\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 194, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/194\hfil Theorems on boundedness of solutions]
{Theorems on boundedness of solutions to stochastic
 delay differential equations}

\author[Y. N. Raffoul, D. Ren \hfil EJDE-2016/194\hfilneg]
{Youssef N. Raffoul, Dan Ren}

\address{Youssef N. Raffoul \newline
Department of Mathematics,
University of Dayton,
Dayton, OH 45469-2316, USA}
\email{yraffoul1@udayton.edu}

\address{Dan Ren \newline 
Department of Mathematics,
University of Dayton,
Dayton, OH 45469-2316, USA}
\email{dren01@udayton.edu}

\thanks{Submitted March 18, 2016. Published July 18, 2016.}
\subjclass[2010]{34F05, 60H10, 65Q10}
\keywords{Stochastically bounded; bounded in probability;  Lyapunov functional;
\hfill\break\indent  inequalities; stochastic Volterra integro-differential system}

\begin{abstract}
 In this report, we provide general theorems about boundedness or  bounded
 in probability of solutions to nonlinear delay stochastic differential systems.
 Our analysis is based on the successful construction of suitable Lyapunov
 functionals. We offer several examples as application of our theorems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Stochastic differential delay equations (SDDEs) have a wide application 
in natural or man-made systems. They arise from an approximation to a 
partial differential equations that describes, e.g., diffusion on some 
reacting substance or a traveling wave in some medium, see \cite{Ashok}. 
Moreover, time delay stochastic systems are an important aspect in the 
modeling of genetic regulation due to slow biochemical reactions such as 
gene transcription and translation, and protein diffusion between the cytosol 
and nucleus, for a reference, see \cite{Parmar}. 
Recently, (SDDEs) have been extensively used in the study of population 
dynamics and for more on this we refer to \cite{Gard1} and \cite{Gard2}. 
For more reading on the subject of stochastic systems, we refer to  
Kushner \cite{Kushner}, Mao \cite{Mao-01,Mao-02}, Hasminskii \cite{Hasminskii} 
and the references therein. The report of \cite{Thygesen} presents an 
interesting survey of Lyapunov functions techniques in stochastic differential 
equation.

Let $B(t)=(B_1(t),B_2(t), \ldots , B_m(t))^T$ be an $m$-dimensional 
standard Brownian motion defined on a complete probability space 
$(\Omega, \mathbf{F}, \mathbf{P} )$ and
$$
x(t)=(x_1(t), x_2(t), \ldots , x_n(t))^T \in \mathbb{R}^n.
$$
Let $h$ be a positive constant and we will consider the $n$-dimensional stochastic 
differential equation (SDE):
\begin{equation}\label{1.1}
\begin{aligned}
dx(t)&=f\Big(t, x(t), x(t-h), \int_{t-h}^t  A(t, s) h(s, x(s)) ds \Big)dt \\
&\quad + g(t, x(t), x(t-h))dB(t),
\end{aligned}
\end{equation}
for  $t\in\mathbb{R}^+$,
with a given deterministic continuous  initial condition 
$\phi: [-h, 0] \to \mathbb{R}^n $, where
$A: \mathbb{R}^+\times \mathbb{R}^+ \to \mathbb{R}$,
$f: \mathbb{R}^+ \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}   \to \mathbb{R}^n $,
$g: \mathbb{R}^+ \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^ {n\times m}$ and
$h: \mathbb{R}^+ \times \mathbb{R}^n \to \mathbb{R}$, are all continuous functions on their
own domains.

It should cause no difficulty to start our solution at any time $t_0$, but 
for simplicity, we consider the case when $t_0 = 0$ in this paper, i.e., 
$x(t) = \phi(t)$ for all $t\in[-h, 0]$, and $x$ satisfies the SDE \eqref{1.1} 
starting from time 0.

Let  $C^{1,2}(\mathbb{R}^+ \times \mathbb{R}^n; \mathbb{R}^+)$ 
denote the set of all non-negative  functions that are twice continuously 
differentiable in the first variable and continuously differentiable 
in the second variable. Take an arbitrary function $V(t, x)$ in the set, 
and define the operator 
$\mathcal{L}V: \mathbb{R}^+ \times \mathbb{R}^n \times 
\mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ as
\begin{equation}\label{1.3}
\begin{aligned}
\mathcal{L} V(t, x, y, z ) 
&=  V_t(t, x) + \sum_{i=1}^n V_{x_i}(t, x)f_i(t, x, y, z) \\
&\quad + \frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n  \sum_{k=1}^m
 V_{x_ix_j}(t, x) g_{ik}(t, x, y)g_{jk}(t, x, y),
\end{aligned}
\end{equation}
where $f_i$ is the $i$th component of vector $f$ and $g_{ij}$ is the $ij$-entry 
of matrix $g$.

To avoid confusion, we will denote $\mathcal{L} V(t, x, y, z)$, the operator
 applied to $V(t, x)$, as $\mathcal{L}V(t, x)$. However, readers should 
keep in mind that $\mathcal{L}V(t, x)$ actually depends on $y$ and 
$z$ through $f$ and $g$.

In this article we say Lyapunov functional instead of Lyapunov function 
to indicate the presence of the variable $x$ in the integrand.

 This research is an extension of \cite{R2}, in which the authors developed 
a general theory for the analysis of solutions of the nonlinear systems 
of stochastic differential equation of the form
\begin{equation*}
dx(t) = f(t, x(t)) dt +  g(t, x(t))dB(t), \quad t\geq 0   .
\end{equation*}
with initial condition $x(0)$ being a constant.

In our analysis, we obtain inequalities from which we can deduce boundedness 
in probability of the solutions of (SDDEs) \eqref{1.1}.
In obtaining our inequalities we resort to the suitable construction of Lyapunov 
functionals. Thus, this paper provides step by step instruction on how these 
Lyapunov functionals can be easily constructed so that under suitable conditions, 
the Lyapunov functional is decreasing along the solutions of the (SDDEs) .
In addition, our results will be applied to nonlinear stochastic systems with 
the function $f$ containing terms of the form $x^n$ where $n$ is positive and 
rational. For a comprehensive review and recent results  of stochastic differential 
and integro-differential equations, we refer the reader to the excellent monograph
\cite{mb} and to the references therein. For more on (SDDEs), we refer to 
the survey paper \cite{Ivanov}.


\section{Boundedness of solutions}

In this section, we use non-negative definite Lyapunov
functionals and establish sufficient conditions to obtain
boundedness in probability results on all solutions $x$ of \eqref{1.1}.
The use of an initial function instead of an initial point allows us to 
 observe the past performance of the random variables over a longer period.

Frist, we state some notation and assumptions which will be needed 
for the rest of the paper.

