\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 92, pp. 1--36.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2004/92\hfil Stability of stochastic equations]
{Stability of stochastic functional differential equations and the
W-transform}
\author[R. Kadiev, A. Ponosov\hfil EJDE-2004/92\hfilneg]
{Ramazan Kadiev, Arcady Ponosov} % in alphabetical order
\address{Ramazan Kadiev \hfil\break
Department of Mathematics \\
Dagestan State University \\
Makhachkala 367005, Russia}
\email{dgu@dgu.ru}
\address{Arcady Ponosov \hfil\break
Institutt for matematiske realfag and teknologi \\
NLH, Postboks 5003\\
N-1432 \AA s, Norway}
\email{arkadi@nlh.no}
\date{}
\thanks{Submitted May 12, 2004. Published July 27, 2004.}
\subjclass[2000]{93E15, 60H30, 34K50, 34D20}
\keywords{Stability; stochastic differential equations with aftereffect;
\hfil\break\indent
integral transforms}
\begin{abstract}
The paper contains a systematic presentation of how the
so-called ``W-transform'' can be used to study stability of
stochastic functional differential equations.
The W-transform is an integral transform which typically is
generated by a simpler differential equation (``reference equation'')
via the Cauchy representation of its solutions
(``variation-of-constant formula''). This other equation is
supposed to have prescribed asymptotic properties (in this paper:
Various kinds of stability). Applying the W-transform to the
given equation produces an operator equation in a suitable
space of stochastic processes, which depends on the asymptotic
property we are interested in. In the paper we justify this
method, describe some of its general properties, and
illustrate the results by a number of examples.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{example}{Example}[section]
\newtheorem{remark}{Remark}[section]
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
Stability analysis for stochastic delay and functional
differential equations is usually based on the classical
Lyapunov-Krasovskii-Razumikhin method (see e.g. \cite{Mao-book}
and \cite{t1}), where one tries to find a suitable Lyapunov
function (a Lyapunov-Krasovskii functional) which ensures the
prescribed stability property. Another way is more
straightforward: it uses direct estimates on solutions
\cite{Mao-book}.
On the other hand, a recent progress in stability theory for
deterministic functional differential equations shows (see e.g.
\cite{a2}) that at least for linear delay equations it seems to be
more convenient to use special integral transforms to study
various asymptotic properties.
The idea of using the W-transform in stability theory was
originally proposed by Berezansky in his pioneer work
\cite{Ber}. Later on, this idea was developed by him and his
collaborators in a series of papers (see \cite{ABSC}). The method
can be briefly outlined as follows. Instead of studying stability
of a given linear delay equation with respect to the initial function, one
first moves the initial function over to the right hand side of
the equation. By this, one arrives at another property called in
the literature "admissibility of pairs of spaces" (see e.g.
\cite{Massera}). One proves then that any kind of stability with respect to
the initial function is implied by admissibility of certain pairs
of functional spaces. To check admissibility one choose a simpler
equation (called ``a reference equation"), which already has the
required property of admissibility. This new equation gives then
rise to an integral transform (traditionally called "the
W-transform") which, when applied to the original equation,
produces an integral equation of the form $x-\Theta x=f$. If the
latter equation is solvable (for instance, if $\|\Theta\|<1$),
then admissibility, and hence stability, is proved.
In this paper, we try to extend this method to the case of
stochastic functional differential equations. We exploit the
scheme that (in the deterministic case) was developed in
\cite{ABSC} replacing functional spaces by certain spaces of
stochastic processes. This enables us to put the study Lyapunov
stability (e.g. asymptotic stability and exponential stability) of
stochastic delay equations into a unified framework in a natural
way. As we show, this method covers more general stochastic
functional differential equations and produces sufficient
stability results in an efficient way.
Let us remark that the main purpose of the present paper is to
give a theoretical justification of the W-method in connection
with stochastic stability. That is why all the examples we present
are illustrative and are not compared with the stability results
which can be found in the literature. A more specific analysis of
some stochastic delay equations, including a comparison with the
existing stability criteria, will be a subject of a forthcoming
paper.
\section{Notation and main assumptions}\label{sect1}
\subsection*{Basic notation}
Let $(\Omega , {\it \mathcal{F}}, ({\it \mathcal{F}}_t)_{t\ge 0},
{\bf P})$ be a complete filtered probability space (see e. g.
\cite[p.~9]{l1}), $Z :=(z^1, \dots, z^m)^T$ be a $m$-dimensional
semimartingale \cite[p.~73]{l1} on it (we distinguish column
vectors $(a^1, \dots a^n)^T$ and row vectors $(a^1, \dots a^n)$).
In the sequel, we let $|\cdot|$ denote the norm in $\mathbb{R}^n$;
$\mathbb{R}^{k\times n}$ will be a linear space consisting of all
real $k \times n$-matrices with the norm $\|.\|$ that agrees with
the chosen vector norm in $\mathbb{R}^n$. We write $\bar 0$ for the
zero column vector in $\mathbb{R}^n$, the symbol $\bar E$ denotes
the unit matrix, while ${\bf E}$ stands for the expectation.
For convenience, we denote by $\lambda ^*$ the complete measure on
an interval $I$, generated by a nondecreasing function $\lambda(t)$
$(t\in I) $.
The following linear spaces of stochastic processes will be used
in the sequel:
$L^n(Z)$ consists of predictable $n\times m$-matrix functions
defined on $[0, \infty )$ with the rows that are locally
integrable with respect to the semimartingale $Z$, see e. g. \cite{j1};
$k^n$ consists of $n$-dimensional $\mathcal{F}_0$-measurable
random variables (we set also $k:=k^1$);
$D^n$ consists of $n$-dimensional stochastic processes on $[0,
\infty )$, which can be represented in the following form:
$$
x(t) = x(0) + \int_0^tH(s)dZ(s)\quad (t\ge 0),
$$
where $x(0) \in k^n$, $H \in L^n(Z)$;
$L_q^\lambda $ consists of scalar functions defined on $[0,
\infty )$, which are $q$-integrable ($1 \le q < \infty $) with respect to
the measure $\lambda ^*$, generated by a nondecreasing function
$\lambda(t)$ $(t\in [0, \infty))$;
$L_{\infty}^\lambda $ consists of scalar functions defined on
$[0, \infty )$, which are measurable and a. s. bounded with respect to
the measure $\lambda ^*$, generated by a nondecreasing function
$\lambda(t)$ $(t\in [0, \infty))$;
$L_q$ stands for $L_q^\lambda $ in the case when $\lambda (t) =
t$ ($1 \le q \le \infty $).
In addition, we will implicitly assume that the real numbers
$p, q$ satisfy the inequalities $1 \le p < \infty$, $1 \le q
\le \infty$.
The following notational agreement will be used in the sequel:
$\int_{a}^b = \int_{[a,b]}$ (otherwise we will write
$\int_{(a,b]}$, or $\int_{(a,b)}$ etc.
The variation of a function over a closed interval $[a,b]$ will be
denoted by $\bigvee _{[a,b]}$, and we will also write
$\bigvee _{(a,b]}$ for $\lim_{\delta \to
0+}\bigvee _{[a+\delta, b]}$ and $\bigvee _{[a,b)}$
for $\lim_{\delta \to 0+}\bigvee _{[a,
b-\delta]}$, respectively.
Now we are able to formulate the main assumption on the
semimartingale $Z(t)$. In what follows we always assume that the
semimartingale $Z(t)$ ($t\in [0, \infty)$) can be represented as a
sum
\begin{equation}
\label{MainAssumptionOnZ}
Z(t) = b(t) + c(t) \quad (t\ge 0),
\end{equation}
where $b(t)$ is a predictable stochastic process of locally
finite variation and $c(t)$ is a local squire-integrable
martingale \cite[p.~28]{l1} such that all the components of the
process $b(t)$ as well as the predictable characteristics $(t)$, $1\le i,j\le m$ of the process $c(t)$ \cite[p. 48]{l1}
are absolutely continuous with respect to to a nondecreasing function
$\lambda :[0, \infty ) \to \mathbb{R}_+$. In this case, we
can write
\begin{equation}\label{Coefficients-a-b-c}
b^i = \int_0^{\cdot}a^id\lambda , \quad
\langle c^i, c^j\rangle = \int_0^{\cdot}A^{ij}d\lambda , \quad
i, j = 1, \dots, m.
\end{equation}
For example, $\lambda(t)=t$ for It\^{o} equations. Without loss of
generality, it will be convenient in the sequel to assume that the
first component of the semimartingale $Z(t)$ coincides with
$\lambda (t)$, i.e. $z^1 (t)= \lambda (t)$. Clearly, we can always
do it adding, if necessary, a new, $(m+1)$-th component to the
$m$-dimensional semimartingale $Z(t)$.
It is known \cite{j1} that under the assumption
(\ref{MainAssumptionOnZ}) the space $L^n(Z)$ can be described as a
set of all predictable $n \times m$-matrices $H(t)=[H^{ij}(t)]$,
for which
\begin{equation}\label{SpaceL^n(Z)}
\int_0^t(|Ha| + \|HAH^\top \|)d\lambda < \infty \ \
\mbox{a. s.}
\end{equation}
for any $t \ge 0$. Here
\begin{equation}\label{Vectors-a-A}
a := (a^1, \dots, a^m)^T, \ \ \ A := [A^{ij}].
\end{equation}
Note that $a $ is an $m$-dimensional column vector and $A$ is a
$m\times m$-matrix.
Under the above assumptions we can also write $\int_0^tHdZ = \int_0^tHdb + \int_0^tHdc$. Moreover,
we can describe the space $D^n$ as a set consisting of all
$n$-dimensional adapted stochastic processes on $[0, \infty )$,
the trajectories of which are right continuous and have left hand
limits for all $t\in [0, \infty)$ and almost all $\omega$ (the
so-called "cadlag processes"). In addition, the following estimate
holds:
\begin{equation}\label{1}
\Big({\bf E}\Big|\int_0^tHdZ\Big|^{2p}\Big)^\frac 1{2p} \le
\Big({\bf E}\Big(\int_0^t|Ha|d\lambda \Big)^
{2p}\Big)^\frac 1{2p} + c_p\Big({\bf E}\Big(\int_0^t\|HAH^T\|d\lambda \Big)^p\Big)^\frac 1{2p},
\end{equation}
where $c_p$ is a certain positive constant depending on $p$ (see
e. g. \cite[ p. 65]{l1}).
Given $H = [H^{ij}]\in L^n(Z)$ and $a$, $A$ defined in
(\ref{Vectors-a-A}), we will write
\begin{equation}\label{Coefficients-a-A-plus}
a^+ := (|a^1|, \dots, |a^m|)^T, \quad A^+ := [|A^{ij}|], \quad
H^+ :=[|H^{ij}|].
\end{equation}
Studying different kinds of stochastic stability requires
different spaces of stochastic processes which are listed below.
\subsection*{Main spaces}
Assume that we are given: \\
a scalar nonnegative function $\xi$, defined on $[0, \infty
)$ and locally integrable with respect to the measure $\lambda^* $
generated by a nondecreasing function $\lambda$ ($\lambda$ is the
same as in (\ref{Coefficients-a-b-c}));\\
a positive scalar function $\gamma (t)$ ($t \in [0, \infty)$).
\begin{remark} \label{rmk2.1}\rm
In what follows we silently adopt the following convention: if in
a definition, a theorem etc. $\gamma(t)$ is mentioned without any
comments, then it is only assumed to be a positive scalar
function. Otherwise, additional properties of $\gamma$ will be
explicitly described.
\end{remark}
These functions are involved in the definitions of almost all
spaces we are going to use in the sequel. Both are crucial for our
considerations as they are responsible for the asymptotic behavior
of the solutions.
\begin{gather*} %\label{MainSpaces}
k_p^n = \{\alpha : \alpha \in k^n, \|\alpha \|_{k_p^n} :=
({\bf E}|\alpha |^p)^{1/p} < \infty \}; \\
M_p^\gamma = \{x: x \in D^n, \|x\|_{M_p^\gamma } := (\sup_{t\ge 0}{\bf E}|\gamma
(t)x(t)|^p)^{1/p} < \infty \} \quad (M_p^1 = M_p); \\
\begin{aligned}
\Lambda_{p, q}^n(\xi ) &= \{H: H \in L^n(Z), ({\bf E}|Ha|^p)^{1/p}\xi ^{q^{-1}-1}\\
&\quad +({\bf E}\|HAH^\top \|^{p/2})^{1/p}\xi ^{q^{-1}-0.5} \in L^\lambda_q\};
\end{aligned}\\
\begin{aligned}
\Lambda_{p, q}^{n+}(\xi ) =&\{H: H \in L^n(Z),
({\bf E}|H^+a^+|^p)^{1/p}\xi ^{q^{-1}-1} \\
&+ ({\bf E}\|H^+A^+(H^+)^\top \|^{p/2})^{1/p}\xi ^{q^{-1}-0.5} \in L^\lambda_q\}.
\end{aligned}
\end{gather*}
The following parameters are involved in the above definitions:
The numbers $p, q$ are assumed to satisfy the inequalities $1
\le p < \infty $, $1 \le q \le \infty$;
$a$, $A$ are defined by (\ref{Vectors-a-A});
$a^+$, $A^+$, $H^+$ are given by (\ref{Coefficients-a-A-plus}).
In the last two spaces the norms are given by
\begin{gather*}
\|H\|_{\Lambda_{p, q}^n(\xi)} :=
\|({\bf E}|Ha|^p)^{1/p}\xi ^{q^{-1}-1}\|_{L_q^\lambda } +
\|({\bf E}\|HAH^\top \|^{p/2})^{1/p}\xi ^{q^{-1}-0.5}\|_{L_q^\lambda },
\\
\|H\|_{\Lambda_{p, q}^{n+}(\xi)} :=
\|({\bf E}|H^+a^+|^p)^{1/p}\xi ^{q^{-1}-1}\|_{L_q^\lambda } +
\|({\bf E}\|H^+A^+(H^+)^\top \|^{p/2})^{1/p}\xi
^{q^{-1}-0.5}\|_{L_q^\lambda }.
\end{gather*}
\subsection*{Operators and equations}
\begin{definition}\label{defVolterra} \rm
An operator $V:D^n \to L^n(Z)$ is called Volterra (see
\cite{k4}) if for any stopping time \cite[p. 9]{l1} $\tau\in
[0,\infty)$ a. s. and any $x,y\in D^n$ such that $x(t)=y(t)$
($t\in [0,\tau]$ a.s.) one has $(Vx)(t)=(Vy)(t)$ ($t\in [0,\tau]$
a.s.).
\end{definition}
\begin{definition}\label{k-linearity} \rm
An operator $V:D^n \to L^n(Z)$ is called $k$-linear if
$$V(\alpha_1x_1 + \alpha_2x_2) = \alpha_1Vx_1 + \alpha_2Vx_2$$
for any $\alpha_i \in k$, $x_i \in D^n$, $i= 1, 2$.
\end{definition}
This property exclude ``global" operations, like expectation, from
the coefficients of the equation, and therefore determines the
pathwise way of describing solutions.
\begin{remark}\label{remK-linearity} \rm
If $V$ is continuous with respect to natural topologies in the spaces $D^n$
and $L^n(Z)$, then one can show that $k$-linearity follows from
the usual linearity (with respect to $\mathbb{R}$).
\end{remark}
The central object of this paper is a stochastic functional
differential equation
\begin{equation}\label{2}
dx(t) = [(Vx)(t) + f(t)]dZ(t) \quad (t \ge 0),
\end{equation}
where $f \in L^n(Z)$ and $V: D^n \to L^n(Z)$ is a
$k$-linear and Volterra (in the sense of Definition
\ref{defVolterra}) operator.
In \cite{Pon93} it is shown that \eqref{2} covers
linear stochastic delay equations, linear stochastic
integro-differential equations, linear stochastic neutral
equations - all with driven semimartingales etc. It can look a
little bit confusing as \eqref{2} does not depend on
the values $x(t)$ for $t<0$. In fact, this dependence can be
incorporated into the right-hand side as it is demonstrated in the
following example.
\begin{example}\label{Ex-privodimost6} \rm
Consider a linear scalar stochastic differential equation of the
form
\begin{equation}\label{3.5}
dx(t) = [a(t)(Tx)(t) + g(t)]dZ(t) \quad (t\ge 0)
\end{equation}
with the prehistory condition
\begin{equation}\label{3.6}
x(s) = \varphi(s)\quad (s<0),
\end{equation}
where $(Tx)(t)=\int_{(-\infty,t)}d_sR(t,s)x(s)$ is the distributed
delay operator.
Under natural assumptions on the right-hand side (see
\cite{Pon93}) this equation can be reduced to the form \eqref{2}
if one sets
\begin{equation}\label{3.7}
(Vx)(t):=a(t)\int_{[0,t)}d_sR(t,s)x(s) \quad \mbox{and} \quad
f(t):=a(t)\int_{(-\infty,0)}d_sR(0,s)\varphi(s)+ g(t).
\end{equation}
\end{example}
In addition to \eqref{2} we consider the associated
homogeneous equation ($f\equiv 0$).
\begin{equation}\label{2homogen}
dx(t) = (Vx)(t)dZ(t) \ \ (t \ge 0).
\end{equation}
Using $k$-linearity of the operator $V$, we immediately obtain the
following result.
\begin{lemma}\label{Lemma1}
Let for any $x(0) \in k^n$ there exists the only solution (up to a
${\bf P}$-null set) $x(t)$ of \eqref{2}. Then one has
the following representation (``the Cauchy representation'') of the
solutions
\begin{equation}\label{3}
x(t) = X(t)x(0) + (Kf)(t) \quad (t \ge 0),
\end{equation}
where $X(t)$ ($X(0) = \bar E$) is an $n\times n$-matrix, the
columns of which are the solutions of the linear homogeneous
equation (\ref{2homogen}) (``the fundamental matrix"), while $K: L^n(Z) \to D^n$ is a $k$-linear operator (``the Cauchy
operator") such that $(Kf)(0) = 0$ and $Kf$ satisfies \eqref{2}.
\end{lemma}
In what follows we will always consider equation \eqref{2}
under the uniqueness assumption, i.e. existence, for any $x(0) \in
k^n$, of the unique (up to a $P$-null set) solution $x(t)$ of this
equation. In other words, according to Lemma \ref{Lemma1}, the
representation \eqref{3} is silently assumed to be fulfilled in
all further considerations.
\section{$\mathbf{ M_p^\gamma}$ - stability }\label{Section2}
\begin{definition}\label{defClassicStability} \rm
The zero solution of the linear homogeneous equation
(\ref{2homogen}) is called
\begin{itemize}
\item[(a)] $p$-stable if for an arbitrary $\varepsilon >0$ there exist
$\eta=\eta(\varepsilon)>0$ such that
$$
{\bf E} |X(t)x(0)|^p\le \varepsilon \ \ (t\ge 0)
$$
for all $x(0)\in\mathbb{R}^n$, $|x(0)|< \eta$.
