All submissions of the EM system will be redirected to **Online Manuscript Submission System**. Authors are requested to submit articles directly to **Online Manuscript Submission System** of respective journal.

^{1}Faculty of Science, Alexandria University, Egypt

^{2}Faculty of Science, Damanhour University, Egypt

- Corresponding Author:
- El-Sayed AMA

Faculty of Science Alexandria University, Egypt

**E-mail:**[email protected]

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

**Visit for more related articles at** Research & Reviews: Journal of Statistics and Mathematical Sciences

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 X_{0} and the (non-random data) coefficients of the nonlocal condition ak. Also, a stochastic differential equation with the integral condition will be considered.

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

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 (t_{1}) W (t_{2})] is a Gaussian random variable for all t_{1}, t_{2} ∈ 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

(2)

where X_{0} is a second order random variable independent of the Brownian motion W (t) and a_{k} are positive real integers. The existence of a unique mean square solution will be studied. The continuous dependence on the random data X_{0} and the nonrandom data a_{k} will be established. The problem (1) with the integral condition will be considered.

(3)

Let C = C(I, L_{2}( Ω)) be the class of all mean square continuous second order stochastic process with the norm

Throughout the paper we assume that the following assumptions hold

**(H1)** The function is mean square continuous.

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

such that the function f satisfies the mean square Lipschitz condition

**(H3)** There exists a positive real number m_{1} such that

Now we have the following lemmas.

This completes the proof.

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

(4)

where

**Proof.** Integrating equation (1), we obtain

and

then

and

then

Hence

Now define the mapping

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

From assumption (ii) we have

then we have

So,

using assumptions and lemma 2.1, we get

which proves that F : C → C.

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

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)

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

**Definition 4.1** The solution X ∈ C of the nonlocal problem (1)-(2) is continuously dependent (on the data X_{0}) if such that implies that

Here, we study the continuous dependence (on the random data X_{0}) 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 X_{0}.

**Proof.** Let

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

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

Using our assumptions, we get

then

This completes the proof.

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

**Definition 4.2** The solution X ∈ C of the nonlocal problem (1)-(2) is continuously dependent (on the coefficient a_{k} of the nonlocal condition) if such that implies that

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 a_{k} of the nonlocal condition.

**Proof. Let**

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

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

and

and

Then

Using our assumptions we get

then

Hence

This completes the proof.

Let a_{k} = v(t_{k}) − v(t_{k} −1), τ_{k} ∈ (t_{k}−1, t_{k}), where (0 < t1 < t2 < t3 < …< T).

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

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

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

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

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

- Boucherif AA. First-order differential inclusions with nonlocal initial conditions. Appl MathLett.2002;15:409-414.
- Boucherif A and Precup R. On the nonlocal initial value problem for first order differential equations. FixedPoint Theory. 2003;4:205-212.
- Byszewski L and Lakshmikantham V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract cauchy problem in a banach space. Applicable Anal.1991;40:11-19.
- El-Sayed AMA, et al. Uniformly stable solution of a nonlocal problem of coupled system of differential equations.Differ Equ Appl. 2013;5:355-365.
- El-Sayed AMA, et al. Existence of solution of a coupled system of differential equation with nonlocal conditions.Malaya J Math. 2014;2:345-351.
- El-Sayed AMA and Ameen I. Continuation of a parameterized impulsive differential equation to an internal nonlocal cauchy problem. Alexandria J Math. 2011.
- El-Sayed AMA and Bin-Tahir EO. An arbitrary fractional order differential equation with internal nonlocal and integral conditions.Advances in Pure Mathematics. 2011;1:59-62.
- Adomian G. Stochastic System, Academic Press, 1983.
- Bharucha-TeidAT. Fixed point theorems in probabilistic analysis. Bull Amer Math Soc. 1976;82:641-657.
- El-Tawil MA. On the application of mean square calculus for solving random differential equations. ElectronicJ Math Anal Appl.2013;1:202-211.
- El-Tawil MA and Sohaly MA. Mean square numerical methods for initial value random differential equations. Open J Discrete Math.2011;1:66-84.
- Itoh S. Random fixed point theorems with an application to random differential equations in banach spaces.J Math Anal Appl.1979;6:261-273.
- PhilipseAP. Notes on Brownian motion. Utrecht University Debye Institute Van t Ho Laboratory, 2011.
- Platen E.An introduction to numerical methods for stochastic differential equations, ActaNumerica. 1999;8:195244.
- Soong TT. Random differential equations in science and engineering. MathSci Eng. 103:1973.
- Zhu W, et al. Exponential stability of stochastic differential equation with mixed delay. JAppl Math. 2014;2014:1-11.