Let $E^{\phi}$ denote the conditional expectation operator associated 
with the probability measure $\mathbf{P}$ given $x(t) = \phi(t)$ for all 
$t\in [-h, 0]$, i.e., 
$E^{\phi}(\cdot) = E (\cdot \mid x(t) = \phi(t), t\in [-h, 0] )$.
Let $\|\cdot\|$ denote the Euclidean norm for a vector in $\mathbb{R}^n$.


\begin{definition} \label{def2.1} \rm
A continuous function $ W: \mathbb{R}^+ \to \mathbb{R}^+$ is called a wedge, if
$W(0)=0$, $ W(s)>0$ for $ s>0$, and $W$ is strictly increasing.
\end{definition}

In this article  a wedge is always denoted as $W$ or
$W_{i}$ where $i$ is a positive integer.
We sue the following assumption:
\begin{itemize}
\item[(A1)] For any fixed $t\in\mathbb{R}^+$, the following condition
holds for all $i \in\{1, \dots, n\}$ and $k \in\{1, \dots, m\}$:
 \begin{equation} \label{e2.2}
 E^{\phi} \Big[\int_{0}^{t} V^2_{x_i}(t, x(s))g^2_{ik}(s, x(s), x(s-h))\,ds \Big]
<\infty.
 \end{equation} 
\end{itemize}
A special case of the general condition \eqref{e2.2} is the following condition.
\begin{itemize}
\item[(A1')]  There exists a deterministic function $\nu(t)$ such that for all 
$t\in\mathbb{R}^+, x\in \mathbb{R}^n, y\in\mathbb{R}^n$, and all $i \in\{1, \dots, n\}$
and $k \in\{1, \dots, m\}$:
  \begin{equation} \label{e2.3}
  \|V_{x_i}(t, x)g_{ik}(t, x, y)\| \leq \nu(t)
  \end{equation}
and  for any fixed $t\in\mathbb{R}^+$,
\begin{equation}\label{e2.4}
\int_{0}^{t}  \nu^2(s) \,ds  <\infty.
\end{equation}
\end{itemize}

\begin{theorem} \label{thm2.2}
Under assumption {\rm (A1)},  suppose that there exists a
continuously differentiable Lyapunov functional
 $$
V(t, x)\in C^{1,2}(\mathbb{R}^+ \times \mathbb{R}^n; \mathbb{R}^+)
$$
such that for all $x\in\mathbb{R}^n, y\in\mathbb{R}^n, z\in\mathbb{R}$ and $t\in\mathbb{R}^+$:
\begin{gather}\label{e2.5}
 W (\|x\|)\leq V(t, x), \\
\label{e2.6}
\mathcal{L} V(t, x) \leq -\alpha(t)V^{q}(t, x) + F(t), \\
\label{e2.7}
V(t, x) -V^{q}(t, x) \leq \gamma,
\end{gather}
where $\gamma$ and $q$ are constants with $\gamma\geq 0$, $q \geq 1$, 
and  $\alpha(t)$ and $F(t)$ are positive continuous functions.
Then all solutions $x$ of \eqref{1.1} satisfy the following inequality for 
any given $t\in\mathbb{R}^+$:
\begin{equation}\label{e2.8}
E^{\phi}(\|x(t)\|) \leq W^{-1}\Big[e^{-\int^t_{0}\alpha(u)du} V(0, \phi(0))  
+ \int^t_{0} e^{-\int^t_{u} \alpha(s) ds} \big[\gamma \alpha(u) + F(u)\big]du\Big].
\end{equation}
\end{theorem}

\begin{proof}
Apply It\^{o}'s formula to $e^{\int_0^t \alpha(s) d s} V(t, x(t))$:
\begin{align*}
& d \left(e^{\int_0^t \alpha(s) d s} V(t, x(t))\right)  \\
&=  e^{\int_0^t \alpha(s) d s} (\alpha(t) V(t, x(t)) 
+ \mathcal{L} V(t, x(t))) d t + d M(t)
\end{align*}
with 
\[
M(t) =  \int_0^t  e^{\int_0^u \alpha(s) d s}
 \sum_{i=1}^n V_{x_i}(u, x(u))\sum_{k=1}^m g_{ik}(u, x(u), x(u-h))dB_k(u),
\]
and (for short notation)
\[
\mathcal{L} V(t, x(t))
=\mathcal{L} V\big(t, x(t), x(t-h), \int_{t-h}^t A(t, s) h(s, x(s)) ds \big).
\]
Integrate both sides from 0 to $t$:
\begin{align*}
& e^{\int_0^t \alpha(s) d s} V(t, x(t)) - V(0, x(0)) \\
&=  \int_0^t e^{\int_0^u \alpha(s) d s} (\alpha(u) V(x(u), u)
  + \mathcal{L} V(x(u), u)) d u + M(t) \\
&\leq   \int_0^t e^{\int_0^u \alpha(s) d s} (\alpha(u) (V(x(u), u) - V^{q}(x(u),u))
  + F(u) ) d u + M(t) \quad (by\ \eqref{e2.6})\\
&\leq   \int_0^t e^{\int_0^u \alpha(s) d s} (\alpha(u) \gamma + F(u) ) d u 
 + M(t) \quad (by\ \eqref{e2.7})
\end{align*}
Since $V(0, x(0)) = V(0, \phi(0))$ , it follows that
\begin{align*}
V(t, x(t)) &\leq e^{-\int_0^t \alpha(s) d s}  V( 0, \phi(0)) \\
 &\quad +  \int_0^t e^{-\int_u^t \alpha(s)\,ds} (\gamma \alpha(u) + F(u) ) d u 
+ e^{-\int_0^t \alpha(s) d s} M(t) 
\end{align*}
Taking expectation $E^{\phi}$ on both sides, and noting that
$E^{\phi}\{ e^{-\int_0^t \alpha(s) d s} M(t) \}=0$ under (A1), we obtain
\begin{equation}
E^{\phi}[ V(t, x(t))]  
\leq e^{-\int_{0}^t \alpha(s)\,ds} V( 0, \phi(0))
 + \int_{0}^t e^{-\int_{u}^t \alpha(s)\,ds} \Big( \gamma \alpha(u)  +F(u)\Big) 
\,du.
\end{equation}
Finally, since $W$ is convex, by Jensen's inequality for expectation, we have
\begin{equation*}
W(E^{\phi} [\|x(t)\|] ) \leq E^{\phi} [W(\|x(t)\|)] \leq E^{\phi}[ V(t, x(t))].
\end{equation*}
The proof is completed by noting that $W$ is strictly increasing.
\end{proof}

Considering another situation, we have the following theorem.


\begin{theorem} \label{thm2.3}
 Under assumption {\rm (A1)}, suppose there exists a
continuously differentiable Lyapunov functional
$$
V(t, x)\in C^{1,2}(\mathbb{R}^+ \times \mathbb{R}^n ; \mathbb{R}^+)
$$
such that for all $t\in\mathbb{R}^+$,
\begin{gather}\label{2.6}
W_1(\|x(t)\|) \leq V(t, x(t)) \leq W_2(\|x(t)\|)+
\int^t_{-h}\varphi_{1}(t,s)W_3(\|x(s)\|)ds, \\
\label{2.7}
 \mathcal{L} V(t, x(t))\leq -\alpha_1(t)W_4(\|x(t)\|)
-\alpha_2(t)\int^t_{-h}\varphi_2(t,s)W_5(\|x(s)\|)ds+ F(t)
\end{gather}
for positive continuous functions $\alpha_1(t), \alpha_2(t)$, $F(t)$, and
$\varphi_{i}(t,s)$, $i= 1,2$.

