Inverse-Positive Matrices and Stability Properties of Linear Stochastic Difference Equations with Aftereffect

Abstract

This article examines the stability properties of linear stochastic difference equations with delays. For this purpose, a novel approach is used that combines the theory of inverse-positive matrices and the asymptotic methods developed by N.V. Azbelev and his students for deterministic functional differential equations. Several efficient conditions for p-stability and exponential p-stability $(2\le p<\infty )$ of systems of linear Itô-type difference equations with delays and random coefficients are found. All results are conveniently formulated in terms of the coefficients of the equations. The suggested examples illustrate the feasibility of the approach.

1. Introduction

Stochastic difference equations with aftereffect constitute a significant class of dynamical systems with a wide range of applications in economic, biological, physical modeling, control theory, and computational mathematics. The analysis of the Lyapunov stability, one of the core features of dynamical systems, including stochastic difference equations, is, therefore, very popular in the literature. The dominating approach to study stability is based on Lyapunov functions in the non-delay case and Lyapunov-like functionals, defined on appropriate functional spaces, in the delay case. This approach is very general and in many situations provides explicit stability criteria, especially in the case of differential equations. For a description of this approach in the case of stochastic difference equations and for the comprehensive literature review before 2011, we refer the reader to the monograph [1], while a list of other relevant results obtained after 2011 can be found in the recent paper [2].

On the other hand, it is known from the theory of deterministic differential and difference equations with aftereffect that the Lyapunov approach, despite its generality, has certain drawbacks, as constructing Lyapunov-like functionals may be a hard task in practice. For instance, in the case of linear stochastic difference equations, stability conditions are usually formulated in terms of auxiliary matrix equations, i.e., implicitly. Moreover, several important classes of differential and difference equations are less suitable for a Lyapunov-like analysis. A typical example includes equations with random coefficients and random delays. Note that difference equations considered in [1] only contain deterministic coefficients, which are mostly time-independent. Because of these and other reasons, N. V. Azbelev and his students suggested, in the series of papers and monographs, to increase efficiency of stability analysis for equations with aftereffect by introducing an alternative approach called “the method of auxiliary equations” or “the W-method”. Ideologically, this method is similar to the Lyapunov approach, but instead of seeking a suitable functional, one tries to construct a suitable auxiliary equation, which gives rise to a special integral transform helping to regularize the equations in question and producing verifiable stability criteria. We refer here to the monograph [3], where the reader can find a detailed description of this approach, mostly for linear, but also for nonlinear, functional differential equations, as well as a literature review and comparison with the Lyapunov-like constructions. The W-method was successfully applied to deterministic difference equations with delay in the papers [4,5]. Later on, this approach was extended to stochastic functional difference equations; see [6] and Section 3 below for a detailed description of the method in this case.

The idea of combining the method of auxiliary equations with the techniques based on inverse-positive matrices with application to deterministic differential equations with aftereffect goes back to the paper [7]. The use of such matrices allows, in particular, us to significantly improve estimates at the final stage of the method of auxiliary equations. This approach was successfully applied to stochastic fractional differential equations in [8].

The present report seems to be a first attempt, at least to our knowledge, to extend this combined approach to the case of stochastic difference equations with aftereffect. We decided, only for the sake of simplification, to restrict ourselves to the analysis of linear difference equations of Itô type, although the method of auxiliary equations works in the nonlinear case as well; see [7,8]. It is important to remark that, using this method, we were able to study a broad class of difference equations with random coefficients, where Lyapunov functionals are particularly difficult to construct.

The paper is organized as follows. Basic notation, definitions and formulation of the problem can be found in Section 2. The method of auxiliary equations is described in Section 3. The main results are proven in Section 4, and Section 5 contains some examples. A numerical example illustrating one of the stability criteria is offered in Section 6. Conclusions and a future outlook are given in Section 7.

For the standard definitions and general facts from stochastic analysis, we refer the reader to the monograph [9].

2. Notation, Definitions, and Formulation of the Problem

Let $(\mathrm{\Omega },\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},P)$ be a stochastic basis (a filtered probability space), where $\mathrm{\Omega }$ is a set of elementary events, $\mathcal{F}$ is a $\sigma$-algebra of events on $\mathrm{\Omega }$, ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$ is a right continuous filtration of $\sigma$-subalgebras of $\mathcal{F}$, and P is a complete probability measure on $\mathcal{F}$.