\item[(b)] Asymptotically $p$-stable if it is $p$-stable and $\lim
_{t\to +\infty}{\bf E} |X(t)x(0)|^p=0$ for all
$x(0)\in\mathbb{R}^n$.
\item[(c)] Exponentially $p$-stable if there exist $\bar{c}>0$,
$\beta>0$ such that
$$
{\bf E} |X(t)x(0)|^p\le \bar{c}|x(0)|\exp\{-\beta t\} \ \ (t\ge 0)
$$
for all $x(0)\in\mathbb{R}^n$.
\end{itemize}
\end{definition}
Similarly, we can define stability of solutions to the
nonhomogeneous equation \eqref{2}. Clearly, the representation
\eqref{3} implies that all solutions to \eqref{2} are $p$-stable
(asymptotically $p$-stable, exponentially $p$-stable) if and only
if the zero solution to the homogeneous equation (\ref{2homogen})
is $p$-stable (asymptotically $p$-stable, exponentially
$p$-stable). In the sequel we shall therefore say that the
nonhomogeneous equation \eqref{2} is stable (in a proper sense) if
the zero solution to the homogeneous equation (\ref{2homogen}) is
stable in the same sense.
\begin{theorem}\label{thMp-stability}
\begin{itemize}
\item[\bf (A)]. Equation \eqref{2} is $p$-stable if and only if $X(\cdot)x(0)\in M_p$
for all $x(0)\in\mathbb{R}^n$.
\item[\bf (B)]. Equation \eqref{2} is asymptotically $p$-stable if and only if there exists a function
$\gamma(t)$, for which $\gamma(t) \ge \delta > 0$ and
$ \lim_{t\to +\infty} \gamma (t)= +\infty$, so that
$X(\cdot)x(0)\in M_p^\gamma$ for all $x(0)\in\mathbb{R}^n$.
\item[\bf (C)] Equation \eqref{2} is exponentially $p$-stable if and only if there
exists a number $\beta>0$ such that $X(\cdot)x(0)\in M_p^\gamma$ for
all $x(0)\in\mathbb{R}^n$, where $\gamma(t)=\exp\{\beta t\}$.
\end{itemize}
\end{theorem}
\begin{proof}
The proof is based on the ideas developed in \cite{k1}.
\noindent(A). Letting \eqref{2} be $p$-stable we
suppose that for some $x_0\in\mathbb{R}^n$ the solution $X(\cdot)x_0$
is not in $ M_p$, i. e. for any $K>0$ there exists $t(K)>0$ such
that ${\bf E}|X(t(K))x_0|^p>K$. Taking an arbitrary
$\varepsilon>0$ we find $\eta>0$, which satisfies the definition
of $p$-stability from Definition \ref{defClassicStability}, and
put $K_0=\varepsilon |2x_0/\eta|^p$, $t_0 =t(K_0)$, $x'_0=\eta
|x_0|^{-1}x_0$. Then for the solution $x(t)= X(t) x'_0$ we have
${\bf E}|x(t_0)|^p>\varepsilon$, although $|x(0)|<\eta$. This
contradicts the assumption.
To prove the converse we first note that for each individual
$x(0)\in\mathbb{R}^n$ one has $$\sup_{t\in [0,\infty)}{\bf
E}|X(t)x(0)|^p\le K=K(x(0)).$$ As $X(t)$ is linear and $\mathbb{R}^n$
is finite dimensional, then there is a constant $K'$ which
provides the uniform estimate: $\sup_{t\in [0,\infty)}{\bf
E}\|X(t)\|^p\le K'.$ Hence for any $\varepsilon>0$ we may put
$\eta=(\epsilon/K')^{1/p}$ so that $|x(0)|<\eta$ implies \ ${\bf
E}|X(t)x(0)|^p\le |x(0)|^p{\bf E}\|X(t)\|^p\le\varepsilon$ for all
$t\in [0,\infty)$.
\noindent(B) Assume that \eqref{2} is asymptotically
$p$-stable. We have to find $\gamma(t)$ satisfying conditions
listed in Part (B) of the theorem. First we find a function
$\bar\gamma(t)$ for which a) $0 < \bar\gamma(t) < M$ ($t\in [0,
\infty)$) and b) $\bar\gamma^{-1}(t){\bf E}\|X(t)\|^p\to 1$ as
$t\to +\infty$. This is possible due to $p$-stability of
\eqref{2} and the boundedness of the function ${\bf
E}\|X(t)\|^p$ (see Part A of the proof).
We set now $\gamma(t)=1/\bar\gamma(t)$ and check directly that
$X(\cdot)x(0)\in M_p^\gamma$ for all $x(0)\in \mathbb{R}^n$.
The converse to Part (B) of the theorem is evident: for any
$\gamma(t)$, satisfying conditions listed in Part a) of
Definition \ref{defClassicStability}, the space $M_p^\gamma$ will
be a subspace of the space $M_p$ and $\lim_{t\to
+\infty}{\bf E} |X(t)x(0)|^p=0$ for all $x(0)\in\mathbb{R}^n$.
\noindent (C) The exponential $p$-stability trivially implies that
$X(\cdot)x(0)\in M_p^\gamma$ for all $x(0)\in\mathbb{R}^n$, where
$\gamma(t)=\exp\{\beta t\}$.
Conversely, if $X(\cdot)x(0)\in M_p^\gamma$ for all $x(0)\in\mathbb{R}^n$, where
$\gamma(t)=\exp\{\beta t\}$, then
$$
\sup_{t\in [0,\infty)}(\exp\{\beta t\}{\bf E} |X(t)x(0)|^p)\le c'
$$
for some $c'=c'(x(0))$. As $X(t)$ is linear and $\mathbb{R}^n$ is
finite dimensional we, as in Part (A), find a constant $\bar c$
such that
$$\sup_{t\in [0,\infty)}\left(\exp\{\beta t\}{\bf E} \|X(t)\|^p\right)\le
\bar c. $$ Clearly, this implies the exponential $p$-stability
of \eqref{2}.
\end{proof}
This theorem says that to prove $p$-stability (asymptotic,
exponential $p$-stability) of \eqref{2} we can check
that the solutions of the homogeneous equation (\ref{2homogen})
belong to a certain space of stochastic processes ($M_p$ or
$M_p^\gamma$).
Minding Theorem \ref{thMp-stability}, we introduce now a new
definition of stability which is more convenient for our purposes.
\begin{definition}\label{def1}
Equation \eqref{2} is called {\it $M_p^\gamma $-stable}, if
for any $x(0) \in k_p^n$ we have $X(\cdot)x(0) \in M_p^\gamma $.
\end{definition}
Due to Theorem \ref{thMp-stability} we can now say that:
\begin{itemize}
\item $M_p$-stability (i. e. $M_p^\gamma$-stability with $\gamma=1)$
of equation \eqref{2} implies the Lyapunov $p$-stability of
\eqref{2};
\item $M_p^\gamma$-stability of \eqref{2} with $\gamma$
satisfying $\gamma(t) \ge \delta > 0$ and $ \lim_{t\to +\infty} \gamma (t)=
+\infty$ implies the asymptotic $p$-stability of \eqref{2};
\item $M_p^\gamma$-stability of \eqref{2} with $\gamma
(t) = \exp\{\beta t\}$ (for some $\beta > 0$) implies the
exponential $p$-stability of \eqref{2}.
\end{itemize}
Thus, we have replaced stability analysis of
\eqref{2} by the problem of how resolve this equation in a certain
space of stochastic processes. This observation is crucial for
applying the method based on the $W$-transform, which we are going
to describe now.
As we already have mentioned any $W$-transform comes from an
auxiliary equation, which we call \emph{a reference equation}.
That is why we assume given another equation, similar to
\eqref{2}, but ``simpler". In addition, we assume the asymptotic
properties of the reference equation to be known.
Let the reference equation have the form
\begin{equation}\label{4}
dx(t) = [(Qx)(t) + g(t)]dZ(t) \ \ (t \ge 0),
\end{equation}
where $Q: D^n \to L^n(Z)$ is a $k$-linear Volterra
operator, and $g \in L^n(Z)$. Also for equation \eqref{4} it
is always assumed the existence and uniqueness assumption, i. e.
for any $x(0) \in k^n$ there is the only (up to a ${\bf P}$-null
set) solution $x(t)$ of \eqref{4}. Then, according to
Lemma \ref{Lemma1}, for this solution we have "the Cauchy
representation" $x(t) = U(t)x(0) + (Wg)(t) \ (t \ge 0)$, where
$U(t)$ is the fundamental matrix of the associated homogeneous
equation, and $W$ is the corresponding Cauchy operator.
Let us rewrite equation \eqref{2} in the form
$$
dx(t) = [(Qx)(t) + ((V-Q)x)(t) + f(t)]dZ(t) \quad (t \ge 0),
$$
or, alternatively,
$$
x(t) = U(t)x(0) + (W(V-Q)x)(t) + (Wf)(t) \quad (t \ge 0).
$$
Denoting $W(V - Q) = \Theta_l$, we obtain the operator equation
\begin{equation}\label{OperatorEquation}
((I - \Theta_l)x)(t) = U(t)x(0) + (Wf)(t) \quad (t \ge 0).
\end{equation}
Here and in the sequel by invertibility of the
operator $(I - \Theta_l): M_p^\gamma \to M_p^\gamma $ we
mean that this operator is a bijection on the space $M_p^\gamma $.
\begin{remark} \rm
The letter "l" in $\Theta_l$ stands for ``left", which indicates
that the W-transform is applied to the equation from the
left hand side. Equivalently, one can apply the W-transform from
the right. This will lead to a similar theory and a similar
operator $\Theta_r$. This approach is not studied in this paper.
However, we are planning to develop it in one of the forthcoming
papers.
\end{remark}
The following important result is proved in \cite{k4}.
\begin{theorem}[\cite{k4}] \label{Th1}
Let the reference equation \eqref{4} be $M_p^\gamma $-stable and
the operator $\Theta_l$ act in the space $M_p^\gamma$. Then, if
the operator $(I - \Theta_l): M_p^\gamma \to M_p^\gamma$
is invertible, then the equation \eqref{2} is $M_p^\gamma$-stable.
\end{theorem}
Let us stress that, in general, there is no direct dependence
between stability of the equation in question and invertibility of
the operator equation (\ref{OperatorEquation}). For instance, in
certain situations the operator $\Theta_l$ even does not act in
the corresponding space of stochastic processes, while both
equations \eqref{2} and \eqref{4} are stable in this space.
Nevertheless, if \eqref{2} is stable, then there
always is at least one (in fact, infinitely many) stable reference
equations, for which the operator $\Theta_l$ will act in the
related space of stochastic processes and $I - \Theta_l$ will be
invertible there. For instance, one can choose equation
\eqref{4} be identical to (or be in the vicinity of) the initial
equation \eqref{2}.
This observation and Theorem \ref{Th1} imply the following
stability criterion based on the W-transform.
\begin{corollary}\label{Cor1}
Equation \eqref{2} is $M_p^\gamma $-stable if and only if
there exists a reference equation \eqref{4}, which is $M_p^\gamma
$-stable and which gives rise to the invertible operator $(I -
\Theta_l): M_p^\gamma \to M_p^\gamma $.
\end{corollary}
Among the assumptions imposed on the initial equation
\eqref{2} and the reference equation \eqref{4} one is more
involved than the others when applying Theorem \ref{Th1}. It is
invertibility of the operator $(I - \Theta_l): M_p^\gamma
\to M_p^\gamma $. A reasonable method to check this
requirement in practice is to estimate the norm of the operator
$\Theta_l$ in the space $M_p^\gamma $.
Thus, from Theorem \ref{Th1} we obtain the following simple
result.
\begin{corollary}\label{Cor2}
Assume that there is a $M_p^\gamma $-stable reference equation
\eqref{4}, for which the operator $\Theta_l$ act in the space
$M_p^\gamma$ and $\|\Theta_l\|_{M_p^\gamma} < 1$. Then
\eqref{2} is $M_p^\gamma $-stable.
\end{corollary}
In what follows we will need a more explicit description of the
$W$-transform (and the corresponding reference equation
\eqref{4}), which is summarized in the assumptions below:
\begin{itemize}
\item[\bf (R1)] The fundamental matrix $U(t)$ to \eqref{4}) satisfies
$\|U(t)\| \le \bar c,$ where $\bar c \in\mathbb{R}_+$.
\item[\bf (R2)] The W-transform coming from \eqref{4}) has the form
\begin{equation}\label{5}
(Wg)(t) = \int _0^tC(t,s)g(s)dZ(s) \quad (t \ge 0),
\end{equation}
where $C(t,s)$ is an $n\times n$-matrix defined on
$G := \{(t,s): t \in [0,\infty ), \; 0 \le s \le t\}$,
and satisfies
\begin{equation}\label{5-bis}
\|C(t,s)\| \le \bar{\bar c}\exp \{-\alpha \Delta v\},
\end{equation}
where $v(t) = \int_0^t\xi (\zeta )d\lambda (\zeta )$,
$\Delta v = v(t) - v(s)$ for some $\alpha>0, \bar{\bar c}>0$.
\end{itemize}
\begin{example}\label{remCauchyRepresent} \rm
Let a reference equation be given by
\begin{equation}\label{ModelODE}
dx(t)=\left(A(t)x(t-)+g_0(t)\right)d\lambda(t) +
\sum_{i=2}^{m}g_i(t)dz^i(t) \quad (t\ge 0),
\end{equation}
where $A(t)$ is an $n\times n$-matrix with locally
$\lambda$-integrable entries (in this case
$(Qx)(t)=(A(t)x(t),\bar 0, \dots, \bar 0)$).
In this case it is straightforward that
the kernel $C(t,s)$ in \eqref{5} is of the form
$C(t,s)=U(t)U^{-1}(s)$. Note also that if we set
$A(t)=-\alpha\xi\bar E$, then conditions (R1) and (R2)
will be fulfilled.
A more involved example of a reference equation is given by
\begin{equation}\label{ModelDDE}
dx(t)=\Big(\int_{[0,t)}d_s\mathcal{R}(t,s)x(s)+g_0(t)\Big)d\lambda(t)
+ \sum_{i=2}^{m}g_i(t)dz^i(t) \quad (t\ge 0),
\end{equation}
where the entries $r_{jk}(t,s)$ of a (non-random) $n\times
n$-matrix $\mathcal{R}(t,s)$, defined on the set $G$ from (R2), are of bounded variation in $s$ and, in addition,
$\bigvee_{s\in [0, t] }r_{jk}(t,s)$ ($t\in [0, \infty)$)
are locally $\lambda$-integrable for all $1\le j,k \le n$. In this
case, the representation \eqref{5} is again valid, but there is no
direct relations between $C(t,s)$ and the fundamental matrix
$U(t)$. The estimate (\ref{5-bis}) can be obtained in special
cases (see \cite{a2} and \cite{g1} for details). One particular
case of this reference equations is used in this paper (see
Section \ref{sect4}).
\end{example}
For more examples of reference equations \eqref{4}, giving
rise to the W-transforms of the form \eqref{5} and which satisfy
all the additional assumptions listed above, see \cite{k2}.
In the rest of this section we will be concerned with the
$M_p^\gamma $-stability of \eqref{2} with
\begin{equation}\label{SpecificGamma}
\gamma (t) = \exp \Big\{\beta \int_0^t\xi (s)d\lambda
(s)\Big\} \quad (t\ge 0),
\end{equation}
where $\beta $ is some positive number satisfying $\beta <\alpha$
(see (R2) for the notation). This specific weight $\gamma$
comes naturally from the W-transform satisfying (R1)-(R2).
We wish to use such a W-transform and the corresponding weight
$\gamma$ in order to prove two main results of this section
(Theorems \ref{Th2} and \ref{Th3}). The first theorem justifies
the W-method in connection with $M_p^\gamma$-stability (and by
this to the Lyapunov stability of \eqref{2} with respect to
the initial value $x(0)$). The second theorem deals with the
following fundamental problem which is also well-known for
deterministic functional differential equations (see e. g.
\cite{a2}): find conditions, under which the $p$-stability implies
the exponential $p$-stability. We shall prove that it is the case
if the delay function satisfies the so-called ``$\Delta$-condition''
(see Definition \ref{Def3} below). The $\Delta$-condition is
fulfilled if for instance the delays are bounded (see Lemma
\ref{lemSmallDeltaUsl}). Apart from the importance of these two
general facts for the theory of stochastic functional differential
equations, the technique we use to prove them is itself a good
illustration of how the W-transform works in practice.
For further purposes we will need the following technical lemma.
\begin{lemma}\label{Lemma2}
If the reference equation \eqref{4} satisfies the assumption (R2), then $W$, given by \eqref{5}, is a continuous operator from
$(\Lambda_{2p,q}^n(\xi ))^\gamma$ to $M_{2p}^\gamma$, where $2p
\le q \le \infty $ and $\gamma$ is defined by
\eqref{SpecificGamma} for all $\beta$ \ ($0<\beta<\alpha$), the
number $\alpha$ is the same as in \eqref{5-bis}.
\end{lemma}
\begin{proof}
To prove the lemma it suffices to check that
\begin{equation}\label{Proof-lemma2-1}
\|Wg\|_{M_{2p}^\gamma} \le \bar c \|g\|_{(\Lambda_{2p,q}^n(\xi
))^\gamma} \quad (\bar c \in \mathbb{R}_+),
\end{equation}
if $g \in (\Lambda_{2p,q}^n(\xi ))^\gamma$.
From the definition of the space $M_{2p}^\gamma$,
$$
\|Wg\|_{M_{2p}^\gamma} = \|\gamma Wg\|_{M_{2p}} = \|\gamma \int
_0^{\cdot}C(\cdot,s)g(s)dZ(s)\|_{M_{2p}}.
$$
We are to show that
\begin{equation}\label{Proof-lemma2-2}
lg :=\|\int_0^{\cdot}
C(\cdot,s)g(s)dZ(s)\|_{M_{2p}} \le \bar c\|\gamma g\|_{\Lambda
_{2p,q}^n(\xi )},
\end{equation}
where $\bar c$ is some positive number.