Moreover, there exists a non-negative constant $\gamma$ such that  the inequality
\begin{equation}\label{2.8}
\begin{aligned}
&W_2(\|x(t)\|)-W_4(\|x(t)\|) \\
&+ \int^t_{-h}\Big(\varphi_{1}(t,s)W_3(\|x(s)\|)-
\varphi_2(t,s)W_5(\|x(s)\|)\Big)ds \leq \gamma
\end{aligned}
\end{equation}
holds for all $t\in\mathbb{R}^+$.
Then all solutions of \eqref{1.1}  satisfy 
\begin{equation}\label{Thm2.2_Concl}
\begin{aligned}
E^{\phi}(\|x(t)\|)
&\leq W_{1}^{-1}\Big\{e^{-\int^t_{0}\alpha(s)\,ds} V(0, \phi(0)) \\
&\quad + \int^t_{0} e^{-\int^t_{u}\alpha(s)\,ds}\Big( \gamma \alpha(u) +
F(u)\Big)du\Big\}, \quad\forall t\geq 0,
\end{aligned}
 \end{equation}
 where  $\alpha(t) = \min\{\alpha_1(t), \alpha_2(t)\}$.
\end{theorem}


\begin{proof}
Let $x(t)$ be a solution of \eqref{1.1} with $x(t)=\phi(t)$, for
$-h\leq t \leq 0$. Then
\begin{equation} \label{diff_of_thm_2.2}
\begin{aligned}
&d \Big(e^{\int_0^t \alpha(s) d s} V(t, x(t))\Big)  \\
&=  e^{\int_0^t \alpha(s) d s} (\alpha(t) V(t, x(t)) 
+ \mathcal{L} V(t, x(t))) d t + d M(t)
\end{aligned}
\end{equation}
where
\begin{gather*}
M(t) = \int_0^t  e^{\int_0^u \alpha(s) d s} 
\sum_{i=1}^n V_{x_i}(u, x(u))\sum_{k=1}^m g_{ik}(u, x(u), x(u-h))dB_k(u), \\
\mathcal{L}V(t, x(t))=\mathcal{L}V\big(t, x(t), x(t-h), 
\int_{t-h}^t  A(t, s) h(s, x(s)) ds \big).
\end{gather*}
By \eqref{2.6}, \eqref{2.7}, \eqref{2.8} and the fact that 
$\alpha(t) = \min\{\alpha_1(t), \alpha_2(t)\}$, we obtain
\begin{equation}\label{2.9}
\begin{aligned}
&\mathcal{L}V(t, x(t))+\alpha(t) V(t, x(t))  \\
&\leq -\alpha_1(t)W_4(\|x(t)\|)-\alpha_2(t)
 \int^t_{-h}\varphi_2(t,s)W_5(|x(s)|)ds \\
&\quad +\ \alpha(t)W_2(\|x(t)\|)+
\alpha(t)\int^t_{-h}\varphi_{1}(t,s)W_3(\|x(s)\|)ds +F(t)\\
&\leq \alpha(t)\Big[W_2(\|x(t)\|)-W_4(\|x(t)\|)\\
&\quad + \ \int^t_{-h}\Big(\varphi_{1}(t,s)W_3(\|x(s)\|)-
\varphi_2(t,s)W_5(\|x(s)\|)\Big)ds\Big] +F(t)
\\
&\leq \gamma \alpha(t) + F(t).
\end{aligned}
\end{equation}
Integrating \eqref{diff_of_thm_2.2} from $0$  to  $t$, combining with \eqref{2.9}, 
we obtain
\begin{equation*}
e^{\int^t_{0}\alpha(s)\,ds} V(t, x(t)) \leq V(0, \phi(0))
+\int_0^t e^{\int^u_{0}\alpha(s)\,ds} \Big( \gamma \alpha(u) +
F(u) \Big)du + M(t).
\end{equation*}
Dividing by $e^{\int^t_{0}\alpha(s)\,ds}$, taking expectation on both sides, 
and noting that
$$
E^{\phi} \left \{ e^{-\int_{0}^t\alpha(s)\,ds} M(t) \right \}=0$$
in view of (A1), we obtain,
\begin{equation*}
E^{\phi}[ V(t, x(t))]  
\leq e^{-\int^t_{0}\alpha(s)\,ds}V(0, \phi(0))
+\int^t_{0} e^{-\int^t_{u}\alpha(s)\,ds}\Big( \gamma \alpha(u) +
F(u)\Big)du.
\end{equation*}
Finally, since $W_1$ is convex, by Jensen's inequality for expectation, we have
\begin{equation*} 
W_1(E^{\phi}\|x(t)\|) \leq E^{\phi} [W_1(\|x(t)\|)] 
\leq E^{\phi}[ V(t, x(t))].
\end{equation*}
The proof is complete by noting that $W_1$ is strictly increasing.
\end{proof}

We now consider some special 1-dimensional stochastic processes $x$, 
and state relevant results in Theorem \ref{thm2.4} and \ref{thm2.5}. 
A sufficient condition 
for both theorems is stated below:
We sue the following assumptin
\begin{itemize}
\item[(A2)] Assume that there exist positive functions 
$\nu, F: \mathbb{R}^+ \to \mathbb{R}^+$, such that for all
$t\in\mathbb{R}^+, x\in\mathbb{R}, y\in\mathbb{R}$:
\begin{gather} \label{a2}
|x g(t, x, y)|\leq \nu(t),\quad  g^2(t, x, y) \leq F(t), \\
\int_0^t \nu^2(s) d s < \infty, \forall t\geq 0.
\end{gather}
\end{itemize}