The following notation is systematically used in this paper.

• $1\le p<\infty$ is a fixed real number (used in the stability estimates);
• N is the set of all natural numbers;
• $N_{+}=\left\{0\right\}\cup N$;
• Z is the set of all integers;
• $\gamma \left(s\right)(s\in N_{+})$ is a sequence of positive reals, which is supposed to be fixed throughout the paper;
• $|.|$ is a fixed norm in $R^n$;
• $\parallel .\parallel$ is the norm of $n\times n$-matrices consistent with the above norm in $R^n$;
• $\overline{E}$ is the identity $n\times n$-matrix;
• $\overline{e}={(1,\dots ,1)}^T\in R^n$;
• E stands for the expectation on the above stochastic basis;
• $k^n$ is the linear space of n-dimensional ${\mathcal{F}}_0$–measurable random variables;
• $k_p^n=\left\{\alpha :\alpha \in k^n{,\parallel \alpha \parallel }_{k_p^n}:={\left(E\right|\alpha \parallel }^p{)}^{1/p}<\infty \right\}.$
• $({\mathcal{B}}_i,i=2,\dots ,m)$ is the standard $(m-1)$-dimensional Wiener process.

Recall that an $m\times m$-matrix $A={\left(a_{\kappa \lambda }\right)}_{\kappa ,\lambda =1}^m$ is called non-negative if $a_{\kappa \lambda }\ge 0$, $\kappa ,\lambda =1,\dots ,m$, and positive if $a_{\kappa \lambda }>0$, $\kappa ,\lambda =1,\dots ,m$.

Definition 1.

An invertible $m\times m$ matrix B is called inverse-positive (or $\mathcal{M}$-matrix) if the matrix $B^{-1}$ is non-negative.

According to [10], $B={\left(b_{\kappa \lambda }\right)}_{\kappa ,\lambda =1}^m$ will be inverse-positive if $b_{\kappa \lambda }\le 0$ ($1\le \kappa ,\lambda \le m$, $\kappa \ne \lambda$), and one of the following conditions is satisfied:

• The leading principal minors of the matrix B are positive;
• There exist numbers $v_{\kappa }>0$ ($\kappa =1,\dots ,m$) such that either

  $$v_{\kappa }{b}_{\kappa \kappa }>\sum _{\lambda =1(\kappa \ne \lambda )}^m{v}_{\lambda }|b_{\kappa \lambda }|(\kappa =1,\dots ,m),\mathrm{or}{v}_{\lambda }{b}_{\lambda \lambda }>\sum _{\kappa =1(\kappa \ne \lambda )}^m{v}_{\kappa }\left|b_{\kappa \lambda }\right|(\lambda =1,\dots ,m).$$

In particular, if $v_{\kappa }=1$, $\kappa =1,\dots ,m$, then we obtain the class of matrices with strict diagonal dominance and non-positive off-diagonal entries.

In this report, we study stability properties of the system of linear stochastic difference equations with aftereffects of the form

$$\begin{array}{c}x(s+1)=x\left(s\right)+{\displaystyle \sum _{j=-\infty }^s}{A}_1(s,j)x\left(j\right)h+\\ {\displaystyle \sum _{i=2}^m}{\displaystyle \sum _{j=-\infty }^s}{A}_i(s,j)x\left(j\right)({\mathcal{B}}_i\left((s+1)h\right)-{\mathcal{B}}_i\left(sh\right))(s\in N_{+}),\end{array}$$

(1)

with respect to the initial data

$$\begin{array}{cc}\hfill x\left(j\right)=\phi \left(j\right)& (j<0),\hfill \end{array}$$

(2a)

$$\begin{array}{c}\hfill x\left(0\right)=b,\end{array}$$

(2b)

where

• $x={(x_1,\dots ,x_n)}^T$ is an unknown n-dimensional stochastic process;
• $h>0$ is a given real number; in many numerical applications h is sufficiently small, but we do not assume this in our paper;
• $A_i(s,j)$ ($i=1,\dots ,m$, $j\le s$) are $n\times n$ matrices with ${\mathcal{F}}_s$-measurable random entries for each $s\in N_{+}$;
• $\phi \left(j\right)$ ($j<0$) are ${\mathcal{F}}_0$-measurable n-dimensional random variables;
• $b={(b_1,..,b_n)}^T\in k^n$.

For equations without aftereffect the condition (2a) should be omitted.

Definition 2.