We have
\begin{align*}
lg & \le \bar{\bar c}(\sup _{t \ge 0}
({\bf E}(\int_0^t \exp \{-(\alpha-\beta) \Delta v \}
|\gamma(s)g(s)a(s)|d\lambda (s))^{2p})^{1/(2p)} \\
&\quad + c_p \sup _{ t \ge 0}
({\bf E}(\int _0^t \exp \{-2(\alpha-\beta) \Delta v\}
|\gamma(s)g(s)A(s)(\gamma (s)g(s))^\top |d\lambda (s))^p)^{1/(2p)}) \\
& \le \bar{\bar c}(\sup _{t \ge 0 }
(\int _0^t \exp \{-(\alpha -\beta ) \Delta v\}dv(s))^{(2p-1)/2p} \\
&\quad \times (\int _0^t \exp \{-(\alpha-\beta) \Delta v\}
(\xi (s))^{1-2p}{\bf E}|\gamma (s)g(s)a(s)|^{2p}d\lambda (s))^{1/(2p)} \\
&\quad + c_p \sup _{t \ge 0 }
(\int _0^t \exp \{-2(\alpha-\beta) \Delta v\}dv(s))^{(p-1)2p} \\
&\quad \times (\int _0^t \exp \{-2(\alpha-\beta) \Delta v\}
(\xi (s))^{1-p}{\bf E}\|\gamma (s)g(s)A(s) (\gamma (s)g(s))^\top
\|^pd\lambda (s))^{1/(2p)}) \\
& \le \hat c\Big\{\sup_{t \ge 0} (\int _0^t \exp \{-(\alpha-\beta) \Delta v\} (\xi
(s))^{1-2p}{\bf E}|\gamma(s)g(s)a(s)|^{2p}d\lambda (s))^{1/(2p)} \\
&\quad + c_p\sup_{t \ge 0}(\int _0^t \exp \{-2(\alpha-\beta ) \Delta v\} (\xi
(s))^{1-p}{\bf E}\|\gamma(s)g(s)A(s) (\gamma(s)g(s))^\top \|^p\\
&\quad\times d\lambda(s))^{1/(2p)}\Big\},
\end{align*}
where $\hat{c}$ is some positive number.
Here we have used the inequality \noindent
\begin{align*}
&\Big({\bf E}\Big|\gamma \int_0^tC(t,s)g(s)dZ(s)\Big|^{2p}\Big)^{1/(2p)} \\
\le & \bar{\bar c}\Big({\bf E}\Big(\int _0^t \exp \{-(\alpha- \beta) \Delta v\}
|\gamma (s)g(s)a(s)|d\lambda (s)\Big)^{2p}\Big)^{1/(2p)} \\
&\quad + c_p\bar{\bar c}\Big({\bf E}\Big(\int _0^t \exp \{-2(\alpha-\beta)
\Delta v\} \|\gamma (s)g(s)A(s)|(\gamma (s)g(s))^\top \|d\lambda (s)
\Big)^p\Big)^{1/(2p)},
\end{align*}
which follows directly from the estimates \eqref{1} and (\ref{5-bis}).
To obtain further estimates we have to consider three
cases separately: (1) $q> 2p$, $q \ne \infty $; (2) $q=2p$; (3) $q = \infty $.
Let first $q > 2p$, $q \ne \infty $. Then
\begin{align*}
lg & \le \hat c\{\sup _{t \ge 0}[(\int _0^t
\exp \{(-(\alpha-\beta) q/(q-2p)) \Delta v\}dv(s))^{(q-2p)/2pq} \\
&\quad \times (\int _0^t (({\bf E}|\gamma(s)g(s)a(s)|^{2p})^{1/(2p)} (\xi
(s))^{q^{-1}-1})^qd\lambda (s))^{1/q}] \\
&\quad + c_p\sup _{t \ge 0}[(\int _0^t
\exp \{(-2(\alpha-\beta) q/(q-2p)) \Delta v\}dv(s))^{(q-2p)/2pq}\\
&\quad \times (\int _0^t (({\bf E}\|\gamma(s)g(s)A(s)
(\gamma(s)g(s))^\top \|^p)^{1/(2p)} (\xi(s))^{q^{-1}-0.5})^qd\lambda
(s))^{1/q}]\} \\
& \le \bar c\|\gamma g\|_{\Lambda_{2p,q}^n(\xi )}.
\end{align*}
Assume now that $ q=2p$. In this case we derive the estimate
\begin{align*}
lg & \le \hat c \{\sup _{t \ge 0}(\int _0^t
(({\bf E}|\gamma(s)g(s)a(s)|^{2p})^{1/(2p)} (\xi
(s))^{q^{-1}-1})^qd\lambda (s))^{1/q} \\
&\quad + c_p\sup _{t \ge 0} (\int _0^t (({\bf E}\|
\gamma(s)g(s)A(s)(\gamma(s)g(s))^\top \|^p)^{1/(2p)} (\xi
(s))^{q^{-1}-0.5})^qd\lambda (s))^{1/q}\} \\
& \le \bar c\|\gamma g\|_{\Lambda_{2p,q}^n(\xi )}.
\end{align*}
Finally, if $q = \infty $, then we have
\begin{align*}
lg & \le \hat c\{\sup _{t \ge 0}(\int _0^t
\exp \{-(\alpha -\beta) \Delta v\}\xi (s) [({\bf
E}|\gamma(s)g(s)a(s)|^{2p})^{1/(2p)}(\xi (s))^{-1}]^{2p} d\lambda
(s))^{\frac 1{2p}} \\
&\quad + c_p\sup_{t \ge 0}\int_0^t\exp \{-2(\alpha -\beta)\Delta v\}\xi
(s)[({\bf E}\|\gamma(s)g(s)A(s)(\gamma(s)g(s))^\top\|^p \\ \\
&\quad \times (\xi (s))^{-0.5})^{1/(2p)}]^{2p} d\lambda (s))^{1/(2p)}\} \\
& \le \hat c \{{\mathrel {\mathop{\mbox vrai{\,}sup}_{0 \le
t \le \infty }}} [({\bf E}|\gamma(t)g(t)a(t)|^{2p})^{1/(2p)} (\xi
(t))^{-1}]{(1/\alpha )}^{1/(2p)} \\
&\quad + c_p{\mathrel {\mathop {\mbox vrai{\,}sup}_{0 \le t \le
\infty }}} [({\bf E}\|\gamma(s)g(s)A(s) (\gamma(s)g(s))^\top
\|^p)^{1/(2p)}(\xi (t))^{-0.5}] {(1/2\alpha )}^{1/(2p)}\} \\
& \le \bar c\|\gamma g\|_{\Lambda_{2p,q}^n(\xi )}.
\end{align*}
The proof of the lemma is completed.
\end{proof}
\begin{corollary}\label{Cor3}
Assume that the reference equation \eqref{4} satisfies (R2).
Then $W$ given by \eqref{5} is a continuous operator from $\Lambda
_{2p,q}^n(\xi )$ to $M_{2p}$, where $2p \le q \le \infty $.
\end{corollary}
From Lemma \ref{Lemma2} and Theorem \ref{Th1} we obtain the following
result.
\begin{theorem}\label{Th2}
Let $\gamma(t)$
be given by \eqref{SpecificGamma} for some $\beta$
($0<\beta<\alpha$, where $\alpha$ is taken from (\ref{5-bis})).
Assume that the reference equation \eqref{4} is
$M_{2p}^\gamma$-stable. Assume also that the operators $V$ and $Q$
from \eqref{2}) and \eqref{4}, respectively, act from
$M_{2p}^\gamma$ to $(\Lambda_{2p,q}^n(\xi ))^\gamma $. Then the
estimate $\|\Theta \|_{M_{2p}^\gamma} < 1$ implies the
$M_{2p}^\gamma $-stability of \eqref{2}, where $2p
\le q \le \infty $.
\end{theorem}
This theorem offers a formal justification of stability analysis
in the case (rather general) when the W-transform is given by
\eqref{5}.
To formulate and prove the second main result of this
section we need some preparations. Below $m_p$ stands for the
space $M_p$ in the scalar case. We assume that the $k$-linear
operator $V$ in \eqref{2} satisfies
$V: M_p \to \Lambda _{p,q}^n(\xi )$.
We will also use the following notation related
to the operator $V$:
\begin{itemize}
\item $Vx = (V_1x, \dots, V_mx)$;
\item $(V^\beta x)(t) := \gamma (t)(V(x/\gamma ))(t)$ , where
$\gamma(t)$ is defined in (\ref{SpecificGamma}).
\end{itemize}
\begin{definition}\label{Def2} \rm
We say that a $k$-linear Volterra operator $\bar V: m_p
\to \Lambda_{p, q}^{1+}(\xi )$ dominates a Volterra
operator $V: M_p \to \Lambda_{p, q}^n(\xi )$ , if 1)
$\bar V$ is positive, i. e. $x \ge 0$ a. s. implies $\bar Vx \ge
0$ a. s., and 2) $(|V_1x|, \dots, |V_mx|) \le \bar V|x|$ a. s. for
any $x \in M_p$.
\end{definition}
\begin{definition}\label{Def3} \rm
We say that a $k$-linear Volterra operator $V: M_p \to
\Lambda_{p, q}^n(\xi )$ satisfies the $\Delta$-condition, if $V$
is dominated by some $k$-linear Volterra operator $\bar V: m_p
\to \Lambda_{p, q}^{1+}(\xi )$ with the following
additional assumption: there exists a number $\beta > 0$, for
which the operator $$(\bar{V}^\beta x)(t) := \gamma
(t)(\bar{V}(x/\gamma ))(t)$$ acts continuously from the space
$m_p$ to the space $\Lambda_{p, q}^{1+}(\xi )$.
\end{definition}
\begin{definition}\label{defSmallDeltaUsl} \rm
Let $X$, $Y$ be two linear spaces consisting of predictable
stochastic processes on $[0, \infty)$, and $T: X\to Y$ be a
$k$-linear Volterra operator. We say that the operator $T$
satisfies {\bf$\delta$-condition} if there exist two positive
numbers $\delta '$, $\delta ''$, $\delta '>\delta ''$, providing
the following implication for all $t\in [0,\infty)$: any $x\in
X$, satisfying $x(\zeta ')=0$ for all $\zeta '\in [0, t]$ such
that $\int_{\zeta '}^t\xi(s)d\lambda(s)<\delta '$, also
satisfies $(Tx)(\zeta '')=0$ for all $\zeta ''\in [0, t]$ such
that $\int_{\zeta ''}^t\xi(s)d\lambda(s)<\delta ''$.
\end{definition}
\begin{lemma}\label{lemSmallDeltaUsl}
Assume that a $k$-linear Volterra operator $V: \, M_p\to \Lambda
_{p,q}^n(\xi )$ is dominated by a $k$-linear, bounded and positive
operator $\bar{V}: \, m_p\to \Lambda_{p,q}^{1+}(\xi )$ satisfying
the $\delta$-condition. Then the operator $V$ satisfies the
$\Delta$-condition.
\end{lemma}
\begin{proof}
According to our notation
$$
(\bar{V}^{\beta_0}x)(t)=\Big(\bar{V}\Big(\exp\{\beta_0\int_{\cdot}^t\xi(s)
d\lambda(s)\}x\Big)\Big)(t).
$$
The $\delta$-condition from Definition \ref{defSmallDeltaUsl}
implies that the value $(\bar{V}y)(t)$ depends only on the values
$y(\zeta '')$, where $\int_{\zeta
''}^t\xi(s)d\lambda(s)<\delta ''$ (here $\delta ''$ is again taken
from Definition \ref{defSmallDeltaUsl} and $\zeta ''\in[0,t]$),
and for these $\zeta ''$ we have $\exp\{\beta_0\int_{\zeta
''}^t\xi(s)d\lambda(s)\}\le \exp\{\beta_0\delta ''\}$. This leads
to the following estimate
$$
\bar{V}^{\beta_0}x\le \bar{V}(\exp\{\beta_0\delta
''\}|x|)=\exp\{\beta\delta ''\}\bar{V}|x|
$$
almost everywhere.
\end{proof}
In examples below we use equations with a discrete delay as
reference equations. The next definition describes the
corresponding operators.
\begin{definition}\label{defVnutrSuperp} \rm
Given a measurable function $g: [0,\infty)\to \mathbb{R}$ such that
$g(t)\le t$ ($t\in [0,\infty)$) and a row vector $G=(G_1, G_2,
\dots, G_m)$, where $G_i=G_i(t)$ are all predictable and nonnegative
stochastic processes, we define the weighted shift operator $GS_g$
by $(GS_gx)(t)=G(t)(S_gx)(t)$, where
\begin{equation}\label{Vnutrsuperp}
(S_gx)(t) = \begin{cases}
x(g(t)), & \mbox {if } g(t) \ge 0\, , \\
0, &\mbox {if } g(t) <0 \,.
\end{cases}
\end{equation}
\end{definition}
Clearly, $GS_g: D^1\to L^1(Z)$.
To check the $\delta$-condition from Definition
\ref{defSmallDeltaUsl} for weighted shifts we will use special
conditions on $g$. We will also need some new notation: for a
given measurable function $g: \, [0,\infty)\to \mathbb{R}$ we will
write
\begin{equation}\label{chi}
\chi_g(t) = \begin{cases}
1, & \mbox {if } g(t) \ge 0 \,,\\
0, & \mbox {if } g(t) <0\,.
\end{cases}
\end{equation}
\begin{definition}\label{defFuncDeltaUsl} \rm
We say that a measurable function $g: [0,\infty)\to \mathbb{R}$
satisfies the $\delta$-condition if there exists $\delta >0$ such
that $\int_{\chi_g(t)g(t)}^t\xi(s)d\lambda(s)<\delta $ for
all $t\in [0, \infty)$.
\end{definition}
\begin{example}\label{exVnutrSuperp} \rm
If a measurable function $g: [0,\infty)\to \mathbb{R}$ satisfies
the $\delta$-condition from Definition \ref{defFuncDeltaUsl}, then
the weighted shift operator $GS_g: D^1\to L^1(Z)$ satisfies the
$\delta$-condition from Definition \ref{defSmallDeltaUsl}.
To see this, we notice that according to Definition
\ref{defFuncDeltaUsl} there exists $\delta >0$ such that
$$
\int_{\chi_g(t)g(t)}^t\xi(s)d\lambda(s)<\delta \ \ \mbox{for
all} \ \ t\in [0, \infty).
$$
Setting $\delta '=2\delta$, $\delta
''=\delta$ and taking arbitrary $t\in [0,\infty)$ and $x\in D^1$,
for which $y(\zeta ')=0$ for all $\zeta '\in [0, t]$ satisfying
$\int_{\zeta '}^t\xi(s)d\lambda(s)<\delta '$, we have to
check that $(GS_gx)(\zeta '')=0$ a.s. for all $\zeta ''\in [0, t]$
such that $\int_{\zeta ''}^t\xi(s)d\lambda(s)<\delta ''$.
This follows from the equality $(S_gx)(\zeta '')=0$ a.s., or
equivalently, from the estimate $\int_{\chi_g(\zeta
'')g(\zeta '')}^t\xi(s)d\lambda(s)<\delta '$. But this is implied
by
$$
\int_{\chi_g(\zeta '')g(\zeta '')}^t\xi(s)d\lambda(s)=
\int_{\chi_g(\zeta '')g(\zeta '')}^{\zeta
''}\xi(s)d\lambda(s) + \int_{\zeta ''}^t\xi(s)d\lambda(s)
<\delta + \delta '' =\delta '.
$$
Note also that if $\lambda =t$ (i.e. $\lambda^*$ is the standard
Lebesgue measure), then $\xi(t)\equiv 1$, and the
$\delta$-condition for $g$ takes the following form:
$t-g(t)\le\delta \ (t\ge 0)$, i. e. the delay will be bounded.
\end{example}
The concluding result of this section explains when the usual
stability of solutions implies the exponential and asymptotic
stability. We present here only a general principle, postponing
all further discussions and examples until the last section.
\begin{theorem}\label{Th3}
Let equation \eqref{2} and the reference equation \eqref{4}
satisfy the following assumptions:
\begin{itemize}
\item The operators $V,Q$ act as follows: $V,Q: M_{2p} \to\Lambda_{2p,q}^n (\xi )$,
where $2p\le q<\infty$
\item The reference equation \eqref{4} is $M_{2p}$-stable and
satisfies condition (R2)
\item The operator $V$ satisfies the $\Delta$-condition.
\end{itemize}
If now the operator $(I - \Theta_l): M_{2p}\to M_{2p}$ is
continuously invertible, then \eqref{2} is
$M_{2p}^\gamma $- stable, where $\gamma$ is defined in
(\ref{SpecificGamma}) with some $\beta > 0$.
\end{theorem}
\begin{remark}\label{Rem1} \rm
Note that under the assumptions of Theorem \ref{Th3} the operator
$\Theta_l$ does act in the space $M_{2p}$ (see Corollary
\ref{Cor3}).
\end{remark}
\begin{proof}[Proof of Theorem \ref{Th3}]
First of all, we notice that \eqref{2} is
$M_{2p}^\gamma$-stable if and only if the equation
\begin{equation}\label{6}
\begin{aligned}
dy(t) & = \exp \{\beta \int_0^t\xi (\zeta )d\lambda
(\zeta )\} \Big[\Big(V\Big(\exp \big\{-\beta \int
_0^{\cdot}\xi (\zeta )d\lambda (\zeta
)\big \}y\Big)\Big)(t)+f(t)\Big]dZ(t) \\
&\quad + \beta \xi (t)y(t)d\lambda (t) \quad (t \ge 0)
\end{aligned}
\end{equation}
is $M_{2p}$-stable. Hence, in order to prove the theorem it is
sufficient to show the existence of a positive number $\beta $,
for which \eqref{6} will be $M_{2p}$-stable. From
Theorem \ref{Th1} it follows that if the operator $\Theta_l^\beta
$ acts in the space $M_{2p}$, and the operator $(I -
\Theta_l^\beta ):M_{2p} \to M_{2p}$ is invertible for some
$\beta >0$, then \eqref{6} will be $M_{2p}$-stable
for this $\beta$. Here $\Theta_l^\beta $ is a $k$-linear operator
defined, according to our previous notational agreements, by
\begin{equation}\label{Theta-l-beta}
(\Theta_l^\beta x)(t) := \gamma (t)(\Theta_l(x/\gamma ))(t),
\end{equation}
so that, in particular, $\Theta_l^0 = \Theta_l$.
Using the assumption of the theorem saying that the operator $V$
satisfies the $\Delta $-condition, we obtain a number
$\beta_0 > 0$, for which the operator $\Theta_l^\beta $ acts
continuously in the space
$M_{2p}$ for all $0 \le \beta \le \beta_0$. This fact follows
from Corollary \ref{Cor3} and a simple observation that if the
operator $V$ satisfies the $\Delta $-condition, then the operator
$\bar V^{\beta } $ acts from the space $m_p$ to the space $\bar
\Lambda_{2p,\infty }^{1+}(\xi )$ and, moreover, it is bounded for
all $0 \le \beta \le \beta_0$. If now we check that
\begin{equation}\label{Proof-Th3-1}
\|\Theta_l^\beta - \Theta_l\|_{M_{2p}} \to 0,
\end{equation}
when $\beta \to 0$, then the operator $(I - \Theta_l^\beta
): M_{2p} \to M_{2p}$ will also be invertible for some
$\beta > 0$. Clearly, in this case the continuous extension of the
operator $(I - \Theta_l^\beta ):M_{2p} \to M_{2p}$ to the
completion of the space $M_{2p}$ will be invertible as well. The
fact that the operator $(I - \Theta_l^\beta ): M_{2p} \to
M_{2p}$ will be invertible can be derived from the observation
that the solution of the equation $(I - \Theta_l^\beta )x =g$
belongs to the space $D^n$ if $g \in M_{2p}$, while the
intersection of the space $D^n$ with the completion of the space
$M_{2p}$ coincides with the space $M_{2p}$.