First we consider a particular form of \eqref{1.1} given in the (SDDE),
\begin{equation}\label{c1}
dx(t)= \Big(a(t) x(t) + b(t) x(t-h) \Big)dt + g(t, x(t), x(t-h))dB(t), \quad  t\geq 0,
\end{equation}
with continuous initial condition $\phi $, i.e., any solution of the above SDDE 
satisfying $ x(t)=\phi(t)$, $t \in [-h, 0]$.  We have the following theorem.

 \begin{theorem} \label{thm2.4} 
Suppose that {\rm (A2)} holds for some functions $\nu$ and $F$. 
We define the function
\begin{equation}\label{F1}
  \xi(t) = \frac{e^{\int^t_0 2a(s)ds}}{1 + 2h\int^{t + \tau}_t e^{\int^u_0  2a(s)ds} 
du},
 \end{equation}
for some constant $\tau \geq 0$, such that for all $t\in\mathbb{R}^+$:
 \begin{equation}\label{F2}
 |b(t)| \leq k \xi(t),
\end{equation}
for some positive constant $k$, and
\begin{equation}\label{F3}
2a(t) +   2k \xi(t) \leq - \alpha(t),
\end{equation}
for some positive continuous function $\alpha(t)$.
Then all solutions of \eqref{c1} satisfy
\begin{equation}\label{2.23}
E^{\phi} (|x(t)|)\leq  \Big\{e^{-\int_{0}^t\alpha(s)ds} V(0)
+\int^t_{0}e^{-\int^t_{u} \alpha(s)
ds} F(u) du\Big\}^{1/2},
 \end{equation}
with
$$
V(0) = \phi^2(0) + k \xi(0) \int^{0}_{-h} \phi^{2}(s)ds.
$$
\end{theorem}

\begin{proof}
For $t \geq 0,$ define a process $V$ via:
 $$
V(t)=x^2(t) + k \xi(t)\int^t_{t-h}x^2(s)ds.
$$
Apply It\^{o}'s formula to $e^{\int_0^t \alpha(s) d s} V(t)$:
\begin{equation}\label{2.24}
d \Big(e^{\int_0^t \alpha(s) d s} V(t)\Big) 
= e^{\int_0^t \alpha(s) d s} (\alpha(t) V(t) + \mathcal{L} V(t)) d t + d M(t)
\end{equation}
where
\begin{gather}\label{F4}
\begin{aligned}
\mathcal{L}V(t)
&=  (2a(t) + k \xi(t))x^2(t) + 2b(t)x(t)x(t-h) - k \xi(t)x^2(t-h) \\
&\quad +k \xi^{\prime}(t)\int^t_{t-h}x^2(s)\;ds + g^2(t, x(t), x(t-h)),
\end{aligned} \\
M(t) =  2 \int_0^t  e^{\int_0^u \alpha(s) d s} x(u) g(u, x(u), x(u-h))dB(u).
\end{gather}
Moreover, by (A2), \eqref{F1}, \eqref{F2}, and the fact that
\begin{align*}
\xi'(t) &= 2a(t) \xi(t) + 2 k \xi^2(t) - 2 k \xi^2(t)e^{\int_t^{t+\tau} 2a(s)\ ds}\\
& \leq 2a(t) \xi(t) + 2 k \xi^2(t) 2b(t)x(t)x(t-h)\\
& \leq |b(t)| (x^2(t) + x^2(t-h)),
\end{align*}
equation \eqref{F4} reduces to
\begin{equation}\label{F5}
\begin{aligned}
\mathcal{L}V(t)
&\leq \Big(2a(t)+|b(t)| + k \xi(t)\Big)x^2(t) + \Big(|b(t)| - k\xi(t)\Big)x^2(t-h)\\
&\quad + \ k\Big(2a(t)\xi(t) + 2k \xi^2(t)\Big)\int^t_{t-h}x^2(s)ds + F(t)\\
&\leq \Big(2a(t)+2 k \xi(t)\Big)x^2(t) + k\xi(t)\Big(2a(t) 
 + 2k \xi(t)\Big)\int^t_{t-h}x^2(s)ds + F(t) \\
&= \Big(2a(t) + 2k \xi(t)\Big)\Big[x^2(t)
+ k \xi(t)\int^t_{t-h}x^2(s)ds\Big] + F(t)\\
&\leq -\alpha(t)V(t) + F(t).
\end{aligned}
\end{equation}

We integrate from $0$ to $t$, divide by $e^{\int_0^t \alpha(s) d s}$,  and then 
take expectation on both sides of \eqref{2.24}. It follows that
\begin{equation}
\begin{aligned}
& E^{\phi} \big(V(t)\big) - e^{-\int_0^t \alpha(s) d s}  V(0)  \\
&\leq E^{\phi} \Big[\int_0^t e^{-\int_u^t \alpha(s) d s} F(u) d u\Big]
+ E^{\phi} \Big[ e^{-\int_0^t \alpha(s) d s} M(t) \Big]
\end{aligned}
\end{equation}


Since under Assumption (A2), $ e^{-\int_0^t \alpha(s) d s} M(t) $ is a martingale 
with expectation 0, we have
\begin{equation}
E^{\phi} \big(v(t)\big) \leq e^{-\int_0^t \alpha(s) d s}  V(0)
+ E^{\phi} \Big[ \int_0^t e^{-\int_u^t \alpha(s) d s} F(u) d u \Big]
\end{equation}
The proof is complete by observing that
$$
E^{\phi}(|x(t)|) \leq (E^{\phi}(x^2(t)))^{1/2} \leq (E^{\phi}(V(t)))^{1/2}.
$$
\end{proof}

Next, we turn our attention to the (SDDE) 
\begin{equation}
\begin{aligned}
\label{d1} dx(t) 
&=  \Big(a(t) x(t) + b(t) x(t-h) + \int^t_{t-h} A(t, s) f(s,x(s)) ds\Big)dt \\
&\quad + g(t, x(t), x(t-h))dB(t), \quad  t\geq 0,
\end{aligned}
\end{equation}
with given continuous  initial condition $\phi $, such that 
$ x(t)=\phi(t)$, $t \in [-h, 0]$.

