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PA new approach based on triangular functions for solving n-dimensional stochastic differential equations

  M. Asgari

 Department of Engineering, Abhar Branch, Islamic Azad University,  Abhar, Iran

Abstract

 In this article, we prepare a new numerical method based on triangular functions for solving n-dimensional stochastic differential equations. At first stochastic operational matrices of triangular functions are derived then n-dimensional stochastic differential equations are solved recently. Convergence analysis and numerical examples are prepared to illustrate accuracy and efficiency of this approach.

 Keywords: Brownian Motion; It'' Integral; N-dimensional Stochastic Differential Equations; Stochastic Operational Matrix;  Triangular Functions.

   AMS subject classifications. Primary: 65C30, 60H35, 65C20; Secondary: 60H20, 68U20

1. Introduction

 Mathematical modeling of real world problems causes differential equations involving stochastic Gaussian white noise excitations. Such problems are modeled by stochastic differential equations (SDE). Some authors have presented numerical approachs to solve stochastic differential/integral equations [1-11]. We consider integral form of n-dimensional stochastic differential equation(N-SDE) as follows:

(1)

 where,   is an initial value,   is a n-dimensional Brownian process.   and  ;   are defined on  ,  probability space, and   is unknown function. Also   are It'' integrals.

 Orthogonal triangular functions (TFs) are derived from the block pulse function (BPF) set by Deb et al. [12]. TFs approximation has been applied for the analysis of dynamical systems [13], integral equations [14,15] and integro-differential equations [16].

In Section 2, we review some properties of TFs. In Section 3, stochastic operational matrices of TFs are presented. Section 4 is devoted for solving N-SDE. Section 5 is prepared convergence analysis of the approach. In Section 6 some numerical examples are provided. Finally, Section 7 gives a brief conclusion.

2 . Brief review of TFs

Deb [12] defined  two m-set TFs over the interval [0,T) as follows

where,  , and  .

TFs, are orthogonal, disjoint  and complete [16].

M-set TF vectors can be considered as

and

 , a square integrable function, may be approximated into TF series as:

(2)

 where,   and   for  . The vectors   and   are called the 1D-TF coefficient vectors and  -vector   is defined as:

The operational matrix for integration can be obtained as[12]

where,

and

Let   be a  vector and   be a   matrix; then, it can be concluded that

(3)

 and

(4)

 in which  . Elements of  , a   vector, are equal to the diagonal elements of   Finally, integration of   can be approximated as follows:

 Any two variable function,  , can be approximated by TFs as follows :

where F is a   coefficient matrix of TFs. We put   So, F can be expanded as:

where  ,     and   are approximated by sampling   at points   and   such that   for   So, the following approximations can be obtained

3. Stochastic Operational Matrix of TFs

Stochastic operational matrix of TFs for the It'' integral is derived in this section. We compute   and   as follows:

(5)

 and

(6)

 where   is the unit step function. These integrations can be divided into tree cases. At first consider  :

(7)

 and

(8)

 For   we get:

(9)

 and

(10)

 Finally, for the case of   we get

(11)

 and

(12)

 The result of these tree cases can be expanded in to TF series:

(13)

 and

(14)

 where   and   for  . From Eqs.(7-12) we get

 and for   Finally we can write

where,   and   are   stochastic operational matrices of TFs. These matrices can be obtained as follow:

and

In a similar manner, the It'' integration of   is

where,

and

Then we get

so,

where  , stochastic operational matrix of T(x), is

Finally we can approximate It'' integration of   with TF oparational matrix as:

(15)

4. Solving n-dimensional Stochastic integral equation

 Approximations of   in  TFs domain can be written as:

(16)

(17)

(18)

(19)

 such that 2m-vectors  ,   are stochastic TF coefficient, and   matrices   and   are TFs coefficients matrices. By substituting Eqs.(16-19) in (1) we get

By using (3) we can write

or

finally by using (4) we get

where   and   are   vectors with elements equal to the diagonal entries of   and   respectively.

(20)

 The linear system of equations in (20) can be solved easily.

5. Convergence analysis

 This section prepares convergence analysis of presented approach in  , continous functions in Banach space , with norm   The following error holds for all   that is expanded in TFs series[12] :

(21)

 where   is defined in (2).

Theorem 5.1 Let   and   be the exact solution and approximate solution of (1) respectively and

i)   ,

ii)  ,  , j=1,...,n,

hold then,

  Proof. Let   be the error function of approximate solution   to the exact solution   we can write

(22)

 where,

 For   we get

(23)

 similarly for   

(24)

 From (22), (23) and (24) we conclude

(25)

 where   . Gronwall inequality and  (25) coclude

By substituting  , and increasing  , it implies   as  

6. Numerical examples

This section is devoted for solving some exampeles to show efficacy of presented approach.

Example 1. A linear stochastic integral equation is considered as follows ([10])

(26)

 with the exact solution

 The numerical results for   are shown in Table I.   is the errors mean and   is the standard deviation of errors in   iteration. In addition, we consider  

Table I: xE (Mean) and sE (standard deviation)  for k = 500.

m xE sE

 

 

Example 2. Consider following example[10]:

(27)

 with the exact solution   .

 The numerical results for   are shown in Table II.

Table II: xE (Mean) and sE (standard deviation)  for k = 500 (iteration).

m xE sE

 

 

7. Conclusion

 In presented approach we obtained operational matrices of TFs to solve N-SDE. The properties of the TFs are used to convert the N-SDE to a system of linear algebraic equations. This presented approach reduces cost of computations due to properties of TFs. Also this approach is applied easily to solve N-SDE. Presented examples show good accuracy of this approach.

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