Note that the operator $(\Theta_l^\beta - \Theta_l)$ is given by
$$
((\Theta_l^\beta - \Theta_l)x)(t)
= \int_0^tC(t,s)(V(\gamma (t)/\gamma (.) - 1)x)(s)dZ(s) + \int
_0^tC(t,s)\beta \xi (s)x(s)d\lambda(s).
$$
Then
\begin{align*}
&\|(\Theta_l^\beta - \Theta_l)x\|_{M_{2p}}\\
&\le \sup _{t\ge 0}\Big({\bf E}\Big(\int_0^t|C(t,s)(V(\gamma (t)/\gamma (.) -
1)x)(s)a(s)|d\lambda (s)\Big)^{2p}\Big)^{1/(2p)} \\
&\quad + c_p\sup _{t \ge 0}({\bf E}(\int_0^t\|C(t,s)(V(\gamma
(t)/\gamma (.) - 1)x)(s)A(s)\\
&\quad\times (C(t,s)(V(\gamma (t)/\gamma (.) -
1)x)(s))^{\top}\|d\lambda (s))^{p})^{1/(2p)} \\
&\quad + \sup _{t \ge 0}\Big({\bf E}\Big(\int_0^t|C(t,s)\beta \xi
(s)x(s)|d\lambda (s)\Big)^{2p}\Big)^{1/(2p)} \\
&\le \sup _{t \ge 0}\Big({\bf E}\Big(\int_0^t\exp \{-\alpha
\Delta v\}|(V(\gamma (t)/\gamma (.) - 1)x)(s)a(s)|d\lambda
(s)\Big)^{2p}\Big)^{1/(2p)} \\
&\quad + c_p\sup _{t \ge 0}\Big({\bf E}\Big(\int_0^t\exp \{-2\alpha
\Delta v\}\|(V(\gamma (t)/\gamma (.) - 1)x)(s)A(s)\\
&\quad\times (V(\gamma
(t)/\gamma (.) - 1)x)(s)^{\top}\|d\lambda (s)\Big)^{p}\Big)^{1/(2p)} \\
&\quad + \sup _{t \ge 0}\Big({\bf E}\Big(\int_0^t\exp \{-\alpha \Delta
v\}|\beta \xi (s)x(s)|d\lambda (s)\Big)^{2p}\Big)^{1/(2p)}.\\
\end{align*}
Here we have used the inequality \eqref{1}.
The next estimation step is based on the $\Delta$-condition for
the operator $V$ and the inequality
$$
\gamma (t)/\gamma (s) - 1
\le (\beta /\beta_0) \exp \{\beta_0\Delta v\} \quad (s \in [0, t],
\ 0 \le \beta \le \beta_0),
$$
following from the estimate
\begin{align*}
\beta \nu + \beta ^2\nu ^2/2! + \beta ^3\nu^3/3! + \dots
&\le \beta /\beta_0 + \beta \nu + \beta ^2\nu ^2/2! + \dots \\
&\le (\beta /\beta_0) (1 + \beta _0\nu
+ \beta_0^2\nu^2/2! + \dots) \quad ( \nu >0).
\end{align*}
Using this estimate, we obtain
\begin{align*}
&\|(\Theta_l^\beta - \Theta_l)x\|_{M_{2p}} \\
& \le \sup _{t\ge 0}({\bf E}(\int_0^t\exp \{-\alpha \Delta v\}|(\bar
V(|\gamma (t)/\gamma (.) - 1\|x|))(s)(a(s))^+|d\lambda
(s))^{2p})^{1/(2p)} \\
&\quad + c_p\sup _{t \ge 0}({\bf E}(\int_0^t\exp \{-2\alpha
\Delta v\}\|(\bar V(|\gamma (t)/\gamma (.) - 1\|x|))(s)\\
&\quad \times (A(s))^+(\bar V(|\gamma (t)/\gamma (.) -
1\|x|))(s)^{\top}\|d\lambda (s))^{p})^{1/(2p)} \\
&\quad + \sup _{t \ge 0}({\bf E}(\int_0^t\exp \{-\alpha
\Delta v\}|\beta \xi (s)x(s)|d\lambda (s))^{2p})^{1/(2p)} \\
&\le \sup _{t \ge 0}({\bf E}(\int_0^t\exp \{-\alpha
\Delta v\}|(\beta /\beta_0)|(\bar V_{\beta
_0}|x|)(s)(a(s))^+|d\lambda (s))^{2p})^{1/(2p)} \\
&\quad + c_p\sup _{t \ge 0}({\bf E}(\int_0^t\exp \{-2\alpha
\Delta v\}(\beta ^2/\beta_0^2)\|\\
&\quad\times (\bar V_{\beta _0}|x|)(s)(A(s))^+
(\bar V^{\beta_0}|x|)(s)^{\top}\|d\lambda (s))^{p})^{1/(2p)} \\
&\quad + \sup _{t \ge 0}({\bf E}(\int_0^t\exp \{-\alpha
\Delta v\}|\beta \xi (s)x(s)|d\lambda (s))^{2p})^{1/(2p)} \\
& \le \sup _{t \ge 0}(\int_0^t \exp \{-\alpha
\Delta v\}dv(s))^{(2p-1)/2p}(\beta /\beta_0)\\
&\quad \times \int_0^t\exp \{-\alpha \Delta v\}(\xi
(s))^{1-p}{\bf E}|(\bar V^{\beta_0}|x|)(s)(a(s))^+|^{2p}d\lambda
(s))^{1/(2p)} \\
&\quad + c_p\sup _{t \ge 0}(\int_0^t \exp \{-2\alpha
\Delta v\}dv(s))^{(p-1)/2p}(\beta /\beta_0)\\
&\quad \times \int_0^t\exp \{-2\alpha \Delta v\}(\xi
(s))^{1-p}{\bf E}\|(\bar V^{\beta_0}|x|)(s)(A(s))^+(\bar V_{\beta
_0}|x|)(s)^{\top}\|^pd\lambda (s))^{1/(2p)}\\
&\quad + \beta \sup _{t \ge 0}(\int_0^t \exp \{-\alpha
\Delta v\}dv(s))^{(p-1)/2p}(\int_0^t\exp \{-\alpha \Delta
v\}\xi (s){\bf E}|x(s)|^{2p}d\lambda (s))^{1/(2p)} \\
& \le (\beta /\beta_0)(1/\alpha )^{(2p-1)/2p}\sup _{t \ge
0}(\int_0^t\exp \{-\alpha \Delta v\}(\xi (s))^{1-p}{\bf
E}|(\bar V^{\beta_0}|x|)(s)(a(s))^+|^{2p}\\
&\quad d\lambda (s))^{1/(2p)} \\
&\quad + (c_p\beta /\beta_0)(1/\alpha )^{(p-1)/2p}\sup _{t \ge
0}\int_0^t\exp \{-2\alpha \Delta v\}(\xi (s))^{1-p}\\
&\quad \times {\bf E}\|(\bar V^{\beta_0}|x|)(s)(A(s))^+(\bar V_{\beta
_0}|x|)(s)^{\top}\|^pd\lambda (s))^{1/(2p)} \\
&\quad + \beta (1/\alpha )^{(2p-1)/2p}\sup _{t \ge 0}
(\int_0^t\exp \{-\alpha \Delta v\}\xi (s) {\bf E}|x(s)|^{2p}d\lambda (s))^{1/(2p)}.
\end{align*}
To proceed, we have to consider three cases:
(1) $q> 2p, q \ne \infty $; (2) $q=2p$; (3) $q = \infty $.
Treating each case separately and making use of the last
estimate we obtain, as in the proof of Lemma \ref{Lemma2}, that
$$
\|(\Theta_l^\beta - \Theta_l)x\|_ {M_{2p}} \le (\beta/\alpha
)\|x\|_{M_{2p}} + \beta d\|\bar V^{\beta_0}\|_{\Lambda
_{2p,q}^{1+}(\xi )}
$$
for some positive number $d$. Hence, due to the boundedness of the
operator $$\bar V^{\beta_0}:m_{2p}\to \Lambda
_{2p,q}^{1+}(\xi ),$$ we get
$$
\|(\Theta_l^\beta - \Theta_l)x\|_ {M_{2p}} \le (\beta/\alpha
)\|x\|_{M_{2p}} + \beta \bar d\|x\|_{M_{2p}},
$$
where $\bar d$ is a positive number. From this we deduce that
$\|\Theta_l^\beta - \Theta_l\|_ {M_{2p}} \to 0$ as $\beta
\to 0$. This proves (\ref{Proof-Th3-1}), and as it is was
mentioned above, this suffices to complete the proof of the
theorem.
\end{proof}
\section{Admissible pairs of spaces and stability with respect to
the initial function}\label{Section3}
Another name for admissibility of pairs of spaces is stability
under constantly acting perturbations. Roughly speaking, given a
pair $(B_1, B_2)$ of spaces of stochastic processes, one calls it
{\it admissible} for a linear stochastic differential equation if
any solution of the equation lies in $B_1$ as soon as the
right-hand side of the equation (``perturbation") lies in $B_2$.
This terminology goes back to Massera and Sch\"{a}fer
\cite{Massera} who studied admissibility for ordinary
deterministic differential equations in Banach spaces. The main
idea of this theory is to connect admissibility and Lyapunov
stability (or the dichotomy of solution spaces). This approach
proved to be particularly useful for deterministic {\it functional
differential equations} \cite{a2}. Stochastic functional
differential equations admissibility was studied in
\cite{k2,k4}, and in this paper we continue those studies.
To outline this method in brief, let us again look at Example
\ref{Ex-privodimost6}. Suppose we want to study Lyapunov
stability of the solutions of (\ref{3.5}) with respect to
the initial function (\ref{3.6}). The usual
Lyapunov-Krasovskii-Razumikhin method suggests that we rewrite
(\ref{3.5}) as an equation in a Banach space of all initial
functions $\varphi$ (usually it is the space $C[-h, 0]$). A
detailed description of this approach in the case of stochastic
differential equation can e.g. be found in the monographs
\cite{Mao-book,t1}.
Another way is presented in \cite{a2} and developed in
\cite{k2,k4} for the case of stochastic delay differential
equations. The idea is to rewrite (\ref{3.5}) in a
different manner, namely in the form \eqref{2} with $V$ and $f$
defined in (\ref{3.7}), as it is described in Example
\ref{Ex-privodimost6}. By this, the initial function $\varphi$
will be included in the right-hand side of the equation, and
stability of (\ref{3.5}) with respect to $\varphi$ will be
reduced to a particular case of the general admissibility problem
for the functional differential equation \eqref{2}. This approach
is flexible and efficient, especially in the case of linear
equations. In its practical use, it is common to exploit the
W-transform as an additional tool.
The main objective of this section is to demonstrate how this
approach, in combination with the general results and techniques
developed in the previous section, can be utilized to derive
stability of stochastic delay differential equations with respect
to the initial function $\varphi$.
We start with some more notation.
Let $B$ be a linear subspace of the space $L^n(Z)$ (defined in
Section 1). The space $B$ is assumed to be equipped with a norm
$\|.\|_B$. Given a weight $\gamma (t) \ (t\in [0, \infty))$ we set
$B^\gamma $ = $\{f: f \in B, \gamma f \in B\}$, which is a linear
space with the norm $\|f\|_{B^\gamma } := \|\gamma f\|_B$.
For the sake of convenience we will also write $x_f(t,x_0)$ for
the (unique) solution of \eqref{2}. Here $f$ is the
right-hand side of \eqref{2} and $x_0$ is the initial
value of the solution, i. e. $x_f(0,x_0) = x_0$.
\begin{definition}\label{Def4} \rm
We say that the pair $(M_p^\gamma , B)$ is {\it admissible} for
equation \eqref{2} if there exists $\bar c > 0$, for which
$x_0 \in k_p^n$ and $f \in B$ imply $x_f(.,x_0) \in M_p^\gamma $
and the following estimate:
$$
\|x_f(\cdot,x_0)\|_{M_p^\gamma } \le \bar c(\|x_0\|_{k_p^n} +
\|f\|_B).
$$
\end{definition}
By definition the solutions belong to $M_p^\gamma$
whenever $f\in B$ and $x_0 \in k_p^n$ and depend continuously on
$f$ and $x_0$ in the appropriate topologies. The choice of spaces
is closely related to the kind of stability we are interested in.
The first two results in this section describe assumptions on the
reference equation that are to be checked if one wants to exploit
the W-transform to study admissibility.
\begin{theorem}\label{Th4}
Assume that the reference equation
\eqref{4} satisfies conditions (R1)-(R2). If the
operator $I - \Theta_l$ acts in the space $M_p^\gamma $ and has a
bounded inverse in this space, then the pair $(M_p^\gamma ,
B^\gamma )$ is admissible for \eqref{2}.
\end{theorem}
\begin{proof}
Under the above assumptions, $U(\cdot)x_0 \in
M_p^\gamma $ whenever $x_0 \in k_p^n$ and
$$
x_f(t,x_0) =((I - \Theta_l)^{-1}(U(\cdot)x_0))(t) + ((I - \Theta_l)^{-1}Wf)(t)
\quad (t\ge 0)
$$
for an arbitrary $x_0 \in k_p^n$, $f \in B^\gamma$. Taking the norms
and using (R1)-(R2) for the reference
equation, we arrive at the inequality
$$
\| x_f(\cdot,x_0)\|_{M_p^\gamma } \le \bar c(\|x_0\|_{k_p^n} +
\|f\|_{B^\gamma }),
$$
which holds for any $x_0 \in k_p^n$, $f \in B^\gamma $. Here $\bar c$
is some positive number. This means that the pair
$(M_p^\gamma , B^\gamma )$ is admissible for \eqref{2}.
\end{proof}
If, in addition, we have the $\Delta$-condition from Definition
\ref{Def3}, then we can prove more.
\begin{theorem}\label{Th5}
Let $\gamma$ be defined by
(\ref{SpecificGamma}), the assumptions of Theorem \ref{Th3} be
fulfilled and the reference equation \eqref{4} satisfy the
condition (R1). Then the pair $(M_{2p}^\gamma, (\Lambda
_{2p,q}^n(\xi ))^\gamma)$ is admissible for \eqref{2}
for some $\beta> 0$.
\end{theorem}
\begin{proof} First we note that the pair
$(M_{2p}^\gamma, (\Lambda_{2p,q}^n(\xi ))^\gamma)$ is admissible
for \eqref{2} if and only if the pair $(M_{2p},
(\Lambda_{2p,q}^n(\xi ))$ is admissible for the modified equation
\eqref{6}. The latter can be proved if we check that the
assumptions of Theorem \ref{Th5} imply the assumptions of Theorem
\ref{Th4} for the modified equation \eqref{6}, where we put
$\gamma=1$, $B=\Lambda_{2p,q}^n(\xi )$ and use $2p$ instead of
$p$ (so that $M_p$ becomes $M_{2p}$).
Then we check that there exists $\beta_0>0$ such that the operator
$$I-\Theta_l^\beta : M_{2p}\to M_{2p},$$
where $\Theta_l^\beta $ was defined in (\ref{Theta-l-beta}), has a
bounded inverse for all $0<\beta<\beta_0$. According to the proof
of Theorem \ref{Th3} we have
$\|\Theta_l^\beta-\Theta_l\|_{M_{2p}}\to 0$ as $\beta\to 0$, so
that the operator $I-\Theta_l^\beta$ is invertible for
sufficiently small $\beta>0$.
To see that the operator $W$ is continuous from the space
$\Lambda_{2p,q}^n(\xi )$ to the space $M_{2p}$ we apply Corollary
\ref{Cor3}. Summarizing, we conclude that for some $\beta> 0$ the
pair $(M_{2p}^\gamma , (\Lambda_{2p,q}^n(\xi ))^\gamma )$ is
admissible for \eqref{2}.
\end{proof}
We are now ready to investigate Lyapunov stability with respect to the
initial function $\varphi$. In the previous section we studied
stability with respect to the initial value $x(0)$. The difference
between these two stabilities can again be explained by virtue of
Example \ref{Ex-privodimost6}. In the initial condition
(\ref{3.6}) there is no formal difference between all the
``prehistory" values of the solution $x(s), s\le 0$. In fact, if we
change the value of the initial function $\varphi(s)$ for one (or
even countably many) $s<0$, then the solution $x(t), t> 0$ will
not be changed. If we, however, change the value $\varphi(0)$,
then the solution will be different, that is the instants $s=0$
and $s<0$ {\it are} different. This observation explains roughly
why it is reasonable to treat the function $\varphi(s), s<0$ and
$\varphi(0)=x(0)$ separately. That is why we rewrite the delay
equation (\ref{3.5}) with the initial condition (\ref{3.6}) as the
functional differential equation \eqref{2}. This idea proved to be
fruitful in many cases (see e. g. \cite{a2,k2} and
references therein).
In our paper we exploit this approach to study Lyapunov stability
wit respect to the initial function with the help of the theory of
admissible pairs of spaces and the W-method.
Generalizing Example \ref{Ex-privodimost6}, we consider a linear
stochastic differential equation with distributed delay of the
form
\begin{equation}\label{7}
\begin{gathered}
dx(t) = (\hat Vx)(t)dZ(t) \quad (t \ge 0),\\
x(\nu ) = \varphi (\nu ) \quad (\nu < 0),
\end {gathered}
\end{equation}
where
\begin{gather*}
(\hat Vx)(t) = (\int_{(-\infty, t)}d_s
\mathcal{R}_1(t,s)x(s), \dots, \int_{(-\infty, t)}d_s \mathcal{R}_m(t,s)x(s)), \\
\mathcal {R}_i(t,s) =\sum_{j=0}^{m_i}Q_{ij}(t)r_{ij}(t,s).
