Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations

and Applied Analysis 3 Thus, we have E sup t∈[t0 ,T] 󵄨󵄨󵄨󵄨xn+1 (t) − xn (t) 󵄨󵄨󵄨󵄨 2 ≤ 3(L 2 1 E sup t∈[t0,T] 󵄨󵄨󵄨󵄨xn (t) − xn−1 (t) 󵄨󵄨󵄨󵄨 2 + (T − t0) 2 L 2 E sup t∈[t0 ,T] 󵄨󵄨󵄨󵄨xn (t) − xn−1 (t) 󵄨󵄨󵄨󵄨 2


Introduction
Differential equations, which are not solved for the derivative, have found diverse applications in many fields.Examples of equations of this type are Lagrange equations of classical mechanics or Euler equations.
Consideration of real objects under the influence of random factors leads to nonlinear stochastic differential equations, which are not solved for stochastic differential.Such equations were introduced by Kolmanovskii and Nosov in [1] for construction of stochastic analogues of neutral functional differential equations.Works [1][2][3][4][5] were devoted to the problems of existence, uniqueness, and properties of solutions of neutral stochastic differential (delay) equations in finite dimensional spaces.Existence of solutions for such equations in Hilbert spaces was studied in papers [6][7][8][9].In the paper [9] the author considered a stochastic equation without delay.In the monograph of Kolmanovskiȋ and Shaȋkhet [10] conditions were obtained for optimality in control problems for these equations.
In this paper we will study the existence and uniqueness of solutions for a class of nonlinear stochastic differential equations, which are not solved for the stochastic differential.
In this work we use the method of successive approximations to establish existence and uniqueness (pathwise) of the solution of (1).We study its probability properties.We prove that () is diffusion process and find coefficients of diffusion.

Main Results
Firstly we prove the theorem of existence and uniqueness of the solution.for ,   ∈   ,  ≥  0 .
Proof.To find a solution of the integral equation (2), we use the method of successive approximations.We start by choosing an initial approximation  0 () = .
We prove that a solution exists on this interval.From (4) it follows that where  1 is a constant independent of  0 , , and .] . Therefore, The estimation of the stochastic integral follows from its properties.Similar to [11, p.20], we have Abstract and Applied Analysis 3 Thus, we have Since  < 1, it then follows that the series converges.This implies the uniform convergence with probability 1 of the series on the interval [ 0 , ].Its sum is ().So   () converges to some random process.Every   () is continuous with probability 1. Whence it follows that the limit () is continuous with probability 1, too.
Then prove uniqueness of this continuous solution.Assume that there exists a second continuous solution () The last inequality implies the estimation So, Use the Gronwall-Bellman inequality to find the estimation where  is a constant independent of .Applying the Fatou lemma to the last inequality and assuming that  → ∞, we have Since || 2 ≤ ∞, then it follows from [8] that || 2 < ∞ for every  ∈ [ 0 , ].Thus existence, uniqueness, and boundedness of the second moment of the solution () are proved on [ 0 , ].Since the constant  is dependent only on  −  0 and || 2 < ∞, and () is independent of ()−() for  ⩾ , then by similar manner we can prove the existence and uniqueness of the solution of the IVP with initial conditions (, ()) on the interval [,  1 ], where  1 is chosen such that the inequality This procedure can be repeated in order to extend the solution of (1) to the entire semiaxis  ⩾  0 .The theorem is proved.
Notes.The existence and uniqueness of the solution can be obtained as corollary from work [1], where this result was proved for an SDE of neutral type by replacing a condition for Lipschitz constant  1 < (1/6) 3/4 with more weak condition  1 < 1.But by using our method, paths of obtained solution are continuous with probability 1.Otherwise, in the pointed work only the measurability of the solution and boundedness of its second moment were stated.Now, we state some probability properties of the solution obtained in Theorem 2. We prove that under assumptions of Theorem 2, the solution of ( 1) is a Markov random process.Moreover, if the coefficients are continuous then it is a diffusion process.We will find its diffusion coefficients.Theorem 3.Under conditions (1)-( 2) of Theorem 2 the solution () of (1) is a Markov process with a transition probability defined by where  , () is a solution of that equation where  ⩾  ⩾  0 ,  ∈ Since the process  ,() () is also a solution of this equation, then () =  ,() () with probability 1.
As for the rest, the proof is the same as the proof of the theorem for ordinary stochastic equations [11].The theorem is proved.
We have a corollary from this theorem.(2) if functions , , and  are periodic functions with period , then a transition probability is periodic function; that is, ( + , ,  + , ) = (, , , ).Now, we investigate conditions for which the solution of (1) is a diffusion process.For this we must find an additional estimate.

Corollary 4 .
Suppose that conditions of Theorem 2 are satisfied.Then(1) if functions , , and  are independent of , then the solution () is a homogeneous Markov process;