By using successive approximation, we prove existence and uniqueness result for a class of nonlinear stochastic differential equations. Moreover, it is shown that the solution of such equations is a diffusion process and its diffusion coefficients are found.

1. 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 [15] 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 [69]. In the paper [9] the author considered a stochastic equation without delay. In the monograph of Kolmanovskiĭ and Shaĭkhet [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.

Statement of the Problem. Let us consider a nonlinear stochastic differential equation: which is not solved for the stochastic differential.

Here , and   are -dimensional vector functions; is a   matrix and is an -dimensional Wiener process with independent components. Let be a random vector in , such that . Assume that does not depend on , .

Definition 1. An -dimensional process is said to be the solution of (1) if   and holds for every with probability .
A solution is said to be unique if for any continuous solutions such that one has for all  .
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.

2. Main Results

Firstly we prove the theorem of existence and uniqueness of the solution.

Theorem 2. Assume that   is a continuous function; are measurable functions for and satisfy the following conditions:(1) there exists a constant  , such that for ;(2) there exist constants and such that andfor , , .

If , then there exists a unique continuous solution of (1) with probability for all . Moreover it has a bounded second moment such that   for all .

Proof. To find a solution of the integral equation (2), we use the method of successive approximations. We start by choosing an initial approximation .
At the next step, Consequently, Consider an interval such that . Note that it can be always achieved by choosing a sufficiently small and by the conditions on .
We prove that a solution exists on this interval.
From (4) it follows that where is a constant independent of , , and .
Next, estimate Therefore, The estimation of the stochastic integral follows from its properties. Similar to [11, p.20], we have Thus, we have Since , it then follows that the series converges.
This implies the uniform convergence with probability of the series on the interval . Its sum is . So converges to some random process. Every is continuous with probability . Whence it follows that the limit is continuous with probability , too.
Then prove uniqueness of this continuous solution. Assume that there exists a second continuous solution of (1). Denote  by the random variable which equals if ,   and it equals otherwise. Then Therefore, The last inequality implies the estimation By the Gronwall-Bellman inequality and the above inequality together, we have So, Probabilities at the right-hand side of the above inequality approach to zero as , because   and are continuous with probability . Therefore and are stochastic equivalent.
We conclude that Hence, the existence and uniqueness of the solution are proved on .
Next, we show boundedness of a second moment of the solution. Denote again by the indicator of the set Then Similarly, we have 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 , then it follows from [8] that   for every .
Thus existence, uniqueness, and boundedness of the second moment of the solution are proved on . Since the constant is dependent only on and , 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 , where is chosen such that the inequality .
This procedure can be repeated in order to extend the solution of (1) to the entire semiaxis . 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   with more weak condition . But by using our method, paths of obtained solution are continuous with probability . 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 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 .

Proof. As in Theorem 2, we solve (1) by the method of successive approximations. It can be shown that    is completely defined by the nonrandom initial value and the process for , which are independent of . Since is a solution of (2), it is measurable. Hence is independent of and events from . Note that, from the uniqueness of the solution for , it follows that it is a unique solution of the equation Since the process is also a solution of this equation, then with probability .
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.

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;(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.

Lemma 5. Let be a solution of (2) such that . If conditions of Theorem 2 are satisfied and is continuous for , , then an inequality holds. Here is a constant dependent only on , and .

Proof. Denote by the indicator of the set Similarly to finding estimation (24), we have where points dependence on .
The last estimate follows from Hölder’s inequality and properties of stochastic integral. From inequality (30) and condition for , we have where depends only on .
Further, the right-hand side of (31) can be estimated by where and  .
Using the lemma from [11, p.38], we obtain where depends only on , , and .
Now, assume that   and obtain estimate (28). This completes the proof of the Lemma.

Theorem 6. If conditions of Theorem 2 are satisfied and functions , , , , , and , , are continuous for , . Functions , , and satisfy Lipschitz condition with respect to in neighborhood of every point . Then the solution of (1) is diffusion process with and diffusion matrix with

Proof. Take any point from a region , . Consider a stochastic ITO equation in a closed neighborhood of this point
From the conditions of the theorem, it follows that there exists a closed neighborhood of the point  , such that coefficients of the equation are Lipschitz with respect to . Fix this neighborhood. Extend these coefficients on the all region , such that they remain continuous by both variables, Lipschitz and linear with respect to
Then the equation with these extended coefficients has a unique solution of the IVP    for  .
From ITO formula and coincidence of coefficients of (36) and (37) in this neighborhood of the point , one can show that processes and coincide in this neighborhood.
It is known that under these assumptions the process is diffusion. Its diffusion coefficients in the point are defined by coefficients of (36).
Next, show that has the same diffusion coefficients. To do this it is sufficient to estimate the following limits and use estimation (28) where is chosen such that the point is laying in this neighborhood.
We get Similarly, obtain the second limit in (38): So, the existence of limits in (38) is proved for every point and fixed , chosen for this point. It is clear that from (28), these limits exist for every and they are independent of . From the above and the definition of diffusion process [12, p.67] we can finish the proof. The theorem is proved.