\end{gather*}
Equation \eqref{7} can be rewritten in the form \eqref{2} by
putting
\begin{equation}\label{Eq-operators-V-f}
\begin{gathered}
(Vx)(t) = (\int_{[0, t)}d_s \mathcal{R}_1(t,s)x(s),
\dots,\int_{[0, t)}d_s \mathcal{R}_m(t,s)x(s)), \\
f(t) =(\int_{(-\infty, 0)}d_s \mathcal{R}_1(t, s)\varphi (s), \dots,
\int_{(-\infty, 0)}d_s \mathcal{R}_m(t,s) \varphi(s)),\\
\end{gathered}
\end{equation}
where $Q_{ij}$ are $n\times n$-matrices with the entries being
predictable stochastic processes and $r_{ij}$ are scalar functions
defined on $\{(t,s): t \in [0, \infty ),\ -\infty < s \le t\}$ for
$i=1,\dots,m$; $j=0,\dots,m_0$.
Let
\begin{gather*}
H_0^i(t) {= \sum_{j=0}^{m_i}\|Q_{ij}(t)\| \bigvee_{s\in(-\infty, 0)} r_{ij}(t,s)},
\\
H_1^i(t) = \sum_{j=0}^{m_i}\|Q_{ij}(t)\| \bigvee_{s\in[0,t]} r_{ij}(t,s)
\quad (i = 1,\dots,m), \\
H_j {= (H_j^1, \dots, H_j^m)} \quad (j = 0, 1).
\end{gather*}
Equation \eqref{7} will be considered under the assumption
$$
\int_0^t(|H_ja^+| + \|H_jA^+H_j^\top\|)d\lambda <
\infty \quad \mbox{a. e. for any } t \ge 0, \; j=0, 1.
$$
This implies, in particular, that $H_j\in L^n(Z)$ (compare the
last inequality with (\ref{SpaceL^n(Z)})). The initial function
$\varphi $ will be a stochastic process such that $f \in L^n(Z)$.
An example of such $\varphi$ is given by a stochastic process on
$(-\infty, 0)$ which is independent of the semimartingale $Z(t)$
and which has a. s. essentially bounded trajectories with respect to to
the measure $\lambda ^*$, generated by the function $\lambda(t)$.
If these assumptions are satisfied, then the operator $V$ in
equation \eqref{2}, defined by the first formula in
(\ref{Eq-operators-V-f}), will be $k$-linear and Volterra and act
from the space $D^n$ to the space $L^n(Z)$. In addition, for any
$x(0) \in k^n$ there will be the unique (up to a $P$-null set)
solution of \eqref{2} (remember that
\eqref{2} is equivalent to \eqref{7}). For the proof
of these results see \cite{k2}.
As a particular case of equation \eqref{7} we obtain
stochastic differential equations with ``ordinary", or concentrated
delay. Another name is difference-differential stochastic
equations. By this we mean the following object:
\begin{equation}\label{9}
\begin{gathered}
dx(t) = (\tilde{V}x)(t)dZ(t) \quad (t \ge 0), \\
x(\nu ) = \varphi (\nu ) \quad (\nu < 0),
\end {gathered}
\end{equation}
where
\begin{equation}\label{9-bis}
(\tilde{V}x)(t) = \Big(\sum_{j=0}^{m_1}\tilde{
Q}_{1j}(t)x(h_{1j}(t)), \dots, \sum_{j=0}^{m_m}\tilde{
Q}_{mj}(t)x(h_{mj}(t))\Big).
\end{equation}
Here $ h_{ij}$ are $\lambda ^*$-measurable functions, for
which
$$ h_{ij}(t) \le t\; (\lambda ^*-\mbox{a. e.}) \;
\mbox{for } t \in [0, \infty ), \; i = 1,\dots,m, \; j =0,\dots,m_i;
$$
$\tilde{Q}_{ij}$ are $n\times n$-matrices with the
entries that are predictable stochastic processes for all $i =
1,\dots,m$, $j = 0,\dots,m_i$; $\varphi $ is a stochastic process
which is independent of the semimartingale $Z(t)$.
The assumptions imposed on the general delay equation \eqref{7}
can easily be adjusted to its particular case \eqref{9}. The
details can be found in \cite{k2}. Here we just outline briefly how
equation \eqref{9} can be represented in the form \eqref{7}
and then formulate the assumptions on the coefficients. We set
$${\mathcal{R}_i(t,s) = \sum_{j=0}^{m_i}\tilde{
Q}_{ij}(t)r_{ij}(t,s)},$$ where $\tilde{ Q}_{ij}$ are the matrices
from (\ref{9-bis}) and $r_{ij}$ is the indicator (the
characteristic function) of the set
$$
\{(t,s): t \in [0, \infty ), \ h_{ij}(t) \le s\le t\},
$$
defined on $t \in [0, \infty ),\ s\in (-\infty, t]$ for
$i = 1,\dots,m$, $j =0,\dots,m_i$. By this,
equation \eqref{9} is rewritten in the form \eqref{7} and this
leads automatically to the following assumptions on the
coefficients of \eqref{9}:
$$
\int_0^t(|HA^+| + \|HA^+H^\top\|)d\lambda < \infty \quad
\mbox{ a. s. for any } t \ge 0,
$$
where
$$
H = (H^1, \dots, H^m), \quad H^i := \sum_{j=0}^{m_i}\|\tilde{ Q}_{ij}\|
\quad (i = 1, \dots, m);
$$
the initial function $\varphi $ is a stochastic process with
trajectories which are a. s. essentially bounded on $[0, \infty)$
with respect to the measure $\lambda ^*$.
In what follows we treat equation \eqref{9} as a special case
of \eqref{7}.
\begin{remark}\label{Rem2} \rm
The assumptions on the initial function $\varphi $ do not imply,
in general, that $\varphi $ should be cadlag. It is an important
observation for what follows as we are going to use a weaker
topology (the $L^p$-topology) on the set of all $\varphi$.
Moreover, we do not treat the solution $x(t)$ on $t\in [0,
\infty)$ as a continuation of the stochastic process $\varphi $.
This is an essential feature of the theory of functional
differential equations presented in \cite{a2} as it offers more
possibilities to choose a suitable topology in the space of
initial functions. A similar idea was also used in \cite{m1} to
define the Lyapunov exponents for stochastic flows associated with
certain linear stochastic functional differential equations. This
was motivated by the fact that Ruelle's multiplicative ergodic
theorem, which is needed to define the Lyapunov exponents,
requires the topology of a Hilbert space instead of the uniform
topology on the space of initial functions.
If we, nevertheless, want the solutions $x(t)$ of
\eqref{7} (or \eqref{9}) to be continuations of the initial
functions $\varphi (t)$, then we can easily treat this situation
as a particular case of the more general setting described above.
First of all we have to require that $\varphi (t)$ should be
cadlag (or continuous, if the semimartingale $Z(t)$ is
continuous). In addition, we set the continuity condition at
$t=0$, i. e. we demand that
$$x(0) = \lim_{\delta \to 0-}\varphi (\delta ).
$$ By this, the solution will be cadlag (or continuous) for all $t$.
\end{remark}
Now we describe different kinds of stability of solutions of
\eqref{7} and \eqref{9} which we intend to study in this
paper. The definitions below are classical, up to some small
adjustments, and can be found in many monographs (see e. g.
\cite{k5,m1,t1}).
In the next definition we use the following notation: $
x(t,x_0,\varphi)$ stands for the solution of \eqref{7}, with the
initial function $\varphi $, such that $ x(0,x_0,\varphi) = x_0$.
\begin{definition}\label{Def5}\rm
The zero solution of \eqref{7} (resp. of \eqref{9}) is called:
\begin{itemize}
\item $p$-{\it stable} with respect to the initial function,
if for any $\varepsilon > 0$ there exists $\eta (\varepsilon )> 0$
such that the inequality
$$
E|x_0|^p + \mathop {\rm vrai\,sup}_{\nu < 0}{\bf E}|\varphi (\nu
)|^p {< \eta }
$$
($\mathop{\rm vrai \, sup}$ is the essential sup with respect to
the measure $\lambda ^*$) implies the estimate
$$
{\bf E}|x(t,x_0,\varphi)|^p \le \varepsilon \quad (t \ge0)
$$
for any $\varphi (\nu )$, $\nu < 0$ and $x_0 \in k_p^n$;
\item {\it asymptotically} $p$-{\it stable} with respect to the initial
function, if it is $p$-stable with respect to the initial function and,
in addition, for any $\varphi (\nu )$, $\nu < 0$ and $x_0 \in
k_p^n$ such that
$$
{\bf E}|x_0|^p + \mathop{\rm vrai\,sup}_{\nu < 0}
{\bf E}|\varphi (\nu )|^p <\infty
$$
one has
$\lim_{t \to +\infty } {\bf E}|x(t,x_0,\varphi)|^p = 0$;
\item {\it exponentially}
$p$-{\it stable} with respect to the initial function, if there exist
positive constants $\bar c$, $\beta $ such that
$$
{\bf E}|x(t,x_0,\varphi)|^p \le \bar c({\bf E}|x_0|^p + \mathop{\rm vrai\,sup}_{\nu<0}{\bf E}|\varphi (\nu )|^p)\exp
\{-\beta t\} \ \ (t \ge 0)
$$
for any $\varphi (\nu )$, $\nu < 0$ and $x_0 \in k_p^n$.
\end{itemize}
\end{definition}
It is easy to see that $p$-stability (resp. asymptotic
$p$-stability, exponential $p$-stability) of the zero solution of
\eqref{7} with respect to the initial function implies
$p$-stability (resp. asymptotic $p$-stability, exponential
$p$-stability) of the zero solution of the homogeneous equation
(\ref{2homogen}), corresponding to \eqref{7}, with respect to the initial value $x(0)$. The converse is, in general, not
true, even in the case of deterministic delay equations (see e. g.
\cite{a2}).
The notions of admissibility and stability with respect to the initial
function are close to each other. In the following lemma we
assume, when treating admissibility, that \eqref{7}
is rewritten in the form \eqref{2}.
\begin{lemma}\label{Lemma3}
Assume that for any $\varphi $ such that ${\mathop{\rm vrai\,sup}_{\nu<0}{\bf E}|\varphi (\nu )|^p <
\infty }$ the stochastic process $f$ defined in
(\ref{Eq-operators-V-f}) belongs to a normed subspace $B$ of the
space $L^n(Z)$, the norm satisfying
$$
\|f\|_B \le K \mathop{\rm vrai\,sup}_{\nu <0}({\bf E}|\varphi (\nu )|^p)^{1/p},
$$
where $K$ is a positive constant. If the pair $(M_p, B)$ is
admissible for \eqref{2}, corresponding to \eqref{7},
then the zero solution of \eqref{7} is $p$-stable with respect to
the initial function.
\end{lemma}
\begin{proof}
Under the assumptions of the lemma, we have
\begin{align*}
\|x_f(\cdot,x_0)\|_{M_p} &\le \hat c(\|x_0\|_{k_p^n} + \|f\|_B)\\
& \le \hat c(\|x_0\|_{k_p^n}
+ K\mathop{\rm vrai\,sup}_{\nu < 0}({\bf E}|\varphi (\nu )|^p)^{1/p}) \\
& \le \bar c(\|x_0\|_{k_p^n} + \mathop{vrai\,sup}_{\nu < 0}
({\bf E}|\varphi (\nu )|^p)^{1/p}),
\end{align*}
where $\hat c, \bar c, K$ are some positive numbers. From this,
using the estimate $x(t,x_0, \varphi) = x_f(t,x_0)$, we obtain
$$
\sup_{t \ge 0}({\bf E}|x (t,x_0,\varphi)|^p)^{1/p} \le
\bar c(\|x_0\|_{k_p^n} + \mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi (\nu )|^p)^{1/p}).
$$
This implies $p$-stability of the zero solution of
\eqref{7} with respect to the initial function.
\end{proof}
\begin{remark}\label{remBgamma} \rm
Evidently, in Lemma \ref{Lemma3} one can replace the space $B$ by
the space $B^\gamma $ for any reasonable weight $\gamma$. Then
admissibility of the pair $(M_p^\gamma , B^\gamma )$ for
\eqref{2} with $\gamma (t) = \exp \{\beta t\}, \ \beta >
0$ will imply the exponential $p$-stability of the zero solution
of \eqref{7} with respect to the initial function. The
asymptotic $p$-stability of the zero solution of
\eqref{7} with respect to the initial function can be derived from
admissibility of the pair $(M_p^\gamma , B^\gamma )$ for the
corresponding equation \eqref{2} , if $ \lim_{t
\to +\infty }\gamma (t) = +\infty $ and $\gamma (t) \ge
\delta> 0$, $t \in [0, \infty )$ for some $\delta $.
\end{remark}
\begin{definition}\label{ConditionZ} \rm
We say that the semimartingale $Z(t)$ satisfies condition (Z):
\begin{itemize}
\item[\bf (Z)] If $=0$ for $i \ne j$, so that $\lambda ^* \times P$-
almost everywhere $A^{ij} =0$ ($i \ne j$, $i, j = 1, \dots, m$).
\end{itemize}
\end{definition}
We will subsequently use only semimartingales with condition
(Z).
We first treat equation \eqref{7} including distributed
delays. Wishing to use the W-transform and the related operator
$\Theta_l$ we have to rewrite \eqref{7} in the form \eqref{2}. It
is easily done via the formulas (\ref{Eq-operators-V-f}).
We begin by listing some technical conditions:
\begin{itemize}
\item[\bf (D1)] $1\le p<\infty$, $2p \le q \le \infty$;
$\sup_ {t\in [1,\infty )}(v(t) - v(t - 1)) < \infty$,
where $v(t)=\int_0^t\xi(s)d\lambda(s)$;
\begin{gather*}
\|Q_{ij}\||a^i| \le a^i_j, \quad \|Q_{ij}\||A^{ii}|^{0.5} \le h^i_j
\quad (\lambda ^* \times {\bf P})\mbox{-almost everywhere},\\
a^i_j\times\bigvee _{(-\infty ,\cdot]}r_{ij}(\cdot,s)\xi ^{q^{-1}-1}
\in L^\lambda_q, \quad h^i_j\times\bigvee _{(-\infty
,\cdot]}r_{ij}(\cdot,s)\xi ^{q^{-1}-0.5}
\in L^\lambda_q
\end{gather*}
$(i = 1,\dots,m, \; j = 0,\dots,m_i)$.
\end{itemize}
\begin{theorem}\label{Th6}
Let the semimartingale $Z(t)$ satisfy condition (Z),
the reference equation \eqref{4} satisfy (R1)-(R2), and
equation \eqref{7} satisfy (D1). If now the operator
$(I - \Theta_l): M_{2p} \to M_{2p}$ (constructed for
\eqref{2} corresponding to \eqref{7}) has a
bounded inverse, then the zero solution of \eqref{7}
is $2p$-stable with respect to the initial function.
\end{theorem}
\begin{remark} \rm
Due to Corollary \ref{Cor3} the operator $W$, under the
assumptions of Theorem \ref{Th6}, are continuous from the space
$\Lambda_{2p,q}^n(\xi )$ to the space $M_{2p}$, while the
operator $\Theta_l$, defined in (\ref{OperatorEquation}), acts in
the space $M_{2p}$.
\end{remark}
\begin{proof}[Proof of Theorem \ref{Th6}]
We first go over to the form \eqref{2} of equation \eqref{7},
where $V$ and $f$ are specified by the formulas (\ref{Eq-operators-V-f}).
Applying Theorem \ref{Th4} gives us, under the assumptions of
Theorem \ref{Th6}, admissibility of the pair $(M_{2p}, \Lambda
_{2p,q}^n(\xi ))$ for \eqref{2}. The property of
$2p$-stability of the zero solution of \eqref{7}
with respect to the initial function follows now from Lemma \ref{Lemma3} if
we manage to prove the following property:
For any $\varphi $ such that $\mathop{\rm vrai\,sup}_{\nu<0}{\bf E}|\varphi (\nu )|^p < \infty $ the
function $f$ in equation \eqref{2} belongs to the normed space
$B := \Lambda_{2p,q}^n(\xi )$ and the following estimate holds
$\|f\|_{B} \le K\mathop{\rm vrai\,sup}_{\nu<0}(E|\varphi (\nu )|^{2p})^{1/(2p)}$,
where $K$ is a positive number.
To prove this property, we observe that
\begin{align*}
\|f\|_{B} &= \|(E|fa|^{2p})^{1/(2p)}\xi ^{q^{-1}-1}\|_{L_q^\lambda } +
\|(E\|fAf^\top \|^{p})^{1/(2p)}\xi ^{q^{-1}-0.5}\|_{L_q^\lambda } \\
& \le \|\sum_{i=1}^m\sum_{j=0}^{m_i} \Big({\bf E}\Big(\int_{(-\infty,
0)} a^i_j(\cdot)|\varphi (\tau )|d_\tau \bigvee
_{s\in(-\infty, \tau]}r_{ij}(\cdot,s)\Big)^{2p}\Big)^{1/(2p)} \xi
^{q^{-1}-1}\|_{L_q^\lambda } \\
&\quad + \|\sum_{i=1}^m\sum_{j=0}^{m_i}
\Big({\bf E}\Big(\int_{(-\infty, 0)} (h^i_j(\cdot)|\varphi (\tau
)|)^2d_\tau \bigvee_{s\in(-\infty, \tau]}
r_{ij}(\cdot,s)\Big)^{p}\Big)^{\frac 1{2p}} \xi
^{q^{-1}-0.5}\|_{L_q^\lambda } \\
& \le \mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi
(\nu )|^{2p})^{1/(2p)} (\sum_{i=1}^m\sum_{j=0}^{m_i} \|a^i_j\times\bigvee _{s\in(-\infty,
0)}r_{ij}(\cdot,s ) \xi ^{q^{-1}-1}\|_{L_q^\lambda } \\ \\
& \quad + \sum_{i=1}^m\sum_{i=1}^{m_i} \|h^i_j\times\bigvee
_{s\in(-\infty, 0)}r_{ij}(\cdot,s) \xi
^{q^{-1}-0.5}\|_{L_q^\lambda })\\
&\le K\mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi (\nu )|^{2p})^{1/(2p)}, \\
\end{align*}
where $K$ is a positive constant, then $f \in B$ and
$$
\|f\|_{B} \le K\mathop{\rm vrai\,sup}_{\nu<0}(E|\varphi (\nu )|^{2p})^{1/(2p)}.
$$
This completes the proof.
\end{proof}
Let us now consider discrete delays, that is
equation \eqref{9} with the operators (\ref{9-bis}). The following
assumption will be used.
\begin{itemize}
\item[\bf (D2)] $1\le p<\infty$, $2p \le q \le \infty$;
$\sup_{t\in [1,\infty )} (v(t) - v(t - 1)) < \infty$, where
$v(t)=\int_0^t\xi(s)d\lambda(s)$;
\begin{gather*}
\|\tilde{Q}_{ij}\||a^i| \le \tilde{a}^i_j, \quad
\|\tilde{Q}_{ij}\||A^{ii}|^{0.5} \le \tilde{h}^i_j \quad
(\lambda^* \times {\bf P})\mbox{-almost everywhere}, \\
\tilde{a}^i_j\xi ^{q^{-1}-1} \in L^\lambda_q, \quad \tilde{h}^i_j\xi ^{q^{-1}-0.5}
\in L^\lambda_q \quad (i = 1,\dots,m, j = 0,\dots,m_i).