By a solution of the initial value problem (1), (2), we understand a sequence of n-dimensional random variables $x\left(s\right)(s\in Z)$, which are ${\mathcal{F}}_s$-measurable for all $s\in N_{+}$ and which almost surely (i.e., P–almost everywhere) satisfy System (1) and the conditions (2).

It is easy to see that under these assumptions, the problem (1), (2) has a unique solution, which we further on denote by $x_{\phi }(s,b)(s\in Z)$. It is also obvious that the zero initial conditions (2) give the zero solution of Equation (1).

Definition 3.

We say that the zero solution of Equation (1) is

• p-stable (with respect to the initial data φ and b) if for any $\epsilon >0$ there exists $\delta >0$, for which $E|x_{\phi }{(s,b)|}^p\le \epsilon (s\in N_{+})$ for all ${\mathcal{F}}_0$-measurable b, $\phi \left(j\right)(j<0)$ such that $\underset{j<0}{sup}{E\left|\phi \left(s\right)\right|}^p+{\parallel b\parallel }_{k_p^n}<\delta$;
• Asymptotically p-stable (with respect to the initial data) if it is p-stable and, in addition,
  $\underset{s\to \infty }{lim}E{\left|x_{\phi }(s,b)\right|}^p=0$ as long as $\underset{j<0}{sup}{E\left|\phi \left(s\right)\right|}^p+{\parallel b\parallel }_{k_p^n}<\infty$;
• Exponentially p-stable (with respect to the initial data) if there exist $c>0$ and $\beta >0$ such that the inequality

  $$E|x_{\phi }{(s,b)|}^p\le cexp\{-\beta t\}\left(\underset{j<0}{sup}{E\left|\phi \left(s\right)\right|}^p+{\parallel b\parallel }_{k_p^n}\right)$$
  holds for all $s\in N_{+}$.

In what follows, we omit the phrase “with respect to the initial data” for the sake of brevity.

Let us now introduce some linear spaces to be used in the sequel. Recall that $\gamma \left(s\right)(s\in N_{+})$ is a sequence of positive reals, which is kept fixed.

• l is the space of all n-dimensional sequences of ${\mathcal{F}}_0$-measurable random variables $\phi \left(j\right)$$(j<0)$;
• m is the solution space of Equation (1) defined through all possible initial conditions (2), where $\phi \in l$ and $b\in k^n$;
• $l_p=\left\{\phi :\phi \in {l,\left|\right|\phi \left|\right|}_{l_p}\stackrel{def}{=}\underset{j<0}{sup}{\left(E\right|\phi \left(j\right)|}^p{)}^{1/p}<\infty \right\};$
• $m_p^{\gamma }=\left\{x:x\in {m,\left|\right|x\left|\right|}_{m_p^{\gamma }}\stackrel{def}{=}\underset{s\ge 0}{sup}{\left(E\right|\gamma \left(s\right)x\left(s\right)|}^p{)}^{1/p}<\infty \right\}$;
• $m_p:=m_p^1$ (i.e., if $\gamma \equiv 1$).

Clearly, $l_p$ and $m_p^{\gamma }$ are linear normed subspaces of the linear spaces l and m, respectively. The following definition generalizes Definition 3:

Definition 4.

We say that Equation (1) is $m_p^{\gamma }$-stable if for all $b\in k_p^n$ and $\phi \in l_p$

1 

$x_{\phi }(·,b)\in m_p^{\gamma }$;

2 

There exists $c>0$ such that $\parallel x_{\phi }{(·,b)\parallel }_{m_p^{\gamma }}\le {c(\parallel b\parallel }_{k_p^n}+{\parallel u\parallel }_{l_p}).$

It is easy to verify that

• If $\gamma \left(s\right)=1(s\in N_{+})$, then $m_p^{\gamma }$-stability ($=m_p$-stability) implies p-stability;
• If $\gamma \left(s\right)\ge \delta (s\in N_{+})$ for some $\delta >0$ and $\underset{s\to +\infty }{lim}\gamma \left(s\right)=+\infty$, then $m_p^{\gamma }$-stability implies asymptotic p-stability;
• If $\gamma \left(s\right)=exp\left\{\beta s\right\}(s\in N_{+})$ for some $\beta >0$, then $m_p^{\gamma }$-stability implies exponential p-stability.

