Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion

The theory of stochastic differential equations driven by a fractional Brownian motion (fBm) has been studied intensively in recent years [1], [2], [3], [4], and [5]. The fBm received much attention because of its huge range of potential applications in several fields like telecommunications, networks, finance markets, biology and so on [6], [7], [8], [9]. Moreover, one of the simplest stochastic processes that is Gaussian, self-similar, and has stationary increments is fBm [10]. In particular, fBm is a generalization of the classical Brownian motion, which depends on a parameter


Introduction
The theory of stochastic differential equations driven by a fractional Brownian motion (fBm) has been studied intensively in recent years [1], [2], [3], [4], and [5]. The fBm received much attention because of its huge range of potential applications in several fields like telecommunications, networks, finance markets, biology and so on [6], [7], [8], [9]. Moreover, one of the simplest stochastic processes that is Gaussian, self-similar, and has stationary increments is fBm [10]. In particular, fBm is a generalization of the classical Brownian motion, which depends on a parameter called the Hurst index [9]. It should be mentioned that when , the stochastic process is a standard Brownian motion; when , it behaves completely in a different way than the standard Brownian motion, in particular neither is a semimartingale nor a Markov process. It is a self-similar process with stationary increments and has a long-memory when . These significant properties make fBm a natural candidate as a model for noise in a wide variety of physical phenomena such as mathematical finance, communication networks, hydrology and medicine. The existence and uniqueness of mild solutions for a class of stochastic differential equations in a Hilbert space with a standard, cylindrical fBm with the Hurst parameter in the interval has been studied [11]. Maslowski and Nualart [12] have studied the existence and uniqueness of a mild solution for nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fBm under some regularity and boundedness conditions on the coefficients. Recently, Caraballo and et al [13] investigated the existence and uniqueness of mild solutions to stochastic delay equations driven by fBm with Hurst parameter . An existence and uniqueness result of mild solutions for a class of neutral stochastic differential equation with finite delay, driven by an fBm in a Hilbert space has been investigated [14] in Boufoussi and Hajji. The asymptotic behavior of solutions for stochastic differential equations with fBm has only been investigated by a few authors [13], [14], [15]. Moreover Nguyen [16] has studied the asymptotic behaviors of mild solutions to neutral stochastic differential equations driven by an fBm. Motivated by this consideration in this paper we investigate the existence and uniqueness and asymptotic behaviors of mild solutions for a neutral stochastic differential equation with finite or infinite delays driven by fBm in the following form ; (1.1) where A is an infinitesimal generator of an analytic semigroup of bounded linear operators, in a Hilbert space with norm , denotes an fBm with on a real and separable Hilbert space , are continuous, , , the initial data the space of all continuous functions from to and has finite second moments. Further we assume W and are independent. The main tool of this paper is the fixed point theory which was proposed by Burton [17].

Preliminaries
In this section we first recall the fBm as well as the Wiener integral with respect to it. We also establish some important results which will be needed throughout the paper. Let be a complete probability space and be an arbitrary fixed horizon. Let  We will denote by the reproducing kernel Hilbert space of the fBm. Infact is the closure of set of indicator functions with respect to the scalar product .
The mapping can be extended to an isometry between and the first Wiener chaos and we will denote by by the previous isometry. We recall for their scalar product in is given by For the deterministic function the fractional Weiner integral of with respect to is defined by , where Let be a sequence of two-sided one dimensional standard fBm mutually independent on . When one consider the following series where is a orthonormal basis in , this series does not necessarily converge in the space .Thus we consider a -valued stochastic process given formally by the following series If is a non-negative self-adjoint trace class operator, then this series converges in the space , that is, it holds that then we say that the above is a -valued -cylindrical fBm with covariance operator . For example if is a bounded sequence of non-negative real numbers such that assuming that is a nuclear operator in (ie., ) then the stochastic process is well defined as a -valued -cylindrical fBm. where are real, independent fBm's. This process is a -valued Gaussian, it starts from 0, zero mean and covariance For all and

Existence and uniqueness of a solution
In this section we study the existence and uniqueness of mild solutions for equation (1.1). For this equation we assume that the following conditions hold. is the infinitesimal generator of an analytic semigroup, , of bounded linear operators on . Further we suppose that and that and For some constants and every There exists a positive constants , and such that, for all , There exists a constant such that the function g is -valued and satisfies for all , such that i) ii) iii) The constants and .This procedure can be repeated in order to extend the solution to the entire intervalin finitely many steps. This completes the proof.

Stability analysis
In this section we establish the results for the case of finite delays, then the case of infinite delays can be proved. Our method is based on the contraction mapping principle. In order to prove the required results, we assume the following additional conditions.
The semigroup satisfies such that The function satisfies .
We first consider the case of finite delays, It is to check that is a Banach space endowed with a norm . Without loss of generality, we assume that . We denote the operator on by and To get desired results, it is enough to show that operator has a unique fixed point in . For this purpose, we use the contraction mapping principle.
Step 1. We first verify that . Thus we obtain which proves that and hence, Once again, by applying the Banach fixed point principle, we complete the proof of the theorem.
To illustrate the obtained theory, let us end this section with an example.

Example
Consider the following neutral stochastic partial differential equations with delays driven by a fBm in the following form

Conclusion
In this paper we derived existence, conditions ensuring the exponential decay to zero in mean square of a neutral stochastic delay differential equations driven by fBm. In addition we also established the case of infinite delays which has not yet been discussed in the context of neutral stochastic delay differential equations driven by fBm.