\begin{theorem} \label{thm2.5} 
Assume that {\rm (A2)} holds.
Let $f: \mathbb{R}^+  \times \mathbb{R} \to \mathbb{R}$ be a nonlinear 
continuous function, and  $a, b$ and $A$ are assumed to be continuous on 
their respective domains. Suppose that there exists a positive constant 
$\lambda$ such that for all $t\in\mathbb{R}^+, x\in\mathbb{R}$,
  \begin{equation} \label{d2}
  |f(t,x)| \leq \lambda |x|\,.
  \end{equation}
Let $k $ be a positive constant and assume that there are two positive 
constants $\alpha_1$ and $\alpha_2$ so that for all $t\in\mathbb{R}^+$:
\begin{gather}\label{2.12}
2a(t) + 1 + b^2(t)+ k\int_{t}^{\infty}|A(u,t)|du  \leq -\alpha_1, \\
\label{d4}
1-k\int^{\infty}_t |A(u,t-h)| du \leq 0, \\
\label{2.13}
\lambda^2\int^t_{t-h} |A(t, s)|ds - k  \leq -\alpha_2k.
\end{gather}
Finally, we assume that there exists some positive constant $\beta$, 
such that for all $s, t (s\leq t)$,
\begin{equation} \label{d5}
|A(t, s)| \geq \beta \int^{\infty}_t |A(u, s)| du.
\end{equation}
Then all solutions of \eqref{d1} satisfy
\begin{equation}\label{2.14}
E^{\phi} (|x(t)|)\leq  \Big\{V(0)e^{-\alpha t}
+\int^t_{0}e^{-\alpha(t-u)} F(u)du\Big\}^{1/2},
 \end{equation}
with
\begin{gather*}
\alpha = \min\{\alpha_1 ,\; \alpha_2 \beta\}, \\
V(0) = \phi(0)^2 + k\int^{0}_{-h}\int^{\infty}_{0}|A(u, s)|du \phi^{2}(s)ds.
\end{gather*}
\end{theorem}

\begin{proof}
To see this we define a process
$$
V(t)=x^2(t)+ k\int^{t}_{t-h}\int^{\infty}_{t}|A(u, s)|du x^{2}(s)ds.
$$
Apply It\^{o}'s formula to $e^{\alpha t} V(t)$:
\begin{equation}\label{2.31}
d \left(e^{\alpha t} V(t)\right) 
= e^{\alpha t} (\alpha V(t) + \mathcal{L} V(t)) d t +  d M(t)
\end{equation}
where
\begin{gather*}
M(t) =  2 \int_0^t e^{\alpha s} x(s) g(s, x(s), x(s-h)) d B(s)\\
\begin{aligned}
\mathcal{L}V(t)
&=  2x(t) \Big(a(t) x(t) + b(t) x(t-h) + \int^t_{t-h} A(t, s) f(s,x(s)) ds\Big) \\
&\quad + k x^{2}(t) \int_{t}^{\infty}|A(u,t)|du 
 - k x^{2}(t-h)\int^{\infty}_{t}|A(u,t-h)|du\\
&\quad - k\int^t_{t-h} |A(t, s)|x^2(s) ds + g^2(t, x(t), x(t-h)) \\
&\leq  2a(t) x^{2}(t) + 2b(t)x(t)x(t-h) + 2x(t)\int^t_{t-h} A(t, s) f(s,x(s)) ds\\
&\quad + k x^{2}(t) \int_{t}^{\infty}|A(u,t)|du 
 - k x^{2}(t-h) \int^{\infty}_{t}|A(u,t-h)|du\\
&\quad - k\int^t_{t-h} |A(t, s)|x^2(s) ds + F(t)
\end{aligned}
\end{gather*}
Using the fact that $2ab \leq a^2 + b^2$, and Schwartz inequality we simplify 
the following terms:
\begin{gather*}
2b(t)x(t)x(t-h) \leq b^2(t)x^2(t) + x^2(t-h), \\
\begin{aligned}
2x(t)\int^t_{t-h} A(t, s) f(s,x(s)) ds 
&\leq  x^2(t) + \Big(\int^t_{t-h} A(t, s) f(s,x(s)) ds\Big)^2\\
&\leq  x^2(t) + \lambda^2\Big(\int^t_{t-h} |A(t, s)|^{\frac{1}{2}}
 |A(t, s)|^{\frac{1}{2}} |x(s)| ds\Big)^2\\
&\leq  x^2(t) + \lambda^2\int^t_{t-h} |A(t, s)|ds\int^t_{t-h}|A(t, s)| x^2(s) ds.
\end{aligned}
\end{gather*}
Using the above two inequalities and conditions \eqref{2.12}-\eqref{d5}
 we arrive at
\begin{align*}
\mathcal{L}V(t, x)
&\leq  \Big(2a(t) + 1 + b^2(t) + k\int_{t}^{\infty}|A(u,t)|du \Big)x^2(t) \\
&\quad +  \Big(1 - k\int^{\infty}_{t}|A(u,t-h)|\Big)x^2(t-h)\\
&\quad + \Big(\lambda^2 \int^t_{t-h} |A(t, s)|ds - k\Big) 
 \int^t_{t-h}|A(t, s)| x^2(s) ds + F(t)\\
&\leq  -\alpha_1 x^2(t) - \alpha_2 \beta k \int^{t}_{t-h}
 \int^{\infty}_{t}|A(u, s)|du x^{2}(s)ds + F(t)\\
&\leq  -\alpha V(t, x) + F(t).
\end{align*}
where $\alpha = \min(\alpha_1, \alpha_2 \beta)$.

We integrate from $0$ to $t$, divide by $e^{\alpha t}$, and then take expectation 
on both sides of \eqref{2.31}. It follows that
\begin{equation}
E^{\phi} \big(v(t)\big) - e^{-\alpha t}  V(0)
\leq \int_0^t e^{-\alpha (t-u)} F(u) d u+ E^{\phi} [e^{-\alpha t} M(t)]
\end{equation}
Again, by (A2), $e^{-\alpha t} M(t) $ is a martingale with expectation 0. 
Therefore,
\begin{equation}
E^{\phi} \big(v(t)\big) \leq e^{-\alpha t}  V(0)
+ \int_0^t e^{-\alpha(t-u)} F(u) d u
\end{equation}
The proof is complete by observing that
$$
E^{\phi}(|x(t)|) \leq (E^{\phi}(x^2(t)))^{1/2} \leq (E^{\phi}(V(t)))^{1/2}.
$$
\end{proof}

\section{Examples}

In this section, we provide several examples as the application of the results 
we obtained in the previous section.
To illustrate the application of Theorem \ref{thm2.2}, we consider the following 
two dimensional stochastic system of nonlinear Volterra integro-differential equations.


\begin{example} \label{examp3.1} \rm
Given continuous scalar functions $A_i: \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}$,
$f_i: \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ $(i = 1, 2)$, such that
$|A_1(\cdot, \cdot)| \geq |A_2(\cdot, \cdot)|$, 
$f_1(\cdot, \cdot) \geq 0$ and $|f_2(\cdot, \cdot)| \leq f_1(\cdot, \cdot)$.
Suppose the process $y = (y_1, y_2)$ satisfies the SDDE
\begin{align*}
d y_{1}(t)
&=  \Big(y_2(t)-y_{1}(t)| y_{1}(t)|
- y_{1}(t)\;y_2^2(t)\int^t_{t-h}| A_1(t,s)|
f_1(y_1(s),y_2(s))ds\Big)dt\\
&\quad +  g_{11}(t, y(t), y(t-h))dB_1(t)+ g_{12}(t, y(t), y(t-h))dB_2(t),
\\
d y_2(t) &=  \Big(-y_{1}(t)-y_2(t)| y_2(t)|
 + y_{1}^2(t)\;y_2(t)\int^t_{t-h} A_2(t,s)f_2(y_1(s),y_2(s))ds\Big)dt\\
&\quad + g_{21}(t, y(t), y(t-h))dB_1(t)+ g_{22}(t, y(t), y(t-h))dB_2(t),
\end{align*}
and
$(y_{1}(t), y_2(t))=(\varphi_{1}(t),\varphi_2(t))$,
$(-h \leq t \leq 0)$,
for some given initial continuous functions $\varphi_{1}(t),\varphi_2(t)$.