3. The Method of Auxiliary Equations

As it is mentioned in the introduction, stability properties of Equation (1) are examined in this paper by the method of auxiliary equations, also known as the “the W-method” in the literature; see, e.g., [3]. Unlike the Lyapunov-like algorithms, this method is based on a search of a suitable auxiliary equation rather than a suitable Lyapunov function(al). This auxiliary equation is supposed to satisfy the desired stability property and is used to convert the equation in question into a special integral form (”the W-transform”), in which it is easier to check the property of stability. The method proved to be particularly efficient in the case of delay equations, in which one has to construct functionals on the infinite-dimensional spaces of initial functions in the case of Lyapunov-like methods, and this may be particularly challengeable for stochastic delay equations. The method of auxiliary equations applies to both differential and difference, linear and nonlinear, and deterministic and stochastic equations, as well as equations with fractional derivatives, etc. In this report, we restrict ourselves to the linear case, although the theoretical results presented in this section can be directly extended to the nonlinear case as well.

Along with the initial value problem (1) and (2), let us consider a linear system of functional-difference equations of the form

$$x(s+1)=x\left(s\right)+[\sum _{j=0}^{s}B(s,j)x\left(j\right)+f\left(s\right)]h(s\in N_{+}),$$

(3)

where $h>0$, $B(s,j)$ is an $n\times n$-matrix whose entries are real numbers for all $s\in N_{+}$, $j=0,\dots ,s$, and $f\left(s\right)$ is an n-dimensional ${\mathcal{F}}_s$-measurable random variable for all $s\in N_{+}$. This system serves as an auxiliary equation that is used to transform Equation (1) to an integral form. The choice of different auxiliary equations leads to different transforms and, by this, to different stability criteria.

In addition to the non-homogeneous Equation (3), we consider its homogeneous counterpart

$$x(s+1)=x\left(s\right)+\sum _{j=0}^{s}B(s,j)x\left(j\right)h(s\in N_{+}).$$

(4)

Definition 5.

The $n\times n$-matrix $X(s,\tau )$ $(s,\tau \in N_{+},0\le \tau \le s)$, whose columns are solutions of Equation (4), and which satisfies the equality $X(\tau ,\tau )=\overline{E}$ (the identity $n\times n$-matrix), is called the fundamental matrix of Equation (3).

The lemma below is proven in [4] for the deterministic difference equations, but by making use of the standard measurability argument, we can easily arrive at the following statement.

Lemma 1.

The solution $x\left(s\right)$ of Equation (3) satisfying the initial condition $x\left(0\right)=x_0$ has the representation

$$x\left(s\right)=X(s,0)x_0+\sum _{\tau =0}^{s-1}X(s,\tau +1)f\left(\tau \right)h(s\in N_{+}).$$

(5)

Using this lemma, we can rewrite the initial value problem (1) and (2) in the integral form

$$x\left(s\right)=X(s,0)b+\left(\mathrm{\Theta }x\right)\left(s\right)+\left(C\phi \right)\left(s\right)(s\in N_{+}),$$

(6)

where

$$\begin{array}{c}\left(\mathrm{\Theta }x\right)\left(s\right)={\displaystyle \sum _{\tau =0}^{s-1}}X(s,\tau +1)\left[{\displaystyle \sum _{j=0}^{\tau }}[A_1(\tau ,j)-B(\tau ,j)]x\left(j\right)h+{\displaystyle \sum _{i=2}^m}{\displaystyle \sum _{j=0}^{\tau }}{A}_i(\tau ,j)x\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right],\end{array}$$

$$\begin{array}{c}\left(C\phi \right)\left(s\right)={\displaystyle \sum _{\tau =0}^{s-1}}X(s,\tau +1)\left[{\displaystyle \sum _{j=-\infty }^{-1}}{A}_1(\tau ,j)\phi \left(j\right)h+{\displaystyle \sum _{i=2}^m}{\displaystyle \sum _{j=-\infty }^{-1}}{A}_i(\tau ,j)\phi \left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right].\end{array}$$

The next result can also be verified directly using the corresponding deterministic algorithm from [3].

Theorem 1.

