Continuous Dependence of the Solution of A Stochastic Differential Equation With Nonlocal Conditions

El-Sayed AMA1*, Abd-El-Rahman RO2, El-Gendy M2

1Faculty of Science, Alexandria University, Egypt

2Faculty of Science, Damanhour University, Egypt

Corresponding Author:
El-Sayed AMA
Faculty of Science Alexandria University, Egypt
E-mail: amasyed@yahoo.com

Received date: 21/01/2016; Accepted date: 14/04/2016; Published date: 18/04/2016

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Abstract

In this paper we are concerned with a nonlocal problem of a stochastic differential equation that contains a Brownian motion. The solution contains both of mean square Riemann and mean square Riemann-Steltjes integrals, so we study an existence theorem for unique mean square continuous solution and its continuous dependence of the random data X0 and the (non-random data) coefficients of the nonlocal condition ak. Also, a stochastic differential equation with the integral condition will be considered.

Keywords

Integral condition, Brownian motion, Unique mean Square solution, Continuous dependence, Random data, Non- Random data, Integral condition

Introduction

Many authors in the last decades studied nonlocal problems of ordinary differential equations, the reader is referred to [1-7], and references therein. Also the theory of stochastic differential equations, random fixed point theory, existence of solutions of stochastic differential equations by using successive approximation method and properties of these solutions have been extensively studied by several authors, especially those contain the Brownian motion as a formal derivative of the Gausian white noise, the Brownian motion W (t), t ∈ R, is defined as a stochastic process such that

W (0) = 0; E(W (t)) = 0, E(W (t))2 = t

and [W (t1) W (t2)] is a Gaussian random variable for all t1, t2 ∈ R. The reader is referred to [8,9] and [10-16] and references therein.

Here we are concerned with the stochastic differential equation

dX(t) = f (t, X(t))dt + g(t)dW (t), t ∈ (0, T ] (1)

with the nonlocal random initial condition

Equation (2)

where X0 is a second order random variable independent of the Brownian motion W (t) and ak are positive real integers. The existence of a unique mean square solution will be studied. The continuous dependence on the random data X0 and the nonrandom data ak will be established. The problem (1) with the integral condition will be considered.

Equation (3)

Integral Representation

Let C = C(I, L2( Ω)) be the class of all mean square continuous second order stochastic process with the norm

Equation

Throughout the paper we assume that the following assumptions hold

(H1) The function Equationis mean square continuous.

(H2) There exists an integrable function k : [0, T ] → R+ , where

Equation

such that the function f satisfies the mean square Lipschitz condition

Equation

(H3) There exists a positive real number m1 such that

Equation

Now we have the following lemmas.

Equation

Proof.

Equation

This completes the proof.

Lemma 2.2: The solution of the problem (1) and (2) can be expressed by the integral equation

Equation (4)

where

Equation

Proof. Integrating equation (1), we obtain

Equation

and

Equation

then

Equation

Equation

and

Equation

then

Equation

Hence

Equation

Equation

Now define the mapping

Equation

Then we can prove the following lemma.

Lemma 2.3 F : C → C.

Proof.Let 1 2 X ∈C, t , t ∈ [0, T] such that 2 1 t -t <δ , then

Equation

From assumption (ii) we have

Equation

then we have

Equation

So,

Equation

using assumptions and lemma 2.1, we get

Equation

which proves that F : C → C.

Existence and Uniqueness

For the existence of a unique continuous solution X ∈ C of the problem (1)-(2), we have the following theorem.

Theorem 3.1 Let the assumptions (H1)−(H3) be satisfied. If 2m < 1, then the problem(1)-(2) has a unique solution X ∈ C.

Proof. Let X and X* ∈ C, then

Equation

Hence

If 2m < 1 , then F is contraction and there exists a unique solution X ∈ C of the nonlocal stochastic problem (1)-(2), [2]. This solution is given by (4)

Equation

Continuous Dependence

Consider the stochastic differential equation (1) with the nonlocal condition

Equation

Definition 4.1 The solution X ∈ C of the nonlocal problem (1)-(2) is continuously dependent (on the data X0) ifEquation such that Equation implies that Equation

Here, we study the continuous dependence (on the random data X0) of the solution of the stochastic differential equation (1) and (2).

Theorem 4.2 Let the assumptions (H1) − (H3) be satisfied. Then the solution of the nonlocal problem (1)-(2) is continuously dependent on the random data X0.

Proof. Let

Equation

be the solution of the nonlocal problem (1)-(2) and

Equation

be the solution of the nonlocal problem (1) and (6). Then

Equation

Using our assumptions, we get

Equation

then

Equation

This completes the proof.

Now consider the stochastic differential equation (1) with the nonlocal condition

Equation

Definition 4.2 The solution X ∈ C of the nonlocal problem (1)-(2) is continuously dependent (on the coefficient ak of the nonlocal condition) if Equation such that Equation implies that Equation

Here, we study the continuous dependence (on the random data X0) of the solution of the stochastic differential equation (1) and (2).

Theorem 4.3 Let the assumptions (H1) − (H3) be satisfied. Then the solution of the nonlocal problem (1)-(2) is continuously dependent on the coefficient ak of the nonlocal condition.

Proof. Let

Equation

be the solution of the nonlocal problem (1)-(2) and

Equation

be the solution of the nonlocal problem (1) and (7). Then

Equation

Equation

and

Equation

and

Equation

Then

Equation

Using our assumptions we get

Equation

then

Equation

Hence

Equation

This completes the proof.

Non Local Integral Condition

Let ak = v(tk) − v(tk −1), τk ∈ (tk−1, tk), where (0 < t1 < t2 < t3 < …< T).

Then, the nonlocal condition (2) will be in the form

Equation

From the mean square continuity of the solution of the nonlocal problem (1)-(2), we obtain from [15]

Equation

that is, the nonlocal conditions (2) is transformed to the mean square Riemann-Steltjes integral condition

Equation

Now, we have the following theorem.

Theorem 5.4 Let the assumptions (H1)-(H3) be satisfied, then the stochastic differential equation (1) with the nonlocal integral condition (3) has a unique mean square continuous solution represented in the form

Equation

Proof. Taking the limit of equation (4) we get the proof.

References