\end{gather*}
\end{itemize}
From Theorem \ref{Th6} we have the following result.
\begin{corollary}\label{CorTh6}
Let the semimartingale $Z(t)$ satisfy condition (Z),
the reference equation \eqref{4} satisfy (R1)-(R2), and
\eqref{9} satisfy (D2). If now the operator
$(I - \Theta_l): M_{2p} \to M_{2p}$ (constructed for
equation \eqref{2} corresponding to \eqref{9}) has a
bounded inverse, then the zero solution of \eqref{9}
is $2p$-stable with respect to the initial function.
\end{corollary}
\begin{definition}\label{def3.2.2} \rm
Equation \eqref{7} (Equation \eqref{9}) is called {\it $M_p^\gamma $-stable}
with respect to the initial function, if for all $x_0 \in k_p^n$ and $\varphi $
such that $\mathop{\rm vrai\,sup}_{\nu<0}E|\varphi (\nu )|^p < \infty $ one
has $x (\cdot,x_0,\varphi) \in M_p^\gamma $ and
$$
\| x(\cdot,x_0,\varphi)\|_{M_p^\gamma }{\le \bar c(\|x_0\|_{k_p^n}
+ \mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi (\nu )|^p)^{1/p})},
$$
where $\bar c \in \mathbb{R}_+$.
\end{definition}
Let us stress that, as before, the notion of $M_p^\gamma
$-stability of \eqref{7} with respect to the initial function
covers the classical notions of $p$-stability, exponential
$p$-stability and asymptotical $p$-stability of the zero solution
with respect to the initial function. It is also evident that $M_p^\gamma
$-stability of \eqref{7} with respect to the initial function
implies $M_p^\gamma $-stability of the associated equation in the
form \eqref{2}.
\begin{theorem}\label{Th7}
Let the semimartingale $Z(t)$ satisfy condition (Z),
the reference equation \eqref{4} satisfy (R1)-(R2), and
equation \eqref{7} satisfy \textbf{D1}. If now the operator
$(I - \Theta_l): M_{2p} \to M_{2p}$ (constructed for
equation \eqref{2} corresponding to \eqref{7}) has a
bounded inverse and there exist numbers $\delta_{ij}
> 0$ such that ${r_{ij}(t,s) = 0}$, where $-\infty ~~ 0$.
\end{theorem}
\begin{proof}
As in the previous theorem, we first rewrite
\eqref{7} in the form \eqref{2}. Then we observe that under the
assumptions of the theorem, the operator $V$ in \eqref{2} will act
from $M_{2p}$ to $\Lambda_{2p,q}^n(\xi )$. Due to Lemma
\ref{lemSmallDeltaUsl}, the operator $V: M_{2p}\to
\Lambda_{2p,q}^n(\xi )$ satisfies the $\Delta $-condition. Hence
the assumptions of Theorem \ref{Th5} are satisfied.
We proceed now as in the proof of the preceding theorem, i.e. we
show that for any $\varphi $ such that $ \mathop{\rm vrai\,sup}_{\nu<0}{\bf E}|\varphi (\nu )|^{2p} <
\infty $ the function $f$ in equation \eqref{7}, given by the
formulas (\ref{Eq-operators-V-f}), belongs to the normed space
$B^\gamma$, where $B:= \Lambda_{2p,q}^n(\xi )$, and the
following estimate holds
$$\|f\|_{B^\gamma} \le K\mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi (\nu )|^{2p})^{1/(2p)},$$ $K$
being a positive number. In this case the $M_{2p}^\gamma
$-stability of \eqref{7} with $\gamma (t) = \exp
\{\beta \int_0^t\xi (\nu )d\lambda (\nu )\}$ (for some
$\beta> 0$) is implied Theorem \ref{Th5} and Lemma \ref{Lemma3}.
To check the above estimate on $f$ we observe that
$$
\|f\|_{B^\gamma} = \|({\bf E}|\gamma fa|^{2p})^{1/(2p)}\xi
^{q^{-1}-1}\|_{ L_q^\lambda } + \|({\bf E}\|\gamma fA(\gamma
f)^\top \|^p)^{1/(2p)} \xi ^{q^{-1}-0.5}\|_{L_q^\lambda }.
$$
Now we have
\begin{align*}
&\|({\bf E}|\gamma fa|^{2p})^{1/(2p)}\xi ^{q^{-1}-1}\|_{ L_{q}^\lambda } +
\|({\bf E}\|\gamma fA(\gamma f)^\top \|^p)^{1/(2p)}\xi
^{q^{-1}-0.5}\|_{L_{q}^\lambda }\\
&\le \|\sum_{i=1}^m\sum_{j=0}^{m_i}
\Big({\bf E}\Big(\int _ {(-\infty,0)}\gamma
(\cdot)a^i_j(\cdot)|\varphi (\tau )| d_\tau\bigvee_{s\in(-\infty, \tau]} r_{ij}
(\cdot,s)\Big)^{2p}\Big)^{1/(2p)}
(\xi (\cdot))^{q^{-1}-1}\|_{L_q^\lambda } \\
&\quad + \|\sum_{i=1}^m\sum_{j=0}^{m_i}
\Big({\bf E}\Big(\int_{(-\infty,0)}(\gamma
(\cdot)h^i_j(\cdot)|\varphi (\tau )|)^2 d_\tau \bigvee_{s\in(-\infty,
\tau]} r_{ij}(\cdot,s )\Big)^p\Big)^{1/(2p)}\\
&\quad \times (\xi (\cdot))^{q^{-1}-0.5}\|_{L_q^\lambda } \\
&\le \Big[\sum_{i=1}^m\sum_{j=0}^{m_i} \exp \{\beta
\int_0^{\delta_{ij}}\xi (\nu )d\lambda (\nu )\}
\|a^i_j(\cdot)\times\bigvee_{s\in(-\infty, 0)}r_{ij}(\cdot,s )
(\xi (\cdot))^{q^{-1}-1}\|_{L_q^\lambda } \\
&\quad + \sum_{i=1}^m\sum_{j=0}^{m_i} \exp \{\beta \int
_0^{\delta_{ij}}\xi (\nu )d\lambda (\nu )\}
\|h^i_j(\cdot)\times\bigvee_{s\in(-\infty,
0)}r_{ij}(\cdot,s)
(\xi (\cdot))^{q^{-1}-0.5}\|_{L_q^\lambda }\Big] \\
&\quad \times \mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi (\nu )|^{2p})^{1/(2p)}\\
&\le K\mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi (\nu )|^{2p})^{1/(2p)},
\end{align*}
where $K$ is some positive number. This gives $f \in B^\gamma $ and
$$
\|f\|_{B^\gamma} \le { K\mathop{\rm vrai\,sup}_{\nu<0}({\bf E}|\varphi
(\nu )|^{2p})^{1/(2p)}}.
$$
The theorem is proved.
\end{proof}
From Theorem \ref{Th7} for \eqref{9} we obtain the following result.
\begin{corollary}\label{CorTh7}
Let the semimartingale $Z(t)$ satisfy condition (Z),
the reference equation \eqref{4} satisfy (R1)-(R2), and
equation \eqref{9} satisfy (D2). Assume that the
operator $(I - \Theta_l): M_{2p} \to M_{2p}$
(constructed for \eqref{2} corresponding to \eqref{9}) has a bounded
inverse, and there exist numbers
$\bar \delta_{ij}> 0$ such that
$$
\int_{\chi_{h_{ij}}(t)h_{ij}(t)}^t \xi (\nu )d\nu \le
\bar \delta_{ij}\quad (t \in [0, \infty )),
$$
where $i = 1,\dots,m$, $j = 0,\dots,m_i$, and $\chi_{g}(t)$
was defined in (\ref{chi}).
Then \eqref{9} is $M_{2p}^\gamma $-stable with respect to the
initial function $\gamma (t) = \exp \{\beta \int_0^t \xi(\nu )d\lambda (\nu )\}$
for some $\beta > 0$.
\end{corollary}
\begin{proof}
To apply Theorem \ref{Th7} we notice that under the assumptions,
listed in Corollary \ref{CorTh7}, equation \eqref{9} in the
form \eqref{7} has the following properties: $r_{ij}(t,s) = 0$ if
$-\infty < s \le t - \delta_{ij} < \infty $ ($t\in
[0,\infty)$), where $\delta_{ij} = \inf \{t\in [0, \infty ): \,
\int_0^t\xi (\nu )d\nu> \bar \delta_{ij}\}$ ($i = 1,\dots,m$, $j =0,\dots,m_i$).
\end{proof}
For the sake of completeness we also observe that the estimates on
$h_{ij}(t)$ in the corollary imply the $\delta$-condition on $h_{ij}(t)$
(see Definition \ref{defFuncDeltaUsl}).
\section{Some sufficient conditions for stability of stochastic
delay equations}\label{sect4}
In this section we use the developed theory to derive certain
stability results for specific classes of equations \eqref{7}
and \eqref{9}. We stress, however, that all the examples below are
of an illustrative character. That is why we do not formally
compare them with the known stability criteria (e.g. those
presented in \cite{k5} and \cite{Mao-book} as well as in other
papers not listed in the bibliography). The aim of this paper is
to describe and illustrate an alternative method of studying
stability. A more careful analysis of specific classes is
therefore left to forthcoming papers. Here we only mention that
our approach normally covers more general classes of linear
stochastic functional differential equations than the
Lyapunov-Krasovskii-Razumikhin method does in this case. Moreover,
our method treats different kinds of stability in an unified
framework. Finally, the W-approach seems to give stability
criteria which are {\it different}, i.e. not exactly comparable,
with those which can be obtained with the help of other
techniques. This observation suggests that the W-method should be
one of the additional instruments in ``the stability analysis
toolbox".
To study equation \eqref{7}, we intend to use a special
reference equation of the form \eqref{4} with
\begin{equation}\label{modelEqSect4}
(Qx)(t) = \big(-\int
_{[0,t)}d_s\mathcal{R}(t,s)x(s)d\lambda(s),\bar 0,\dots,\bar 0\big),
\quad \mathcal{R}(t,s) = \sum_{j=0}^lQ_{1j}(t)r_{1j}(t,s).
\end{equation}
Note that we use here the same $Q_{1j}$, $ r_{1j}$
($j=0,\dots,l$)as in \eqref{7}. As we also want the
assumptions (R1)-(R2) to be fulfilled, we require that the
$n\times n$-matrices $Q_{1j}$
should be non-random for $j=1,\dots,l$ (the matrices $Q_{1j}$ ($j>l$)
are still allowed to be random).
Let us now introduce to important constants which are used in what
follows. Assuming that the formula \eqref{5} from condition
(R2) in Section 2 is valid we put
\begin{equation}\label{constantsC1-C2}
C_1 = \sup_{t \ge 0}\int_0^t \xi(s)\|C(t,s)\|d\lambda(s), \quad
C_2 = \sup_{t \ge 0}\int_0^t \xi(s)\|C(t,s)\|^2d\lambda(s).
\end{equation}
We will also use the following notation: if $M$ is an $n\times
n$-matrix function, then we write $\||M\||_{L_q^\lambda } := \|
\,\|M\|\, \|_{L_q^\lambda }$.
We proceed with describing the main assumptions on the
semimartingale $Z(t)$.
\begin{definition}\label{ConditionZ0} \rm
For a semimartingale $Z(t)$ we difne the conndition
\begin{itemize}
\item[\bf (Z0)] The condition (Z) from Definition \ref{ConditionZ} holds and,
$a^1 = 1 $, $A^{11} = 0$, $a^i = 0$ ($i = 2, \dots,m$)
$\lambda ^* \times P$-almost everywhere (see (\ref{Coefficients-a-b-c})).
\end{itemize}
\end{definition}
We note that, in fact, we can always deduce condition
(Z0) from condition (Z) by increasing the
number of the components of the semimartingale $Z(t)$ (and
adjusting the operator $\hat V$ appropriately). This means that
(Z) and (Z0) are equivalent. But in this section
we choose to use (Z0) as it simplifies our calculations.
A typical example we have in mind is given by the semimartingale
coming from It\^{o} equations.
In what follows we will also need some hypotheses on the
coefficients of \eqref{7}.
\begin{itemize}
\item[\bf (D3)]
$ 1\le p<\infty$, $2p \le q \le \infty$; $\mathrel {\mathop {\sup}_
{t\in [1,\infty )}}(v(t) - v(t - 1)) < \infty$, where
$v(t)=\int_0^t\xi(s)d\lambda(s)$;
$Q_{1j}$ $(j=0,\dots,l)$ are non-random;
\begin{gather*}
\|Q_{1j}\| \le a^1_j \quad (\lambda ^* \times {\bf P})\mbox{-almost
everywhere}, \\
a^1_j\times\bigvee_{s\in(-\infty,0)}r_{1j}(\cdot,s)\xi
^{q^{-1}-1} \in L_q^\lambda, \\
\hat a^1_j(\cdot) :=a^1_j(\cdot)\times\bigvee_{s\in[0, \cdot]}r_{1j}
(\cdot,s)\xi ^{q^{-1}-1} \in L_q^\lambda, \quad (j = 0, \dots, m_1), \\
\|Q_{ij}\||A^{ii}|^{0.5} \le h^i_j \quad (\lambda ^*
\times {\bf P})\mbox{-almost everywhere}, \\
h^i_j\times\bigvee_{s\in(-\infty, 0)}r_{ij}(\cdot, s)\xi ^{q^{-1}-0.5} \in
L_q^\lambda, \\
\hat h^i_j(\cdot) = h^i_j(\cdot)\times\bigvee_{s\in[0, \cdot]}r_{ij}(\cdot,s)
\xi ^{q^{-1}-0.5} \in L_q^\lambda \quad (i = 2, \dots, m, j = 0, \dots, m_i).
\end{gather*}
\end{itemize}
We remark that if we replace condition (Z) by
condition (Z0), then condition (D1) becomes
condition (D3).
Our first theorem in this section is of general character and will
in the sequel be used to more specific studies.
\begin{theorem}\label{Th8}
Let the semimartingale $Z(t)$ satisfy condition (Z0),
equation \eqref{7} satisfy condition (D3), the
reference equation \eqref{4}, where $Q$ is given by
\eqref{modelEqSect4}, satisfy (R1)-(R2). Assume also that
for some $l$ ($0 \le l \le m_1$) the following estimate holds:
$$
\rho := C_1^{1-q^{-1}} \sum_{j=l+1}^{m_1}\|\hat
a^1_j\|_{L_q}+ c_pC_2^{0.5-q^{-1}}\sum_{i=2}^m \sum
_{j=0}^{m_i}\|\hat h^i_j\|_{L_q} < 1,
$$
where $C_1$, $C_2$ are given by (\ref{constantsC1-C2}).
Then \eqref{7} is $M_{2p}$-stable with respect to the
initial function. If, in addition, there exist positive numbers
$\delta_{ij}$, $i {= 1, \dots, m}$, $j = 0, \dots, m_i$ such that
$r_{ij}(t,s) = 0$, where $-\infty < s \le t - \delta_{ij} <
\infty $, $t \in [0, \infty )$, $i = 1, \dots, m$, $j = 0, \dots,
m_i$, then \eqref{7} will be $M_{2p}^\gamma $-stable
with respect to the initial function, where $\gamma (t) = \exp\{\beta
v(t)\}$ for some $\beta > 0$.
\end{theorem}
\begin{proof}
The proof of the first statement in the theorem is based on
Theorem \ref{Th6}, while the second statement exploits Theorem
\ref{Th7}.
According to the assumptions of the theorem the operator
$\Theta_l$ for \eqref{7} in the form \eqref{2} acts
in the space $M_{2p}$. Now, if we manage to show that the operator
$(I - \Theta_l): M_{2p} \to M_{2p}$ has a bounded
inverse, then applying Theorems \ref{Th6} and \ref{Th7} will prove
Theorem \ref{Th8}.
To prove the invertibility of the operator $(I - \Theta_l)$ we
check that, under the assumptions of the theorem, the norm of the
operator $\Theta_l$ in the space $M_{2p}$ is less than $1$. In
this case the only continuous extension of the operator $(I -
\Theta_l):M_{2p} \to M_{2p}$ to the completion of the
space $M_{2p}$ in its own norm will be invertible. To see this, we
observe that the equation $(I - \Theta_l)x = g$ will have the
unique solution in the space $D^n$ for all $g \in M_{2p}$, while
the intersection of the completion of the space $M_{2p}$ with the
space $D^n$ coincides with the space $M_{2p}$ by definition. This
will imply the existence of a bounded inverse of the operator
$(I- \Theta_l): M_{2p} \to M_{2p}$.
In the rest of the proof we estimate the norm of the operator
$\Theta_l$, which is given by
$$
(\Theta_lx)(t) = \int_0^t C(t,s) [(Vx)(s)dZ(s) -
\int_{[0, s)}d_\tau \mathcal{R}(s,\tau)x(\tau )d\lambda (s)],
$$
in the space $M_{2p}$. We have
\begin{align*}
&\|\Theta_lx\|_{M_{2p}} \\
& \le \Big(\sup_{t\ge 0}{\bf E}\Big|\int_0^t C(t,s)\Big(\sum_{j=l+1}^{m_1}Q_{1j}(s)
\int_{[0,s)}d_\nu r_{1j}(s,\nu )x(\nu )\Big)d\lambda (s)\Big|^{2p}\Big)^{1/(2p)} \\
&\quad +c_p\Big(\sup _{t \ge 0}{\bf E}\Big(\int_0^t
\|C(t,s)\|^2\sum_{i=2}^m|A^{ii}(s)|
\Big|\int_{[0,s)}d_\nu \mathcal{R}_i(s,\nu )x(\nu )\Big|^2
d\lambda(s)\Big)^p\Big)^{1/(2p)} \\
& \le C_1^{(2p-1)/2p}(\|x\|_{M_{2p}}\sum_{j=l+1}^{m_1}
( \sup_{t\ge0}{\bf E}\int_0^t\|C(t,s)\| \\
& \times (\xi (s))^{1-2p/q}(\hat a^1_j(s))^{2p}d\lambda
(s))^{1/(2p)})+ c_pC_2^{(p-1)/2p}\|x\|_{M_{2p}} \\ \\
& \times \sum_{i=2}^m\sum_{j=0}^{m_i} \Big(\sup
_{t\ge 0}{\bf E}\int_0^t\|C(t,s)\| (\xi
(s))^{1-2p/q}(\hat h^i_j(s))^{2p}d\lambda (s)\Big)^{1/(2p)}.
