A note on the solutions of neutral SFDEs with infinite delay

The main aim of this paper is to discuss the existence and uniqueness of solution of the neutral stochastic functional differential infinite delay equations under a non-Lipschitz condition and a weakened linear growth condition. Furthermore, an estimate for the error between approximate solution and accurate solution is given. MSC:60H05, 60H10.


Introduction
Stochastic differential equations (SDEs) are well known to model problems from many areas of science and engineering. For instance, in , Henderson et al. [] published the Stochastic Differential Equations in Science and Engineering, in , Mao [] published the stochastic differential equations and applications, in , Li and Fu [] considered the stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks.
In recent years, there is an increasing interest in stochastic functional differential equations (SFDEs) (see [, -], and references therein for details).
On the one hand, Kolmanovskii and Myshkis [] introduced the following neutral stochastic differential equations with finite delay: which could be used in chemical engineering and aeroelasticity. Since then, the theory of neutral SDEs has been developed by researchers (see [, , ]).
After, Ren and Xia [] derived an existence and uniqueness of the solution to the following neutral SFDEs under some Carathéodory-type conditions with Lipschitz conditions and non-Lipschitz conditions as a special case: This kind of neutral SFDE has a practical background in a collision problem in electrodynamics. The extra noise B can be regarded as some extra information, which cannot be detected in the electrodynamics systems, but is available to the particular investors. http://www.journalofinequalitiesandapplications.com/content/2013/1/181 Motivated by [], one of the objectives of this paper is to get one proof to the existence and uniqueness theorem for given neutral SFDEs, which contains an improved condition given in []. The other objective of this paper is to estimate on how fast the Picard iterations x n (t) converge the unique solution x(t) of the neutral SFDEs.

Preliminary and notations
Let | · | denote Euclidean norm in R n . If A is a vector or a matrix, its transpose is denoted by A T ; if A is a matrix, its trace norm is represented by |A| = trace(A T A). Let t  be a positive constant and ( , F, P) throughout this paper unless otherwise specified, be a complete probability space with a filtration {F t } t≥t  satisfying the usual conditions (i.e. it is right continuous and F t  contains all P-null sets). Assume that B(t) is a m-dimensional Brownian motion defined on complete probability space, that is B(t) = (B  (t), B  (t), . . . , B m (t)) T . We consider the following spaces: With all the above preparation, consider the following d-dimensional neutral SFDEs: be Borel measurable, and be continuous. Next, we give the initial value of (.) as follows: To be more precise, we give the definition of the solution of the equation (.) with initial data (.).
Definition . R d -value stochastic process x(t) defined on -∞ < t ≤ T is called the solution of (.) with initial data (.), if x(t) has the following properties: http://www.journalofinequalitiesandapplications.com/content/2013/1/181 The following lemma is known as the moment inequality for stochastic integrals which was established by Mao [] and will play an important role in next section.
In order to attain the solution of (.) with initial value (.), we propose the following assumptions: (H) (a) There exists a function (t, u) is locally integrable in t for each fixed u ∈ R + and is continuous, nondecreasing, and concave in u for each fixed t ≥ t  . Moreover, (t, ) =  and if a nonnegative continuous function where D >  is a positive constant, then Z(t) =  for all t  ≤ t ≤ T. (c) For any constant K > , the deterministic ordinary differential equation has a global solution for any initial value u  . (H) For any t ∈ [t  , T], it follows that f (t, ), g(t, ) ∈ L  such that where K is a positive constant. http://www.journalofinequalitiesandapplications.com/content/2013/1/181 (H) Assuming that there exists a positive number K  such that K  < α  ( < α < ) and for any ϕ, ψ ∈ BC((-∞, ]; R d ) and t ∈ [t  , T], it follows that

Existence and uniqueness of the solution
Now we give the existence and uniqueness theorem to (.) with initial value (.) under the Carathéodory-type conditions. In order to obtain the existence of solutions to neutral SFDEs, let x  t  = ξ and x  (t) = ξ (), for t  ≤ t ≤ T. For each n = , , . . . , set x n t  = ξ and define the following Picard sequence We prepare a lemma in order to prove this theorem.

Lemma . Let the assumption (H) and (H) hold. If x(t) is a solution of equation (.)
with initial data (.), then

In particular, x(t) belong to M  ((-∞, T]; R d ).
Proof For each number n ≥ , define the stopping time Obviously, as n → ∞, τ n ↑ T a.s. Let x n (t) = x(t ∧ τ n ), t ∈ (-∞, T]. Then, for t  ≤ t ≤ T, x n (t) satisfy the following equation: Applying the elementary inequality (a + b)  ≤ a  α + b  -α when a, b > ,  < α < , we have where condition (H) has also been used. Taking the expectation on both sides, one sees that Noting that sup -∞<s≤t |x n (s)|  ≤ ξ  + sup t  ≤s≤t |x n (s)|  , we get Consequently, On the other hand, by Hölder's inequality, Lemma ., and the condition (H), one can show that . Assumption (c) indicates that there is a solution u t satisfies that Since E ξ  < ∞, for all n = , , , . . . , we deduce that x n (s)  ≤ u t ≤ u T < ∞. http://www.journalofinequalitiesandapplications.com/content/2013/1/181 Letting t = T, it then follows that Thus, Consequently, the required result follows by letting n → ∞.
Proof of Theorem . To check the uniqueness, let x(t) and x(t) be any two solutions of (.), by Lemma ., Note that One then gets where  < α < . We derive that Therefore, Consequently, On the other hand, one can show that This yields that Hence, By assumption (b), we get Z(t) = . This implies that

Applying the elementary inequality
where condition (H) has also been used. Taking the expectation on both sides, one sees that On the other hand, by Hölder's inequality and the conditions, one can show that where γ  = E ξ  + γ  K(Tt  ), γ  = (Tt  + ). Substituting this into (.) yields that as n → ∞. Hence, taking limits on both sides in the Picard sequence, we obtain that g(s, x s ) dB(s).
The above expression demonstrates that x(t) is a solution of equation (.) satisfying the initial condition (.). So far, the existence of theorem is complete.
In the proof of Theorem ., we have shown that the Picard iterations x n (t) converge to the unique solution x(t) of equation (.). The following theorem gives an estimate on the difference between x n (t) and x(t) under some special condition, and it clearly shows that one can use the Picard iteration procedure to obtain the approximate solutions to equations (.).