Assume that for some sequence of positive numbers $\gamma \left(s\right)$$(s\in N_{+})$ and for any $x\in m_p^{\gamma }$, $\phi \in l_p$, $b\in k_p^n$ Equation (6) satisfies the conditions

$$X(.,0)b\in m_p^{\gamma },\mathrm{\Theta }x\in m_p^{\gamma },C\phi \in m_p^{\gamma },$$

$${\parallel X(.,0)b\parallel }_{m_p^{\gamma }}\le c_1{\parallel b\parallel }_{k_p^n}{,\parallel \mathrm{\Theta }x\parallel }_{m_p^{\gamma }}\le c_2{\parallel x\parallel }_{m_p^{\gamma }}{,\parallel C\phi \parallel }_{m_p^{\gamma }}\le c_3{\parallel \phi \parallel }_{l_p},$$

where $c_1,c_2,c_3$ are some positive constants and $c_2<1$. Then, Equation (1) is $m_p^{\gamma }$-stable.

This result gives a theoretical justification for the method of auxiliary equations. Based on it, several sufficient conditions for p-stability in terms of the parameters of the stochastic difference systems were obtained in the authors’ papers; see, e.g., an overview in [8]. Below, we formulate and prove the theorem, which is crucial for our paper and which connects auxiliary equations with Equation (1) via inverse-positive matrices, rather than via the norm of the operator $\mathrm{\Theta }$. This allows, exactly as in the deterministic case (see [4]), for us to derive more subtle stability criteria and, in addition, to extend the method of auxiliary equations to the nonlinear case; see, e.g., [8].

Put

$$\begin{array}{c}\hfill x\left(s\right)={(x_1\left(s\right),\dots ,x_n\left(s\right))}^T(s\in N_{+}),{\overline{x}}_i^{\gamma }=\underset{s\in N_{+}}{sup}{\left(E|\gamma \left(t\right)x_i{\left(s\right)|}^{2p}\right)}^{1/2p},{\overline{x}}^{\gamma }={({\overline{x}}_1^{\gamma },\dots ,{\overline{x}}_n^{\gamma })}^T\end{array}$$

Assume that using componentwise estimation in (6), we obtain a matrix inequality of the form

$$\begin{array}{c}\hfill \overline{E}{\overline{x}}^{\gamma }\le C{\overline{x}}^{\gamma }+\overline{c}{\parallel b\parallel }_{k_{2p}^n}\overline{e}+\hat{c}{\parallel \phi \parallel }_{l_{2p}}\overline{e},\end{array}$$

where C is some $n\times n$-matrix and $\overline{c},\hat{c}$ are positive numbers. Then, the following statement holds true:

Theorem 2.

If $\overline{E}-C$ is inverse-positive, then Equation (1) is $m_{2p}^{\gamma }$-stable.

Proof. 

As the matrix $\overline{E}-C$ is inverse-positive, the inequality (7) can be rewritten as

$$\overline{E}{\overline{x}}^{\gamma }\le {(\overline{E}-C)}^{-1}(\overline{c}{\parallel b\parallel }_{k_{2p}^n}\overline{e}+\hat{c}{\parallel \phi \parallel }_{l_{2p}}\overline{e}).$$

Thus, we obtain

$$|{\overline{x}}^{\gamma }{|\le K(\parallel b\parallel }_{k_{2p}^n}+{\parallel \phi \parallel }_{l_{2p}}),$$

(7)

where $K=||{(\overline{E}-C)}^{-1}\overline{e}||max\{\overline{c},\hat{c}\}$. Since $x_{\phi }(t,b)=x\left(t\right)$ and $\parallel x_{\phi }{(.,b)\parallel }_{m_{2p}^{\gamma }}\le |{\overline{x}}^{\gamma }|$, we deduce from (7) that $x_{\phi }(.,b)\in m_{2p}^{\gamma }$ and $\parallel x_{\phi }{(.,b)\parallel }_{m_{2p}^{\gamma }}\le {c(\parallel b\parallel }_{k_{2p}^n}+{\parallel \phi \parallel }_{l_{2p}})$ for some $c>0$ and all $b\in k_{2p}^n$ and $\phi \in l_{2p}$. Equation (1) is $m_{2p}^{\gamma }$-stable. □

In the next section, this theorem is applied to Equation (1), which results in sufficient stability conditions of the zero solution of this equation. The conditions are formulated in terms of the equation’s parameters through the property of positive invertibility.

4. Sufficient Stability Conditions

In this section, we adhere to the notation introduced in previous sections and, in addition, for all $i=1,\dots ,m$, $s\in N_{+}$, $j\le s$ we assume that

• $a_{kl}^i(s,j),k,l=1,\dots ,n$ are the entries of the random matrices $A_i(s,j)$ in Equation (1);
• these entries satisfy the estimates $|a_{lk}^i(s,j)|\le {\hat{a}}_{kl}^i(s,j)$ almost surely ($l,k=1,\dots ,n$), where ${\hat{a}}_{kl}^i(s,j),$$k,l=1,\dots ,n$ are some real numbers.