\end{align*}
Now we, as in the proof of Lemma \ref{Lemma2}, should consider
three cases separately. We omit the corresponding calculations
here as they are identical with those in Lemma \ref{Lemma2}.
Accepting this we then obtain, using the above estimate on
$\|\Theta_l x\|_{M_{2p}}$, that $\|\Theta_lx\|_{M_{2p}} \le \rho
\|x\|_{M_{2p}}$.
Since $\rho < 1$, we conclude that $\|\Theta_l \|_{M_{2p}} < 1$, and
the theorem is proved.
\end{proof}
\begin{corollary}\label{corTh8-1}
Let the semimartingale $Z(t)$ satisfy condition (Z0)
and reference equation \eqref{4}, where $Q$ is given by
\eqref{modelEqSect4}, satisfy (R1)-(R2). Equation
\eqref{7} is supposed to have the following property:
\begin{itemize}
\item %($\mathbf{\sharp}$) \
The functions $\xi ^{-1}$, $A^{ii}$, the entries of the matrix $Q_{ij}$ and
the variation $\bigvee_{s\in[0,\cdot]}r_{ij}(\cdot,s)$ are all from the space
$L_\infty^{\lambda}$ for $i = 1, \dots, m$, $j = 0, \dots, m_i$.
\end{itemize}
Also assume that for some $l$ ($0 \le l \le m_1$) one has
the estimate
\begin{align*}
&\sum_{j=l+1}^{m_1}\||Q_{1j}\times\bigvee
_{s\in[0, \cdot]} r_{1j}(\cdot,s)\xi ^{-1}\||_{L^\lambda_\infty } \\
&+ c_p(\sqrt {C_2}/C_1)\sum_{i=2}^m \sum_{j=0}^{m_i}\||Q_{ij}(A^{ii})^{0.5}
\times\bigvee_{s\in[0, \cdot]} r_{ij}(\cdot,s)\xi ^{-0.5}\||_{L^\lambda_\infty}
< 1/C_1,
\end{align*}
where $C_1$, $C_2$ are given by (\ref{constantsC1-C2}). Then
\eqref{7} is $M_{2p}$-stable with respect to the initial function.
If, in addition, there exist positive numbers $\delta_{ij}$, $i
{= 1, \dots, m}$, $j = 0, \dots, m_i$ such that $r_{ij}(t,s) = 0$,
where $-\infty < s \le t - \delta_{ij} < \infty $, $t \in [0,
\infty )$, $i = 1, \dots, m$, $j = 0, \dots, m_i$, then
\eqref{7} will be $M_{2p}^\gamma $-stable with respect to the initial
function, where $\gamma (t) = \exp\{\beta v(t)\}$ for some $\beta> 0$.
\end{corollary}
We remark that the property assumed in Corollary
\ref{corTh8-1}, which describes the assumptions on
\eqref{7}, implies the property (D3).
The next proposition is a particular case of Corollary
\ref{corTh8-1} if we put $\xi(t)\equiv 1$ ($t\in [0,\infty)$.
\begin{corollary}\label{corTh8-2}
Assume that the semimartingale $Z(t)$ satisfies condition
(Z0) and reference equation \eqref{4}, where $Q$ is
given by \eqref{modelEqSect4} with $l=0$, satisfies (R1)-(R2).
Equation \eqref{7} has the following property:
\begin{itemize} %($\mathbf{\flat}$)
\item $z^i = 0$ a. s. ($i = 3, \dots, m$), $m_1 =0$, $m_2 = 0$,
the entries of the matrices $Q_{i0}$ and
$\bigvee_{s\in [0, \cdot]} r_{i0}(\cdot,s)$ belong to the space
$L_\infty ^\lambda $\ ($i =1, 2$).
\end{itemize}
Also assume that one has the estimate
$$
c_p(\sqrt {C_2}/C_1)\||d_2\||_{L^\lambda_\infty }< 1/C_1 ,
$$
where $C_1$, $C_2$ are given by (\ref{constantsC1-C2}) and
$d_2 =Q_{20}\times\bigvee_{s\in [0,\cdot]}r_{20}(\cdot,s)|A^{22}|^{0.5}$.
Then \eqref{7} is $M_{2p}$-stable with respect to the
initial function. If, in addition,
there exist positive numbers $\delta_{i}$ ($i = 1, 2$) such
that $r_{i0}(t,s) = 0$, where $-\infty < s \le t - \delta_{i} <\infty $,
$t \in [0, \infty )$, $i = 1, 2$, then
\eqref{7} will be $M_{2p}^\gamma $-stable with respect to the initial
function, where $\gamma (t) = \exp\{\beta (\lambda (t) - \lambda (0))\}$
for some $\beta > 0$.
\end{corollary}
In the rest of the paper we are concerned with the stability
analysis of equation \eqref{9} with discrete time delays. As
this equation is a particular case of equation \eqref{7}, we
can use Theorem \ref{Th8} to obtain sufficient conditions of
$M_{2p}^{\gamma}$-stability of \eqref{9} with respect to the
initial function.
As we wish the reference equation to be a part of the
studied equation, we define $Q$ in \eqref{4} as follows:
\begin{equation}\label{modelEqSect4discrete}
(Qx)(t) = (\sum_{j=0}^l\ \tilde{\tilde
Q}_{1j}(t)(S_{h_{1j}}x)(t),\bar 0,\dots,\bar 0), \quad
\tilde{\tilde Q}_{ij}(t) = \tilde Q_{ij}(t) \chi_{h_{ij}}(t).
\end{equation}
Shift operators of the form $S_g$ are described in
(\ref{Vnutrsuperp}), while $\chi_g(t)$ is defined by (\ref{chi}).
In (\ref{modelEqSect4discrete}) we again require that the matrices
$\tilde{\tilde Q}_{1j}$ ($j=1,\dots,l$) should be non-random, while
the matrices $\tilde{\tilde Q}_{1j}$ ($j>l$) can be random). This
is to ensure the assumptions (R1)-(R2).
The assumptions on the coefficients of \eqref{9} are
summarized in the following condition
\begin{itemize}
\item[\bf (D4)] $1\le p<\infty$, $2p \le q \le \infty$;
$\sup_{t\in [1,\infty )} (v(t) - v(t - 1)) < \infty$, where
$v(t)=\int_0^t\xi(s)d\lambda(s)$;
$$
\tilde{\tilde Q}_{ij}(t) := \tilde Q_{ij}(t) \chi_{h_{ij}}(t) \quad
(i=1,\dots,m, j=0,\dots,m_i),
$$
where $\chi_g$ is given by (\ref{chi}),
$\tilde{\tilde Q}_{1j}(t)$ are non-random for $j=0,\dots,l$;
$$
\|\tilde{\tilde Q}_{1j}\| \le \tilde a^1_j\quad
(\lambda ^* \times {\bf P})\mbox{-almost everywhere},
$$
$\hat{\hat a}^1_j = \tilde a^1_j\xi ^{q^{-1}-1} \in L_q^\lambda$
for $j = 0, \dots, m_1$,
$$
\|\tilde{\tilde Q}_{ij}\|\,|A^{ii}|^{0.5} \le \tilde
h_j^i\quad (\lambda ^* \times P)\mbox{-almost everywhere},
$$
$\hat{\hat h}_j^i = \tilde h^i_j \xi ^{q^{-1}-0.5} \in L_q^\lambda$
for $i = 2, \dots, m$, $j = 0, \dots, m_i$.
\end{itemize}
Clearly, if condition (Z) is replaced by
condition (Z0), then condition (D2) becomes
condition (D4).
From Theorem \ref{Th8} we now deduce the following corollary.
\begin{corollary}\label{corTh8-4}
Let the semimartingale $Z(t)$ satisfy condition (Z0),
equation \eqref{9} satisfy condition (D4), the
reference equation \eqref{4}, where $Q$ is given by
\eqref{modelEqSect4discrete}, satisfy (R1)-(R2). Assume
also that for some $l$ ($0 \le l \le m_1$) the following estimate
holds:
$$
C_1^{1-q^{-1}} \sum_{j=l+1}^{m_1}\|\hat{\hat
a}^1_j\|_{L_q^\lambda } + c_pC_2^{0.5-q^{-1}}\sum_{i=2}^m
\sum_{j=0}^{m_i}\|\hat{\hat h}^i_j\|_{L_q^\lambda } < 1,
$$
where $C_1$, $C_2$ are given by \eqref{constantsC1-C2}. Then
\eqref{9} is $M_{2p}$-stable with respect to the initial function.
If, in addition, there exist positive numbers $\delta_{ij}$,
$i= 1, \dots, m$, $j = 0, \dots, m_i$ such that $r_{ij}(t,s) = 0$,
where $-\infty < s \le t - \delta_{ij} < \infty $, $t \in [0,
\infty )$, $i = 1, \dots, m$, $j = 0, \dots, m_i$, then
\eqref{9} will be $M_{2p}^\gamma $-stable with respect to the initial
function, where $\gamma (t) = \exp\{\beta v(t)\}$ for some $\beta> 0$.
\end{corollary}
As an important particular case of Corollary \ref{corTh8-4} we
obtain:
\begin{corollary}\label{corTh8-5}
Assume that the semimartingale $Z(t)$ satisfies condition
(Z0) and the reference equation \eqref{4}, where $Q$ is
given by (\ref{modelEqSect4discrete}), satisfies (R1)-(R2).
Then \eqref{9} has the following property:
\begin{itemize}
\item The entries of the matrices $\tilde{\tilde
Q}_{ij}$ and the functions $A^{ii},$ $\xi ^{-1}$ belong to the
space $L_\infty^{\lambda} $ for $i = 1, \dots, m$, $j {= 0, \dots,
m_i}$.
\end{itemize}
Assume also that for some $l$ ($0 \le l \le m_1$) the following
estimate holds:
$$
\sum_{j=l+1}^{m_1}\||\tilde{\tilde Q}_{1j}\xi
^{-1}\||_{L^\lambda_\infty } +
c_p(\sqrt {C_2}/C_1)\sum_{i=2}^m
\sum_{j=0}^{m_i}\||\tilde{\tilde Q}_{ij}(A^{ii})^{0.5}
\xi ^{-0.5}\||_{L^\lambda_\infty} < 1/C_1,
$$
where $C_1$, $C_2$ are given by (\ref{constantsC1-C2}). Then
equation \eqref{9} is $M_{2p}$-stable with respect to the initial function.
If, in addition, there exist positive numbers $\delta_{ij}$, $i
{= 1, \dots, m}$, $j = 0, \dots, m_i$ such that $r_{ij}(t,s) = 0$,
where $-\infty < s \le t - \delta_{ij} < \infty $, $t \in [0,
\infty )$, $i = 1, \dots, m$, $j = 0, \dots, m_i$, then
\eqref{9} will be $M_{2p}^\gamma $-stable with respect to the initial
function, where $\gamma (t) = \exp\{\beta v(t)\}$ for some $\beta> 0$.
\end{corollary}
\begin{remark} \rm
It is convenient to apply Theorem \ref{Th8} and Corollaries
\ref{corTh8-1} - \ref{corTh8-5} if it is known that a part of the
drift operator gives rise to a stable deterministic equation. In
this case we use this deterministic equation as a reference
equation. In order to achieve best possible stability results we
have to find the constants $C_1$ and $C_2$ from
(\ref{constantsC1-C2}), or at least good estimates on these
constants. The exact values are only known in exceptional cases
(like for diagonal ordinary differential systems). But good
estimates on $C_1$ and $C_2$ can easily be found if the constants
$\alpha$ and $\bar{\bar c}$ in (\ref{5-bis}) are known or
estimated.
\end{remark}
In what follows we will restrict ourselves to the case of It\^{o}
delay equations. In this case the semimartingale $Z(t)$ has the
form.
\begin{itemize}
\item[\bf (B)] $Z(t) = (t, {B}^1(t),$ $\dots,{B}^{m-1}(t))^T$,
where ${B}^i, i = 1,\dots,m-1$ are independent standard Wiener
processes.
\end{itemize}
When this condition is satisfied, $\lambda (t) = t$ and the
associated measure $\lambda^*$ becomes the Lebesgue measure which
we denote by $\mu$.
\begin{remark} \rm
It is easy to see that the semimartingale, described in
(B), satisfies condition (Z0) as $a =
(1,0,\dots,0)^T$, and the $m\times m$-matrix $A$ is given by
$A^{ii} =1$ if $i = 2,\dots,m$, and $A^{ij} =0$ otherwise, i.e. if
$i=j=1$ or $i\ne j$, $i,j=1,\dots,m$ (see (\ref{Vectors-a-A})). In
addition, we have that $L^n(Z)$ is a linear space consisting of
$n\times m$-matrices, where all the entries are stochastic
processes on $[0, \infty )$ that are adapted with respect to the given
filtration, and the first column in the matrix are a.s. locally
(Lebesgue) integrable, while the other columns are a.s. locally
squire (Lebesgue) integrable. The space $D^n$ consists now of
adapted stochastic processes on $[0, \infty )$ with a.s.
continuous trajectories.
\end{remark}
We will also use an adjusted reference equation \eqref{4} with
\begin{equation}\label{ModelEqWiener}
\begin{gathered}
Qx)(t) = (-\kappa (S_hx)(t),\bar 0,\dots,\bar 0), \quad \kappa
\mbox{ is an $n\times n$-matrix};\\
h(t) \mbox{ is a $\mu$-measurable function such that $h(t) \le t$
($t \in [0, \infty)$).}
\end{gathered}
\end{equation}
Recall that here $\mu $ is the Lebesgue measure, and the operator
$S_h$ is given by (\ref{Vnutrsuperp}).
As before, we introduce a new condition to summarize assumptions
on the coefficients.
\begin{itemize}
\item[\bf (D5)] $1\le p<\infty$, $2p \le q \le \infty$;
$\sup_{t\in [1,\infty )} (v(t) - v(t - 1)) < \infty$, where
$v(t)=\int_0^t\xi(s)ds$;
$$
\tilde{\tilde Q}_{ij}(t) := \tilde Q_{ij}(t) \chi_{h_{ij}}(t) \quad
(i=1,\dots,m, j=0,\dots,m_i),
$$
where $\chi_g$ is given by (\ref{chi}),
$\tilde{\tilde Q}_{1j}(t)$ ($j=0,\dots,l$) are non-random,
$$
\|\tilde{\tilde Q}_{1j}\|\le \tilde a_j^1 \quad (\mu\times {\bf P})\mbox{-almost
everywhere}
$$
$\hat{\hat a}^1_j = \tilde a^1_j\xi ^{q^{-1}-1} \in L_q $ for $j = 0, \dots, m_1$,
$$
\|\tilde{\tilde Q}_{ij}\| \le \tilde h_j^i \quad(\mu \times P)
\mbox{-almost everywhere,}
$$
$ \hat{\hat h}_j^i= \tilde h^i_j \xi ^{q^{-1}-0.5} \in L_q $
for $i = 2, \dots, m, \quad j = 0, \dots, m_i$.
\end{itemize}
Note that if the semimartingale $Z(t)$ satisfies condition
(B), then (D4) becomes (D5).
Recall that we use the following notation (adjusted for the case
$\lambda(t)= t$, \ $t\in [0, \infty)$): if $M$ is an $n\times
n$-matrix function, then we write $\||M\||_{L_q } := \| \,\|M\|\,\|_{L_q}$.
\begin{theorem}\label{Th9}
Let the semimartingale $Z(t)$ satisfy condition (B),
equation \eqref{9} satisfy condition (D5),
reference equation \eqref{4}, where $Q$ is given by
\eqref{ModelEqWiener}, satisfy (R1)-(R2). Assume also that
there exist a natural number $l$ ($0 \le l \le m_1$) and positive
constants $\vartheta_{j}$ ($j = 0,\dots,l$) such that
$$
\int_{[\chi_h(t)h(t),\ \chi_{h_{1j}}(t)h_{1j}(t)]} \xi(s)ds \le
\vartheta_{j}\quad (j = 0,\dots,l).
$$
Finally, the following estimate is supposed to hold:
\begin{align*}
\rho &:= \ C_1^{1-q^{-1}}\{\||\Big(\sum_{j=0}^l\tilde{\tilde
Q}_{1j} + \xi\kappa \Big)\xi^{q^{-1}-1}\||_{L_q}
+ \sum_{j=0}^l\|\hat{\hat a}^1_j\|_{L_q}(\vartheta_{j}^{1-q^{-1}}
\sum_{j=0}^{m_1}\| \hat{\hat a}^1_j\|_{L_q} \\
&\quad + c_p \vartheta_{j}^{0.5-q^{-1}}\sum_{i=2}^m
\sum_{j=0}^{m_i}\|\hat{\hat h}_j^i\|_{L_q})\\
&\quad + \sum_{j=l+1}^{m_1}\|\hat{\hat a}^1_j\|_{L_q}\} +
c_p C_2^{0.5-q^{-1}}\sum_{i=2}^m
\sum_{j=0}^{m_i}\|\hat{\hat h}_j^i\|_{L_q} < 1,
\end{align*}
where $C_1$, $C_2$ are given by (\ref{constantsC1-C2}). Then
\eqref{9} is $M_{2p}$-stable with respect to the initial function.
If, in addition, there exist positive numbers $\delta_{ij}$, $i
{= 1, \dots, m}$, $j = 0, \dots, m_i$ such that the functions
$h_{ij}(t)$ satisfy the $\delta_{ij}$-condition for $i=1,\dots,m;
j=0,\dots,m_i$ (see Definition \ref{defFuncDeltaUsl}), then
\eqref{9} will be $M_{2p}^\gamma $-stable with respect to the
initial function, where $\gamma (t) = \exp\{\beta v(t))\}$ for
some $\beta> 0$.
\end{theorem}
\begin{proof}
As in Theorem \ref{Th8}, to prove the first part we use Corollary
\ref{CorTh6}, while to prove the second part we apply Corollary
\ref{CorTh7}. Evidently, that under the assumptions of Theorem
\ref{Th9} the operator $V$ for \eqref{9} in the form
\eqref{2} acts from $M_{2p}$ to $\Lambda_{2p,q}^n(\xi )$. By
Corollary \ref{Cor3} this implies that $\Theta_l : M_{2p}
\to M_{2p}$.
Due to Corollaries \ref{CorTh6} and \ref{CorTh7} it suffices to
prove that the operator $I - \Theta_l$ has a bounded inverse in
the space $M_{2p}$.
We show that the norm of the operator $\Theta_l$ in the space
$M_{2p}$ is less than 1. As in the proof of Theorem \ref{Th8}, we
then observe that the only continuous extension of the operator
$(I - \Theta_l): M_{2p} \to M_{2p}$ to the completion of
the space $M_{2p}$ in its own norm is invertible. Indeed, the
equation $(I - \Theta_l)x = g$ has the unique solution in the
space $D^n$ for all $g \in M_{2p}$, while the intersection of the
completion of the space $M_{2p}$ with the space $D^n$ coincides
with the space $M_{2p}$ by definition. This implies the existence
of a bounded inverse of the operator $(I - \Theta_l): M_{2p}\to M_{2p}$.
