989 10.1016/j.camwa.2011.10.079 MR2863484 ZBL1238.65007 Pardoux E. Protter P. Stochastic Volterra equations with anticipating coeffcients The Annals of Probability 1990 18 4 1635 1655 10.1214/aop/1176990638 MR1071815 ZBL0717.60073 Shiota Y. A linear stochastic integral equation containing the extended Itô integral Mathematics Reports, Toyama University 1986 9 43 65 Wen C. H. Zhang T. S. Improved rectangular method on stochastic Volterra equations Journal of Computational and Applied Mathematics 2011 235 8 2492 2501 2-s2.0-79251595060 10.1016/j.cam.2010.11.002 MR2763162 ZBL1221.65023 Higham D. J. Mao X. Stuart A. M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations SIAM Journal on Numerical Analysis 2002 40 3 1041 1063 2-s2.0-0012279718 10.1137/S0036142901389530 MR1949404 ZBL1026.65003 Khodabin M. Maleknejad K. Rostami M. Nouri M. Numerical solution of stochastic differential equations by second order Runge-Kutta methods Mathematical and Computer Modelling 2011 53 9-10 1910 1920 2-s2.0-79951954714 10.1016/j.mcm.2011.01.018 MR2782894 ZBL1219.65009 Kloeden P. E. Platen E. Numerical Solution of Stochastic Differential Equations 1992 Berlin, Germany Springer Applications of Mathematics MR1214374 Mao X. Stochastic Differential Equations and Applcation 1997 Chichester, UK Horwood Publishing Limited Horwood Series in Mathematics and Applications MR1475218 Platen E. An introduction to numerical methods for stochastic differential equations Acta Numerica 1999 8 197 246 10.1017/S0962492900002920 MR1819646 ZBL0942.65004 Janković S. Ilić D. One linear analytic approximation for stochastic integrodifferential equations Acta Mathematica Scientia 2010 30 4 1073 1085 2-s2.0-77954602996 10.1016/S0252-9602(10)60104-X MR2730534 ZBL1240.60153 Padgett W. J. Tsokos C. P. Existence of a solution of a stochastic integral equation in turbulence theory Journal of Mathematical Physics 1971 12 2 210 212 2-s2.0-22944455801 MR0278419 ZBL0212.59702 Subramaniam R. Balachandran K. Kim J. K. Existence of solutions of a stochastic integral equation with an application from the theory of epidemics Nonlinear Functional Analysis and Applications 2000 5 1 23 29 MR1795707 ZBL0967.60065 Maleknejad K. Rahimi B. Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind Communications in Nonlinear Science and Numerical Simulation 2011 16 6 2469 2477 2-s2.0-78651427225 10.1016/j.cnsns.2010.09.032 MR2765199 ZBL1221.65338 Babolian E. Maleknejad K. Mordad M. Rahimi B. A numerical method for solving Fredholm-Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix Journal of Computational and Applied Mathematics 2011 235 14 3965 3971 2-s2.0-79955905619 10.1016/j.cam.2010.10.028 MR2801421 ZBL1219.65158 Maleknejad K. Shahrezaee M. Khatami H. Numerical solution of integral equations system of the second kind by Block-Pulse functions Applied Mathematics and Computation 2005 166 1 15 24 2-s2.0-19044368178 10.1016/j.amc.2004.04.118 MR2145872 ZBL1073.65149 Øksendal B. K. Stochastic Differential Equations: An Introduction with Applications 1998 5th New York, NY, USA Springer MR1619188 Klebaner F. C. Intoduction to Stochastic Calculus with Applications 2005 2nd Melbourne, Australia Monash University MR2160228