In the sequel, the following well-known inequality is utilized:

$$\begin{array}{c}{\left(E{\left|\underset{t}{\overset{t+h}{\int }}\psi \left(\zeta \right)d{\mathcal{B}}_i\left(\zeta \right)\right|}^{2p}\right)}^{1/2p}\le c_p{\left(E{\left|\underset{t}{\overset{t+h}{\int }}{\left|\psi \left(\zeta \right)\right|}^{2}d\zeta \right|}^p\right)}^{1/2p},\end{array}$$

(8)

where explicit formulas for the constant $c_p$ can be found in the monographs [11,12].

The first three theorems below describe the p-stability of solutions of Equation (1), which we formulate in terms of the $m_p$-stability of this equation; see Definition 4.

Theorem 3.

Assume that Equation (1) satisfies the following conditions:

(1) $c_{lk}\stackrel{def}{=}{\displaystyle \sum _{\tau =0}^{\infty }}\left[{\displaystyle \sum _{j=0}^{\tau }}[{\hat{a}}_{lk}^1(\tau ,j)h+c_p{\displaystyle \sum _{i=2}^m}{\displaystyle \sum _{j=0}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)\sqrt{h}\right]<\infty ,k,l=1,\dots ,n;$

(2) ${\hat{c}}_{lk}\stackrel{def}{=}{\displaystyle \sum _{\tau =0}^{\infty }}\left[{\displaystyle \sum _{j=-\infty }^1}[{\hat{a}}_{lk}^1(\tau ,j)h+c_p{\displaystyle \sum _{i=2}^m}{\displaystyle \sum _{j=-\infty }^{-1}}{\hat{a}}_{lk}^i(\tau ,j)\sqrt{h}\right]<\infty ,k,l=1,\dots ,n;$

(3) $\overline{E}-C$, where $C={\left(c_{lk}\right)}_{l,k=1}^n$, is inverse-positive.

Then, Equation (1) is $m_{2p}$-stable.

Proof. 

Let us apply Theorem 2, where we define $B(s,j)=0$ for all $s\in N_{+}$, $j=0,\dots ,s$, so that the fundamental matrix of Equation (3) becomes the identity matrix, i.e., $X(s,\tau )=\overline{E}$ for all $(s,\tau \in N_{+},$$0\le \tau \le s)$. In this case, Equation (6) is converted into the system

$$\begin{array}{c}{x}_l\left(s\right)=b_l+{\displaystyle \sum _{\tau =0}^{s-1}}\left[{\displaystyle \sum _{j=0}^{\tau }}{\displaystyle \sum _{k=1}^n}{a}_{lk}^1(\tau ,j)x_k\left(j\right)h+{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }\sum _{k=1}^n}{a}_{lk}^i(\tau ,j)x_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right]+\\ {\displaystyle \sum _{\tau =0}^{s-1}}\left[{\displaystyle \sum _{j=-\infty }^{-1}\sum _{k=1}^n}{a}_{lk}^1(\tau ,j){\phi }_k\left(j\right)h+\right.\\ \left.{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}\sum _{k=1}^n}{a}_{lk}^i(\tau ,j){\phi }_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right](s\in N_{+},l=1,\dots ,n).\end{array}$$

(9)

Taking into account that $\parallel b_l{\parallel }_{k_{2p}^1}\le {\parallel b\parallel }_{k_{2p}^n}$, $\underset{j<0}{sup}(E|{\phi }_l{\left(j\right)|}^{2p}{)}^{1/2p}\le {\parallel \phi \parallel }_{l_{2p}}$ and applying the inequality (8), we obtain the following estimate from (9):

$$\overline{E}\overline{x}\le C\overline{x}+{\parallel b\parallel }_{k_{2p}^n}\overline{e}+\hat{c}{\parallel \phi \parallel }_{l_{2p}}\overline{e},$$

where C is an $n\times n$–matrix, the entries of which are described in Condition 1) of Theorem 3, while $\hat{c}=max\{{\hat{c}}_{lk},l,k=1,\dots ,m\}$, where ${\hat{c}}_{lk},l,k=1,\dots ,m$ are defined in Condition 2) of Theorem 3. Thus, the statement of Theorem 3 follows from Theorem 2. □

Theorem 4.