In our case, the operator $\Theta_l$ is given by
$$
(\Theta_lx)(t) = \int_0^t C(t,s) ((Vx)(s)dZ(s) -
\kappa\xi(s) (S_hx)(s) ds) \quad (t\ge 0).
$$
For short notation, let us write $\sigma(t)=\chi_h(t)h(t)$,
$\sigma_j(t)=\chi_{h_{1j}}(t)h_{1j}(t)$ ($j=0,\dots,l$).
Then estimating the norm of the operator $\Theta_l$ in the space
$M_{2p}$ gives
\begin{align*}
&\|\Theta_lx\|_{M_{2p}} \\
& \le (\sup_{t\ge 0}{\bf E}|\int_0^t C(t,s)[\sum_{j=0}^l(\tilde{\tilde Q}_{1j}(s)
+ \xi(s)\kappa )(S_hx)(s) \\
&\quad + \sum_{j=0}^l\tilde{\tilde Q}_{1j}(s) \int
_{\sigma_{j}(s)}^{\sigma(s)} dx(\tau)
+ \sum_{j=l+1}^{m_1}\tilde{\tilde Q}_{1j}(s)
(S_{h_{1j}}x)(s)]ds|^{2p})^{1/(2p)} \\
&\quad + c_p(\sup _{t \ge 0}{\bf E}(\int_0^t \|C(t,s)\|^2\sum_{i=2}^m
|\sum_{j=0}^{m_i}\tilde{\tilde Q}_{ij}(s)(S_{h_{ij}}x)(s)|^2
ds)^p)^{1/(2p)} \\
&\le C_1^{(2p-1)/2p}[(\sup_{t\ge 0 }\int_0^t
\|C(t,s)\|\xi(s)\|\sum_{j=0}^l(\tilde{\tilde
Q}_{1j}(s)\xi^{-1}(s) + \kappa \|^{2p}ds)^{1/(2p)}\|x\|_{M_{2p}} \\
&\quad + \sum _{j=0}^l(\sup_{t\ge 0} ({\bf E}|\int_{\sigma_{j}(t)} ^{\sigma(t)}
dx(\tau )|^{2p})^{1/(2p)}) (\sup_{t\ge 0}\int_0^t\|C(t,s)\|\xi(s)({\tilde
a}^1_{j}(s)\xi^{-1}(s))^{2p}ds)^{1/(2p)} \\
&\quad + \|x\|_{M_{2p}}\sum_{j=l+1}^{m_1} (\sup_{t\ge0}\int_0^t\|C(t,s)
\|\xi(s)({\tilde a}^1_{j}(s)\xi^{-1}(s))^{2p}ds)^{1/(2p)}] \\
&\quad + c_p(C_2)^{(p-1)/2p}\|x\|_{M_{2p}}\sum_{i=2}^m\sum
_{j=0}^{m_i} (\sup_{t\ge 0}\int_0^t\|C(t,s)\|^2\xi(s)({\tilde
h}^i_{j}(s)\xi^{-0.5}(s))^{2p}ds)^{1/(2p)}.
\end{align*}
Since $x$ is a solution of equation \eqref{9},
\begin{align*}
\Gamma_k
&:= \sup_{t\ge 0} ({\bf E}|\int_{\sigma_{j}(t)}
^{\sigma(t)}dx(\tau )|^{2p})^{1/(2p)} \\
&\le \sum_{j=0}^{m_1} \sup_{t\ge 0} ({\bf E}|\int_{\sigma_{j}(t)}
^{\sigma(t)}\xi(s) |\tilde{\tilde Q}_{ij}(s)\xi^{-1}
(s)(S_{h_{1j}}x)(s)ds|^{2p})^{1/(2p)}\\
&\quad + c_p\sum_{i=2}^m\sum_{j=0}^{m_i} \sup_{t\ge 0}
({\bf E}|\int_{\sigma_{j}(t)}^{\sigma(t)}\xi(s)|\tilde{\tilde
Q}_{ij}(s)\xi^{-0.5}(s)(S_{h_{ij}}x)(s)|^{2}ds|^p)^{1/(2p)} \\
&\le \vartheta_{k}^{(2p-1)/2p}\sum_{j=0}^{m_1} \sup_{t\ge 0}
({\bf E}|\int_{\sigma_{j}(t)} ^{\sigma(t)}\xi(s) (\|\tilde{\tilde
Q}_{ij}(s)\xi^{-1}(s)\|\, |(S_{h_{1j}}x)(s)|)^{1-2p}ds|)^{1/(2p)}\\
&\quad + c_p\vartheta_{k}^ {(p-1)/2p}\sum_{i=2}^m\sum_{j=0}^{m_i}
\sup_{t\ge 0} ({\bf E}|\int_{\sigma_{j}(t)}^{\sigma(t)}\xi(s)(\|\tilde{\tilde
Q}_{ij}(s)\xi^{-0.5}(s)\|\\
&\quad\times |(S_{h_{ij}}x)(s)|)^{1-p}ds|)^{1/(2p)},
\end{align*}
where $k = 0, \dots, l$.
Now we have to consider three different cases: 1) $q>2p$,
$q\ne\infty$, 2) $q=2p$, 3) $q=\infty$. Fortunately, they can be
treated in a similar way. Let us therefore restrict ourselves to
the first case.
Assuming $q>2p$, $q\ne\infty$ we obtain
$$
\Gamma_k\le \Big(\vartheta_{k}^{1-q^{-1}}\sum_{j=0}^{m_1} \|\hat{\hat a}^1_j\|_{L_q} + c_p\vartheta
_{k}^{0.5-q^{-1}}\sum_{i=2}^m\sum_{j=0}^{m_i}
\|\hat{\hat h}_j^i\|_{L_q }\Big)\|x\|_{M_{2p}}
$$
where $k = 0, \dots, l$.
From this and from the estimates for $\|\Theta_l x\|_{M_{2p}}$ we
conclude that $\|\Theta_lx\|_{M_{2p}} \le \rho \|x\|_{M_{2p}}$.
Since $\rho < 1$, we have $\|\Theta_l \|_{M_{2p}} < 1$. This
completes the proof.
\end{proof}
We apply now Theorem \ref{Th9} to an It\^{o} equation with
unbounded delays. In \eqref{9}, we therefore assume
that
\begin{equation}\label{UnboundedDelay}
h_{ij}(t)=t/\tau_{ij}, \quad \tau_{ij}\ge 1 \; (i=1,\dots,m, \;
j=0,\dots,m_i).
\end{equation}
Such equation are known to have a number of ``strange'' properties,
for instance they are exponentially stable only in exceptional
cases. Applying our general scheme gives, however, asymptotic
stability of such equations in a natural way. This is shown in
Corollaries \ref{CorTh9-1}-\ref{CorTh9-3} below.
\begin{remark}\label{remUnboundedDelay} \rm
In the case of the delays given by (\ref{UnboundedDelay}), the
initial function in \eqref{9} disappears as $h_{ij}(t)\ge 0$ for
all $t\in [0,\infty)$, \ $i=1,\dots,m, \
j=0,\dots,m_i$. That is why for \eqref{9} with the delays (\ref{UnboundedDelay})
it is natural to study its
$M_p^\gamma$-stability in the sense Definition \ref{def1} (which, in turn,
for certain $\gamma$
implies stability properties from Definition \ref{defClassicStability}).
\end{remark}
Below we use the following function which determines asymptotical
properties of the equation we are interested in:
\begin{equation}\label{xi}
\xi(t) =
\mathbf{1}_{[0,r]}(t)+\mathbf{1}_{[r,\infty]}(t)(1/t) \ \ (t\in
[0,\infty)),
\end{equation}
where $r>0$ is some number and $\mathbf{1}_e$ is the indicator of
a set $e$. From Theorem \ref{Th9} we obtain the following result.
\begin{corollary}\label{CorTh9-1}
Let the semimartingale $Z(t)$ satisfy condition (B),
equation \eqref{9} with (\ref{UnboundedDelay}) satisfy
condition (D5), the reference equation \eqref{4}, where $Q$
is given by (\ref{ModelEqWiener}), satisfy (R1)-(R2).
Assume also that there exist a natural number $l$ ($0 \le l \le
m_1$) and a positive real number $r$ such that
\begin{align*}
\rho &:= \ C_1^{1-q^{-1}}\{\||(\sum_{j=0}^l\tilde{\tilde Q}_{1j}
+ \xi\kappa )\xi^{q^{-1}-1}\||_{L_q}
+ \sum_{j=0}^l\|\hat{\hat a}^1_j\|_{L_q}(\vartheta_{j}^{1-q^{-1}}
\sum_{j=0}^{m_1}\| \hat{\hat a}^1_j\|_{L_q} \\
&\quad + c_p \vartheta_{j}^{0.5-q^{-1}}\sum_{i=2}^m
\sum_{j=0}^{m_i}\|\hat{\hat h}_j^i\|_{L_q})\\
&\quad + \sum_{j=l+1}^{m_1}\|\hat{\hat a}^1_j\|_{L_q}\} +
c_p C_2^{0.5-q^{-1}}\sum_{i=2}^m
\sum_{j=0}^{m_i}\|\hat{\hat h}_j^i\|_{L_q} < 1, \\
\end{align*}
where $\xi$ are defined in (\ref{xi}), $C_1$, $C_2$ are given by
(\ref{constantsC1-C2}) and
$ \vartheta_{j}=\max\{\log \tau_{1j},r(1- \tau_{1j}^{-1})\}$,
$j =0,\dots,l$. Then \eqref{9} is $M_{2p}^\gamma $-stable,
where $\gamma (t) = \mathbf{1}_{[0,r]}(t)+\mathbf{1}_{[r,\infty]}(t)(t/r)^\beta$
for some $\beta > 0$.
\end{corollary}
\begin{remark}\label{remTh9-1} \rm
In fact, Corollary \ref{CorTh9-1} gives us the usual asymptotic
$2p$-stability in the sense of Definition
\ref{defClassicStability}.
\end{remark}
\begin{proof}[Proof of Corollary \ref{CorTh9-1}]
Using (\ref{xi}) we easily check that the delay functions
$h_{ij}(t)=t/\tau_{ij}$ satisfy the $\delta_{ij}$-condition with
$\delta_{ij}=\max\{\log \tau_{ij}, r(1- \tau_{ij}^{-1})\}$
$(i=1,\dots,m, \; j=0,\dots,m_i)$. This enables us to use Theorem
\ref{Th9} directly.
\end{proof}
Applying this corollary to the equation
\begin{equation}\label{EqUnboundedDelay-1}
dx(t)=Q(t)\xi^{-1}(t)x(t)dt+\sum_{i=2}^m\sum_{j=0}^{m_i}Q_{ij}(t)x(t/\tau_{ij})dB^{i-1}(t)
\ \ (t\ge 0 ; \ \tau_{ij}\ge 1),
\end{equation}
where $B^i(t)$ ($i=2,\dots, m$) are independent standard Wiener
processes, the $n\times n$-matrix $Q(t)$ has entries from the
space $L_\infty$ and
$$
\|Q_{ij}(t)\|\le q_{ij}(t)\sqrt{\xi(t)}\quad (t\ge 0, \ 2=1,\dots,m, \;
j=0,\dots,m_i)
$$
for some $q_{ij}\in L_\infty$ ($\xi$ is again given by (\ref{xi}))
we obtain from Theorem \ref{Th9} the following result.
\begin{corollary}\label{CorTh9-2}
Assume that there exists $\bar\alpha>0$ such that
$$
\||Q+\bar\alpha\bar
E\||_{L_\infty}+c_p\sqrt{0.5\bar\alpha}\sum_{i=2}^m
\sum_{j=0}^{m_i}\|q_{ij}\|_{L_\infty}<\bar\alpha.
$$
Then (\ref{EqUnboundedDelay-1}) is $M_{2p}^\gamma$-stable with respect to
the initial function, where
$$
\gamma (t) = \mathbf{1}_{[0,r]}(t)+\mathbf{1}_{[r,\infty]}(t)(t/r)^\beta$$ for
some $\beta > 0$.
\end{corollary}
\begin{proof}
As the reference equation we can take
\begin{equation}\label{ModelEqOrdinary}
dx(t)=\left(\mathop{\rm diag}[-\bar\alpha,\dots,-\bar\alpha]\xi(t)x(t)+g_1(t)\right)dt
+\sum_{i=2}^mg_i(t)dB^{i-1}(t) \ \ (t\ge 0).
\end{equation}
It is straightforward that conditions (R1)-(R2) are
satisfied in this case. It is also easy to see that (D5) is
fulfilled.
\end{proof}
The example below illustrates Corollary \ref{CorTh9-2}.
\begin{example} \rm
The equation
\begin{equation}\label{EqUnboundedDelay-2}
\begin{array}{crl}
dx(t) = \left(a\xi^{-1}(t)x(t) + b\xi^{-1}(t)x(t/\tau_0)\right)dt
+ c\xi^{-0.5}(t)x(t/\tau_1)dB(t) \ \ (t \ge 0),
\end{array}
\end{equation}
where $\xi$ is again given by (\ref{xi}), $B(t)$ is a scalar
Wiener process, $a, b, c, \tau_0, \tau_1$ are real numbers
($\tau_0\ge 1, \tau_1\ge 1$) is $M_{2p}^\gamma $-stable, where
$\gamma (t) =
\mathbf{1}_{[0,r]}(t)+\mathbf{1}_{[r,\infty]}(t)(t/r)^\beta$ for
some $\beta > 0$, provided there exists $\bar\alpha>0$ such that
$$
|a+b+\bar\alpha|+c_p|c|\sqrt{0.5\bar\alpha}+(|ab|+b^2)\delta_0
+c_p|bc|\sqrt{\delta_0}<\bar\alpha,
$$
and $\delta_0=\max\{\log h_0, (1-h_0^{-1})r\}$.
\end{example}
Some situations where such a number $\alpha$ does exist are found
in the dissertation \cite[Sect. 3.3]{k2}, where the results are
formulated in terms of coefficients of (\ref{EqUnboundedDelay-2}).
Let us now consider the case of a scalar equation of the form
\begin{equation}\label{EqBoundedDelay-1}
\begin{gathered}
dx(t) = [ax(t) + bx(h_0(t))]dt + cx(h_1(t))dB(t) \quad (t \ge 0),\\
x(\nu ) = \varphi (\nu ) \quad (\nu < 0),
\end{gathered}
\end{equation}
where $B(t)$ is a scalar Wiener process, $a, b, c$ are real
numbers, $h_0, h_1$ are $\mu$measur\-able functions such that
$h_i(t) \le t$ ($t \in [0, \infty )$) for $i = 0, 1$, $\varphi $
is a stochastic process, which is independent of the (scalar)
standard Wiener process $B(t)$.
We will now exploit the following reference equation:
\begin{equation}\label{ModelEqScalar}
dx(t)=\left(-\bar\alpha (S_hx)(t)+g_1(t)\right)dt+g_2(t)dB(t) \ \ (t\ge 0),
\end{equation}
where $\bar\alpha>0$ and $h(t)$ is $\mu$-measurable and $h(t)\le
t$ for all $t\in [0,\infty)$.
We remark that in the works \cite{k1,k2}, the
$M_p^\gamma$-stability of (\ref{EqBoundedDelay-1})
was studied with the help of the reference equations
(\ref{ModelEqScalar}) which was ordinary differential equations,
i.e. when $h(t)\equiv t$.
From Theorem \ref{Th9}, we obtain the following corollary for
equation (\ref{EqBoundedDelay-1}).
\begin{corollary}\label{CorTh9-3}
(1) Assume that there exist positive numbers $\bar\alpha$
and $\delta$ such that the reference equation
\eqref{ModelEqScalar} satisfies $t-h(t)\le\delta$ (for all $t\in
[0,\infty)$) and conditions (R1)-(R2). If
$$
|a+\bar\alpha|+\delta(a^2+|ab|)+c_p\sqrt\delta|ac|+|b|+c_p|c|C_1^{-1}\sqrt{C_2}0$,
$-b\epsilon < \pi /2$; $\tilde h(t)$ is a $\mu $-measurable
function such that $\tilde h(t) \le t$ for all $t \in [0, \infty)$,
$\varphi $ is a stochastic process, which is independent of
the (scalar) standard Wiener process $B(t)$.
Taking in the reference equation (\ref{ModelEqScalar}) with
$\bar\alpha = -b$, $h(t) = t - \epsilon $, we refer to \cite{a2},
where the assumptions (R1)-(R2) are verified.
Corollary \ref{CorTh9-3} yields now the following result.
\begin{corollary}\label{CorTh9-4}
(1). If $2|b| + |c|(\sqrt {C_2}/C_1)c_p < C_1$,
where $C_1$, $C_2$ are given by \eqref{constantsC1-C2}, then the
zero solution of \eqref{EqBoundedDelay-2} is
$2p$-stable with respect to the initial function. If, in addition, there
exists a number $\delta > 0$ such that $t-\tilde h(t) \le \delta$
($t \in [0, \infty )$), then the zero solution of (\ref{EqBoundedDelay-2})
is exponentially $2p$-stable with respect to the
initial function.
\noindent (2). If
$|c|(\sqrt {C_2} /C_1)c_p < C_1$,
where $C_1$, $C_2$ are given by \eqref{constantsC1-C2}, then the
zero solution of \eqref{EqBoundedDelay-2} is
$2p$-stable with respect to the initial function. If, in addition, there
exists a number $\delta> 0$ such that $t-\tilde h(t)\le \delta$
($t \in [0,\infty)$), then the zero solution of
\eqref{EqBoundedDelay-2} is exponentially $2p$-stable with respect to the
initial function.
\end{corollary}
The constants $C_1$ and $C_2$ can only be estimated numerically.
An algorithm of how to find $C_1$ with an arbitrary precision is
presented in \cite{g1}. There are some estimates from this paper
in Tables 1 and 2.
\begin{table}[ht]
\caption{Estimates for $C_1$}
\begin{center}
\begin {tabular}{|c|l|l|l|l|l|l|l|l|}
\hline
$-b\tau$ & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.2 & 1.4 \\
\hline
$C_1$ & 1.001 & 1.164 & 1.262 & 1.510 & 1.840 & 2.290 & 4.620 & 9.740 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\caption{Estimates for $C_2$}
\begin{center}
\begin {tabular}{|c|l|l|l|l|l|l|cl|l|}
\hline $-b\tau $ & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.2 & 1.4 \\
\hline
$C_2$ & 0.754 & 0.843 & 0.948 & 1.075 & 1.233 & 1.434 & 2.666 & 5.833 \\
\hline
\end{tabular}
\end{center}
\end{table}
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\end{document}
~~