Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

: The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufﬁcient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened H¨older condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We ﬁrst show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.


Introduction
In the study of natural science systems, we assume that the system being researched is governed by the causes and results of the principle. A more realistic model would include some of the past and present values, but that involves derivatives with delays as well as the function of the system. These equations have historically been referred to as neutral stochastic functional differential equations, or neutral stochastic differential delay equations [1][2][3][4][5][6].
This kind of probability differential equation is not easy to obtain the solution, but often arises from the study of more than one simple electrodynamic or oscillating system with some interconnection. We ca not ignore the effect of the science systems with time delay. For example, when studying the collision problem in electrodynamics, Driver [7] considered the system of neutral type: z(t) = f 1 (z(t), z(δ(t))) + f 2 (z(t), z(δ(t)))ż(δ(t)), where δ(t) ≤ t. Generally, a neutral functional differential equation has the form Taking into account stochastic perturbations, we are led to a neutral stochastic functional differential equation d[z(t) − D(z t )] = f (z t , t)dt + g(z t , t)dB(t) Neutral stochastic functional differential equations (NSDEs) have been used to model problems in several areas of science and engineering. For instance, in 2007, Mao [5] published stochastic differential equations and applications. After that, the study of the existence and uniqueness theorem for stochastic

Preliminary and Basic Lemmas
The symbol | · | represents the Euclidean norm in R n . If X is a random variable and is integrable with respect to the measure P, then the integral E X is called the expectation of X. The transpose of vector or matrix A is marked as Let t 0 be a positive constant and (Ω, F , P) be a complete probability space with a filtration {F t } t≥t 0 satisfying the usual conditions (i.e., it is right continuous and F t 0 contains all P-null sets) throughout this paper unless otherwise specified.
An m-dimensional Brownian motion defined on complete probability space is denoted by B(t), that is, With the above preparations, consider the following d-dimensional neutral SFDEs: The initial value of the system (1), is a F t 0 -measurable, BC((−∞, 0]; R d )value random variable such that ξ ∈ M 2 ((−∞, 0]; R d ).

X(t) is called a unique solution if any other solution x(t) is indistinguishable with X(t), that is,
The following lemmas are known as special names for the integrals that appeared in [1,5,16] and play an important role in the next section. Lemma 1 ((Stachurska's inequality) ( [16])). Let u(t) and k(t) be nonnegative continuous functions for t ≥ α, and let u(t) ≤ a(t) + b(t) t α k(s)u p (s)ds, t ∈ J = [α, β), where a/b is a nondecreasing function and 0 < p < 1. Then, .

Results
To obtain main results of the solution to Equation (1), we impose following assumptions: where 0 < α ≤ 1, and κ(·) is a concave non-decreasing function from R + to R + such that

Hypothesis 3.
Assuming there exists a positive number K 2 such that 0 < K 2 < 1 and for any ϕ, To demonstrate the generality of our results, we provide an illustration using a concave function κ(·). Let K > 0 and let δ ∈ (0, 1) be sufficiently small. Define They are all concave nondecreasing functions satisfying κ i (u) > 0, for u > 0 and 0+ In particular, the condition in Bae et al. [8] is a special case of our proposed condition (4).
Since our goal was to demonstrate the existence and uniqueness theorem of the solution of the neutral stochastic differential Equation (1) under sufficient conditions, we start with following useful lemmas:
If κ(·) is concave and κ(0) = 0, we can find the positive constants a and b such that κ(u) ≤ a + bu for all u ≥ 0. So, we obtain: . Substituting this into (7) yields: Noting that E sup −∞<s≤t |X n (s)| 2 ≤ E||ξ|| 2 + E sup t 0 ≤s≤t |X n (s)| 2 , we see that: Letting n → ∞ implies the following inequality: We obtain the required inequality. Now, we provide the uniqueness theorem to the solution of Equation (1) with initial data (2).
To obtain the existence of solutions to neutral SFDEs, define X 0 t 0 = ξ and X 0 (t) = ξ(0) for t 0 ≤ t ≤ T. For each n = 1, 2, · · · , set X n t 0 = ξ and, by the Picard iterations, define: for t 0 ≤ t ≤ T. Since our goal was to find the conditions that guarantee the existence of the solution to Equation (1), we start with following useful lemma: Lemma 10. Let the assumptions (4)-(6) hold. Let X n (t) be the Picard iteration defined by (9). Then, where Proof . X 0 (t) ∈ M 2 (−∞, T]; R d . It is easy to find that X n (t) ∈ M 2 ([t 0 , T]; R d ). Note that: where: It follows from Lemma 7 that: Taking the expectation on both sides, we have: We have: Combining (11) and (12), we obtain: Taking the maximum on both sides: Using the elementary inequality (∑ y i ) p ≤ n p−1 ∑ y p i , when p ≥ 1, we have: By Hölder's inequality and the moment inequality, we have: |g(X n−1 s , s) − g(0, s) + g(0, s)| 2 ds.
Now we outline the existence theorem to the solution of Equation (1) with initial data (2) using approximate solutions by means of Picard sequence (9).
growth condition (5), and a contractive condition (6) were used to demonstrate that the probability process is bounded. In Lemma 10, these conditions were used to demonstrate that the Picard iteration is bounded. Therefore, in Theorems 1 and 2, we have proved a existence and uniqueness of a solution to a neural stochastic differential equation in this paper. However, the weakened Hölder condition condition only guarantees the existence and uniqueness of the solution and, in general, the solution does not have an explicit expression except for the linear case. In practice, we therefore often seek the approximate rather than the accurate solution. The questions of continuity and approximate solution (for numerical methods) under a weaker condition of the solution were not addressed in this paper, but we think it may take some time to accomplish this. We want to leave this improvement as an open problem.

Conclusions
In the present paper, we proved a type of existence and uniqueness theorem of a solution of the neutral stochastic differential equation using the weakened conditions when the conditions are in the form of (4)- (6). Our main result does not cover the more general case of existence and uniqueness of the stochastic equation under some weakened conditions. Nevertheless, it is valuable that we showed a type of existence theorem of the solution of the stochastic differential equation with the expanded concept of ordinary differential equations.