Assume that Equation (1) satisfies the following conditions:

(1) The entries of matrix $A_1(s,j)$ in Equation (1) are equal to 0 for $s\in N_{+}$, $j<0$ and are some real numbers for $s\in N_{+}$, $0\le j\le s$;

(2) The fundamental matrix $X(s,\tau )={\left(x_{lk}(s,\tau )\right)}_{l,k=1}^n(s,\tau \in N_{+},0\le \tau \le s)$ of Equation (3), where $B(s,j)\equiv A_1(s,j)$ ($s\in N_{+}$, $0\le j\le s$), admits the estimates $|x_{lk}(s,\tau )|\le {\overline{c}}_{lk}$ ($s\in N_{+}$, $0\le \tau \le s$), where ${\overline{c}}_{lk}$ ($l,k=1,\dots ,n$) are some constants;

(3) $c_{lk}\stackrel{def}{=}{\overline{c}}_l{c}_p{\displaystyle \sum _{\tau =0}^{\infty }}\left[{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)\sqrt{h}\right]<\infty$, where ${\overline{c}}_l\stackrel{def}{=}{\displaystyle \sum _{k=1}^n}{\overline{c}}_{lk}$ ($k,l=1,\dots ,n$);

(4) ${\hat{c}}_{lk}\stackrel{def}{=}{\overline{c}}_l{c}_p{\displaystyle \sum _{\tau =0}^{\infty }}\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}}{\hat{a}}_{lk}^i(\tau ,j)\sqrt{h}\right]<\infty$ ($k,l=1,\dots ,n$);

(5) $\overline{E}-C$, where $C={\left(c_{lk}\right)}_{l,k=1}^n$, is inverse-positive.

Then, Equation (1) is $m_{2p}$-stable.

Proof. 

Let us again use Theorem 2. As

$$X(s,\tau )={\left(x_{ij}(s,\tau )\right)}_{i,j=1}^n(s,\tau \in N_{+},0\le \tau \le s)$$

is the fundamental matrix of Equation (3) with $B(s,j)=A_1(s,j)$ ($s\in N_{+}$, $0\le j\le s$), then Equation (6) converts to

$$\begin{array}{c}{x}_l\left(s\right)={\displaystyle \sum _{k=1}^n}{x}_{lk}(s,0)b_k+{\displaystyle \sum _{\tau =0}^{s-1}\sum _{\sigma =1}^n}{x}_{l\sigma }(s,\tau +1)\times \\ \left[{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }\sum _{k=1}^n}{a}_{lk}^i(\tau ,j)x_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right]+\\ {\displaystyle \sum _{\tau =0}^{s-1}\sum _{\sigma =1}^n}{x}_{l\sigma }(s,\tau +1)\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}\sum _{k=1}^n}{a}_{lk}^i(\tau ,j){\phi }_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right](s\in N_{+},l=1,\dots ,n).\end{array}$$

(10)

As $\parallel b_l{\parallel }_{k_{2p}^1}\le {\parallel b\parallel }_{k_{2p}^n}$, $\underset{j<0}{sup}(E|{\phi }_l{(j)|}^{2p}{)}^{1/2p}\le {\parallel \phi \parallel }_{l_{2p}}$, the inequality (8) gives the following estimate for the solutions of Equation (10):

$$\overline{E}\overline{x}\le C\overline{x}+\overline{c}{\parallel b\parallel }_{k_{2p}^n}\overline{e}+\hat{c}{\parallel \phi \parallel }_{l_{2p}}\overline{e},$$

where C is the $n\times n$-matrix, the entries of which are described in Condition 3) of Theorem 4, $\overline{c}=max\{{\overline{c}}_1,\dots ,{\overline{c}}_n\}$, ${\overline{c}}_l,l=1,\dots ,n$ are also defined by 3), while the entries ${\hat{c}}_{lk},l,k=1,\dots ,m$ of the matrix $\hat{c}=max\{{\hat{c}}_{lk},l,k=1,\dots ,m\}$ are defined in Condition 4) of Theorem 4. Evidently, the statement of Theorem 3 follows now from Theorem 2. □

Theorem 5.