We assume that the functions $g_{ij}: \mathbb{R}^+ \times \mathbb{R} \times
\mathbb{R} \to \mathbb{R}$ ($i, j =1,2$) are given nonlinear continuous functions 
and for some $M(t)$ and $N(t)$:
\begin{gather*}
K^2(t, y(t), y(t-h)) := \sum_{i, j = 1}^2 g^2_{ij}(t, y(t), y(t-h)) \leq  M(t)
\\
\sum_{i, j = 1}^2 y_i(t) g_{ij}^2(t, y(t), y(t-h)) \leq  N(t)
\end{gather*}
where $\int_0^t N(s) ds < \infty$, for all $t\in\mathbb{R}^+$.

Take $V(t, y(t)):= \frac{1}{2}(y_{1}^{2}(t)+y_2^{2}(t))$,
$W(y):= \frac{1}{2}y^2$.
Then $W(|y|) \leq V(y, t)$, and
\begin{align*}
&\mathcal{L}V(t, y(t))\\
&=  - y_{1}^{2}| y_{1}|-y_2^{2}| y_2|+ \frac{1}{2}K^2(t, y(t), y(t-h))
\\
&\quad - y^2_{1}y^2_2\Big(\int^t_{t-h}|A_1(t,s)| f_1(y_1(s),y_2(s))ds
- \int^t_{t-h} A_2(t,s) f_2(y_1(s),y_2(s))ds \Big) \\
& \leq -\big( | y_{1}| ^{3}+| y_2|^{3}\big)
 + y^2_{1}y^2_2\int^t_{t-h}\Big(|A_2(t,s)|
 - |A_1(t,s)|\Big) f_1(y_1(s),y_2(s))ds + \frac{M(t)}{2}\\
& \leq -2\Big[\frac{| y_{1}| ^{3}}{2}+\frac{| y_2| ^{3}}{2}\Big]
 + \frac{M(t)}{2} \\
&= -2\Big[\frac{{(|y_{1}|^{2})}^{3/2}}{2}+\frac{{(|
y_2| ^{2})}^{3/2}}{2}\Big] + \frac{M(t)}{2} \\
&\leq -2\big( | y_{1}| ^{2}+| y_2|^{2}\big)^{3/2} 2^{-3/2} + \frac{M(t)}{2}\\
&=  -2 V^{\frac{3}{2}}(y(t),t) + \frac{M(t)}{2}.
\end{align*}
where  we have used the inequality
 $( \frac{a+b}{2})^{l}\leq \frac{a^{l}}{2}+\frac{b^{l}}{2}$, $a,b>0$, $l>1$.
Here $y_i$  $(i= 1,2)$ are the short notations for $y_i(t)$.
Then $\alpha(t) = 2$, $q = \frac{3}{2}$, $F(t) = \frac{M(t)}{2}$.

Next, by straightforward calculation,
\begin{align*}
 V(t, y) - V^q(t, y) 
&=  V(t, y)- V^{\frac{3}{2}}(t, y) \\
&= y_1^2+y_2^2 -(y_1^2+y_2^2)^{\frac{3}{2}}2^{-3/2} \leq \frac{4}{27}.
\end{align*}
 Hence, we have $\gamma = \frac{4}{27}$.
By Theorem \ref{thm2.2},  all solutions of the above two dimensional stochastic
 system satisfy
\begin{align*}
&E^{\varphi_1,\varphi_2 }\Big[\sqrt{y_{1}^{2}(t)+y_2^{2}(t)}\Big]\\
&\leq  \Big(2 \Big\{ \frac{1}{2}(\varphi_1^2(0) 
+ \varphi_2^2(0)) e^{-\int^t_{0}2ds}+ \int^t_{0}\big[ \frac{4}{27}(2) 
+ \frac{M(u)}{2}\big]e^{-\int^t_{u}2 ds}du \Big\}\Big)^{1/2}
\\
&=  \Big\{e^{-2t} \Big[\varphi_1^2(0) + \varphi_2^2(0) 
+ \int_0^t e^{2u} M(u) \ du \Big] + \frac{8}{27} (1-e^{-2t})\Big\}^{1/2}.
\end{align*}
\end{example}

Next we present an example of Theorem \ref{thm2.3}.

\begin{example} \label{examp3.2} \rm
Let $\phi(t)$ be a given bounded continuous initial function and consider 
the scalar stochastic Volterra integro-differential  equation
\begin{equation}\label{2.15}
\begin{gathered}
d x(t) = \Big(a(t) x(t) +\int^t_{t-h}A(t, s)(x(s))^{2/3} ds\Big)dt 
 + g(t, x(t), x(t-h))dB(t),\quad t \geq 0\;\\
 x(t) = \phi(t)\quad \text{for } -h \leq t \leq 0,
\end{gathered}
\end{equation}
where $g: \mathbb{R}^+  \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$
is a given nonlinear continuous function, which satisfies (A2)
 for some functions $M$ and $F$. Also, the functions
 $a : \mathbb{R}^ + \to \mathbb{R}$ and 
$A:\mathbb{R}^{+}  \times \mathbb{R}^ + \to \mathbb{R}$, 
are assumed to be continuous on their respective domains.
If
\[
-2a(t)-\int^t_{t-h}|A(t, s)|ds -\int_{t}^{\infty}|A(u,t)|du  >0,
\]
 and
$$
\frac{|A(t, s)| }{3} \geq \int_{t}^{\infty}|A(u, s)|du
$$
then all solutions of \eqref{2.15} satisfy inequality \eqref{Thm2.2_Concl}
 with $\gamma = 0$,
where  
$$
\alpha(t) = \min\{-2a(t)-\int^t_{t-h}|A(t, s)|ds
-\int_{t}^{\infty}|A(u,t)|du ,\; 1\},
$$
and
 $$
V(0) = \phi^2(0) + \int^{0}_{-h}\int^{\infty}_{0}|A(u, s)|du \phi^{2}(s)ds.
$$
To see this we define the process
$$
V(t)=x^{2}(t)+ \int^{t}_{-h}\int^{\infty}_{t}|A(u, s)|du x^{2}(s)ds.$$
which satisfy \eqref{2.6} with
$W_1(x) = W_2(x) = W_3(x) = x^2$, 
$\phi_1(t, s) = \int_t^\infty |A(u, s)| du$;
Then along solutions of \eqref{2.15} we have
\begin{align*}
\mathcal{L}V(t)
&=  2x(t)\Big(a(t) x(t) +\int^t_{t-h}A(t, s)(x(s))^{2/3}ds\Big)dt
+ g^2(t, x(t), x(t-h))\\
&\quad +  \int_{t}^{\infty}|A(u,t)|x^{2}(t)du
-\int^{t}_{-h}|A(t, s)|x^{2}(s)ds\\
&\leq  2a(t) x^{2}(t) +2\int^t_{t-h}|A(t, s)|\;|x(t)|(x(s))^{2/3}ds + F(t)\\
&\quad +\ \int_{t}^{\infty}|A(u,t)|x^{2}(t)du
-\int^{t}_{-h}|A(t, s)|x^{2}(s)ds.
\end{align*}
Using the fact that $ab \leq a^2/2 + b^2/2$, the above inequality
simplifies to
\begin{equation}\label{2.16}
\begin{aligned}
\mathcal{L}V(t)
&\leq  2a(t) x^{2} (t)+ \int^t_{t-h}|A(t, s)|(x^2(t)+x^{4/3}(s))ds\\
&\quad + \int_{t}^{\infty}|A(u,t)|x^{2}(t)du
-\int^{t}_{-h}|A(t, s)|x^{2}(s)ds +F(t)
\end{aligned}
\end{equation}
To further simplify \eqref{2.16} we  use Young's inequality, which says 
for any two nonnegative real numbers $w$ and $z$, we have 
$$
wz \leq \frac{w^e}{e} +  \frac{z^f}{f},\quad \text{with }
 \frac{1}{e} +  \frac{1}{f}=1.
$$
Thus, for $e=3$ and $f=3/2$, we obtain
\begin{align*}
\int^t_{t-h}|A(t, s)|x^{4/3}(s)ds
&= \int^t_{t-h}|A(t, s)|^{1/3}|A(t, s)|^{2/3}x^{4/3}(s)ds\\
&\leq  \int^t_{t-h}\Big(\frac{|A(t, s)|}{3} +\frac{2}{3}|A(t, s)|x^2(s)\Big)ds.\\
&\leq  \int^t_{t-h} \frac{|A(t, s)|}{3} ds
  + \int^t_{-h}\frac{2}{3}|A(t, s)|x^2(s)ds.
\end{align*}
Then by substituting the above inequality into \eqref{2.16}, we arrive at
\begin{align*}
\mathcal{L}V(t) 
&\leq  \Big(2a(t)+\int^t_{t-h}|A(t, s)|ds
+\int_{t}^{\infty}|A(u,t)|du\Big)x^2(t)\\
&\quad -\int^t_{-h}\frac{|A(t, s)|}{3}x^2(s)ds 
+ \frac{1}{3}\int^t_{t-h}|A(t, s)|ds + F(t)\\
\end{align*}
Then \eqref{2.7} holds for
\begin{gather*}
 W_4(x) = W_5(x) = x^2,  \quad \phi_2(t, x) = \frac{|A(t, s)|}{3} \\
\alpha_1(t) = -2a(t)-\int^t_{t-h}|A(t, s)|ds -\int_{t}^{\infty}|A(u,t)|du  >0, 
\quad \alpha_2(t) = 1
\end{gather*}
and $F(t)$ replaced by $L(t) = \frac{1}{3}\int^t_{t-h}|A(t, s)|ds +F(t)$.

It is obvious that \eqref{2.8} holds for $\gamma = 0$.
Hence by \eqref{2.14}, all solutions of \eqref{2.15} satisfy:
$$
E^{\phi}(\|x(t)\|) \leq \Big[ e^{-\int^t_{0}\alpha(u)du} V(0, \phi(0))  +
\int^t_{0} e^{-\int^t_{u} \alpha(s) ds} L(u) du \Big]^{1/2}.
$$
where  $\alpha(t) = \min(\alpha_1(t)$, 
\[
\alpha_2(t)) = \min(-2a(t)-\int^t_{t-h}|A(t, s)|ds 
-\int_{t}^{\infty}|A(u,t)|du, 1).
\]
\end{example}


Now we present an example of Theorem \ref{thm2.4}.


\begin{example} \label{examp3.3} \rm
Given some positive constant $c$, let
$$
d x(t) = [-c x(t) - c e^{-2ct} x(t-h)] d t + e^{-c t} 
\frac{\min(1, |x(t-h)|)}{|x(t)| + |x(t-h)|} \ d B(t)
$$
i.e., $a(t) = -c$, $b(t) = -c e^{-2ct}$, 
\[
g(t, x(t), x(t-h)) = e^{-c t} \frac{\min(1, |x(t-h)|)}{|x(t)| + |x(t-h)|}
\]
 in \eqref{c1}.
Then (A2) holds for $M(t) = e^{-c t}, F(t) = e^{- 2c t}$.

Let $\tau = 0$, then $\xi(t) = e^{-2c t}$. Then \eqref{F2} and \eqref {F3} 
hold for $k = c$, $\alpha(t) = 2c(1-e^{-2c t})$. By applying Theorem \ref{thm2.4}, 
\[
E^{\phi} (|x(t)|)\leq  e^{-ct - \frac{1}{2}e^{-2c t}} 
\Big(e\cdot V(0) + \int^t_{0} e^{e^{-2c u}} du \Big)^{1/2},
\]
with $V(0) = \phi(0)^2 + c\int_{-h}^0 \phi^2(s) ds$.
\end{example}

We conclude this paper with an application of Theorem \ref{thm2.5}.


\begin{example} \label{examp3.4} \rm
 Given some positive constant $\beta$, let
\begin{equation}
\begin{aligned}
d x(t) &=  \Big( -2\big( e^{\frac{t+h}{\beta} } + 1\big)  x(t) 
 +  x(t-h)  +    \int_{t-h}^t    \frac{2}{\beta} e^{-t/beta}  
 |x(s)|     d s   \Big) dt \\
&\quad  +  e^{-\alpha t/2} \frac{\min(1, |x(t-h)|)}{|x(t)| + |x(t-h)|} d B(t)
\end{aligned}
\end{equation}
where $\alpha$ is defined later as in \eqref{eq: alpha}. 
Obviously, $a(t) = -2\big( e^{\frac{t+h}{\beta} } + 1\big)$, 
$b(t) = 1$, $A(t, s) = e^{-\frac{1}{\beta}(t-s)}$, 
$f(t, x) = \frac{2}{\beta} e^{-t/beta} |x|$.

Then \eqref{d2}, \eqref{d4} and  \eqref{d5} hold for $\lambda = \frac{2}{\beta}$ 
and $k = \frac{4}{\beta} e^{h/\beta}$. Now consider for \eqref{2.12}:
\begin{align*}
2a(t) + 1 + b^2(t)+ k\int_{t}^{\infty}|A(u,t)|du 
= -4\left( e^{\frac{t+h}{\beta} } + 1\right) + 2+ 4e^{\frac{h}{\beta}} \leq -2
\end{align*}
which implies that \eqref{2.12} holds for $\alpha_1 = 2$

Then we consider, for \eqref{2.13},
\begin{equation*}
\lambda^2\int^t_{t-h} |A(t, s)|ds - k 
= \frac{4}{\beta}e^{\frac{h}{\beta}} 
\big( -e^{-\frac{2h}{\beta}}  + e^{-\frac{h}{\beta}} - 1 \big) 
\leq \frac{4}{\beta}e^{\frac{h}{\beta}} \big( - \frac{3}{4} \big)
\end{equation*}
implying that \eqref{2.13}  holds for $\alpha_2 = 3/4$.
Therefore,
\begin{equation}\label{eq: alpha}
\alpha = \min\{\alpha_1, \alpha_2 \beta\} 
= \min\{2, \frac{3\beta}{4}\}
\end{equation}
By Theorem \ref{thm2.5}, straightforward calculation suggests that for any $t\in\mathbb{R}^+$,
$$
E^{\phi}(|x(t)|) \leq e^{-\alpha t/2} (V(0) + t )^{1/2}
$$
with $ V(0) = \phi^2(0) + 4 \int_{-h}^0 e^{\frac{s+h}{\beta}} \phi^2(s) d s$.
\end{example}


\begin{thebibliography}{99}

\bibitem{Ashok} D. Bazeiaa, A. Dasb, L. Losanoa,   M. J. Santosa;
\emph{Traveling wave solutions of nonlinear partial differential equations}, 
Applied Mathematics Letters, Volume 23, Issue 6, June 2010, Pages 681--686

\bibitem{Hasminskii}  R. Z. Hasminskii;
\emph{Stochastic Stability of  Differential Equations,} 
Sijthoff and Noordhoff, 1980.

\bibitem{Gard1} T. C. Gard; 
 \emph{Persistence in stochastic food web models},
Bull. Math. Biol., 46 (1984), pp. 357--370

\bibitem{Ivanov} A. F. Ivanov, Y. I. Kazmerchuk,  A. V. Swishchuk;
\emph{Theory, stochastic stability and applications of stochastic delay 
differential equations: A survey of results}, to appear in Differential 
Equations Dynam. Systems Citations. 

\bibitem{Gard2} T. C. Gard;
\emph{Stability for multispecies population models in random environments},
Nonlinear Anal., 10 (1986), pp. 1411--1419

\bibitem{Kushner}    H. J. Kushner;
\emph{Stochastic Stability and Control,} Academic Press, 1967.

\bibitem{Mao-01}   X. Mao;
\emph{Exponential Stability of  Stochastic Differential Equations,}
 Marcel Dekker, 1994.

\bibitem{Mao-02}  X. Mao;
\emph{Stochastic Differential Equations and Applications,} Horwood, 1997.

\bibitem{Thygesen}  U. H. Thygesen;
\emph{A Survey of Lyapunov Techniques for Stochastic Differential Equations}, 
IMM Technical Report nc. (1997).
Downloaded from http://www.imm.dtu.dk.

\bibitem{JJ} J. J. Levin, J. A. Nohel;
\emph{On a system of integro-differential equations occurring in reactor dynamics},
 J. Math. Mech.,  9(1960), pp. 347-368.

\bibitem{R} J. Luo;
\emph{Fixed points and stability of neutral stochastic delay differential equations},
J. Math. Anal. Appl., Volume 334, Issue 1, 1 October 2007, Pages 431--440.

\bibitem{ka}  S. Kato;
\emph{Existence, uniqueness and continuous
dependence of solutions of delay-differential equations with
infinite delay in a Banach space}, J. Math. Anal. Appl., Vol. 195, No 1, (1995).

\bibitem{mb} A.  Majda, M. Branicki;
\emph{Lessons in uncertainty quantification for turbulent dynamical systems},
 Discrete Continuous Dyn. Sys., Vol. 32, No. 9, September 2012, pp. 3133-3221.

\bibitem{Mao} Xuerong Mao, Chenggui Yuanb, Jiezhong Zouc;
\emph{Stochastic differential delay equations of population dynamics}, 
Journal of Mathematical Analysis and Applications
Volume 304, Issue 1, 1 April 2005, Pages 296-320

\bibitem{Mao2007} Xuerong Mao;
\emph{Stochastic differential equations and applications}, Elsevier, 2007.

\bibitem{ML} R. K. Miller;
\emph{On Volterra's populations equation},
 Scholar Commons, University of South Carolina, 14 (1996), pp. 446-452.

 \bibitem{OG} M. N. O\v{g}uzt\"{o}reli;
emph{Time-Lag Control Systems},
 Academic Press, New York, 1966.

\bibitem{Parmar} K. Parmar, K. B. Blyuss, Y. N. Kyrychko, S. J. Hogan;
 \emph{Time-delayed models of gene regulatory networks}, 
Computational and Mathematical Methods in Medicine
Volume 2015 (2015), Article ID 347273, 16 pages
http://dx.doi.org/10.1155/2015/347273

\bibitem{ra} Y. N.  Raffoul;
 \emph{Boundedness in nonlinear functional differential equations with 
applications to Volterra integro-differential equations}, 
J.  Integral Equations  Appl. Vol. 16, No. 4, Winter (2004).

\bibitem{R2}  Y. N. Raffoul,  Ruihua L. H.;
\emph{Boundedness and exponential stability of highly nonlinear stochastic 
differential equations,}
 Electron. J. Diff. Eqns.,  Vol. 2009 (2009), No. 143, pp. 1--10.

\bibitem{ZL} Chunmei Zhang, Wenxue Li,  Ke Wang;
\emph{Stability and boundedness of stochastic Volterra integro-differential 
equations with infinite delay,} Journal of Applied Mathematics
Volume 2013 (2013), Article ID 320832, 9 pages

\end{thebibliography}

\end{document}