Assume that Equation (1) satisfies the following conditions:

(1) The entries of the matrix $A_1(s,j)$ in Equation (1) are equal to 0 for $s\in N_{+}$, $j<0$ and are some real numbers for $s\in N_{+}$, $0\le j\le s$;

(2) The fundamental matrix $X(s,\tau )={\left(x_{lk}(s,\tau )\right)}_{l,k=1}^n(s,\tau \in N_{+},0\le \tau \le s)$ of Equation (3) with $B(s,j)=A_1(s,j)$ ($s\in N_{+}$, $0\le j\le s$) admits the following estimates: $|x_{lk}(s,\tau )|\le {\overline{c}}_{lk}{\lambda }_l^{s-\tau }$ ($s\in N_{+}$, $0\le \tau \le s$), where ${\overline{c}}_{lk},0<{\lambda }_l<1$ ($l,k=1,\dots ,n$) are some constants;

(3) $c_{lk}\stackrel{def}{=}{\displaystyle \frac{{\overline{c}}_l{c}_p\sqrt{h}}{1-{\lambda }_l}}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{i=2}^m}{\displaystyle \sum _{j=0}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)\right]<\infty$, where ${\overline{c}}_l\stackrel{def}{=}{\displaystyle \sum _{k=1}^n}{\overline{c}}_{lk}$ ($k,l=1,\dots ,n$);

(4) ${\hat{c}}_{lk}\stackrel{def}{=}{\displaystyle \frac{{\overline{c}}_l{c}_p\sqrt{h}}{1-{\lambda }_l}}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}}{\hat{a}}_{lk}^i(\tau ,j)\right]<\infty$ ($k,l=1,\dots ,n$);

(5) $\overline{E}-C$, where $C={\left(c_{lk}\right)}_{l,k=1}^n$, is inverse-positive.

Then, Equation (1) is $m_{2p}$-stable.

Proof. 

As before, Theorem 2 comes into the service. Equation (3) with $B(s,j)=A_1(s,j)$ ($s\in N_{+}$, $0\le j\le s$) gives rise to the fundamental matrix

$$X(s,\tau )={\left(x_{lk}(s,\tau )\right)}_{l,k=1}^n(s,\tau \in N_{+},0\le \tau \le s),$$

which converts Equation (6) to

$$\begin{array}{c}{x}_l\left(s\right)={\displaystyle \sum _{k=1}^n}{x}_{lk}(s,0)b_k+{\displaystyle \sum _{\tau =0}^{s-1}\sum _{\sigma =1}^n}{x}_{l\sigma }(s,\tau +1)\times \\ \left[{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }\sum _{k=1}^n}{a}_{lk}^i(\tau ,j)x_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right]+\\ {\displaystyle \sum _{\tau =0}^{s-1}\sum _{\sigma =1}^n}{x}_{l\sigma }(s,\tau +1)\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}\sum _{k=1}^n}{a}_{lk}^i(\tau ,j){\phi }_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right](s\in N_{+},l=1,\dots ,n).\end{array}$$

(11)

The estimates $\parallel b_l{\parallel }_{k_{2p}^1}\le {\parallel b\parallel }_{k_{2p}^n}$, $\underset{j<0}{sup}(E|{\phi }_l{\left(j\right)|}^{2p}{)}^{1/2p}\le {\parallel \phi \parallel }_{l_{2p}}$ and the inequality (8) reshape Equation (11) into

$$\begin{array}{c}{\overline{x}}_l\le {\overline{c}}_l{\parallel b\parallel }_{k_{2p}^n}+\underset{s\in N}{sup}\left[{\displaystyle \sum _{\tau =0}^{s-1}}{\overline{c}}_l{\lambda }_l^{s-\tau -1}\left[{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }\sum _{k=1}^n}{\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}{\overline{x}}_k\right]\right]+\\ \underset{s\in N}{sup}\left[{\displaystyle \sum _{\tau =0}^{s-1}}{\overline{c}}_l{\lambda }_l^{s-\tau -1}\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}\sum _{k=1}^n}{\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}{\parallel \phi \parallel }_{l_{2p}}\right]\right]\le \\ \underset{s\in N}{sup}\left[{\displaystyle \sum _{\tau =0}^{s-1}}{\lambda }_l^{s-\tau -1}\right]\left[{\overline{c}}_l{c}_p\sqrt{h}{\displaystyle \sum _{k=1}^n}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)\right]{\overline{x}}_k\right.+\\ {\overline{c}}_l{c}_p\sqrt{h}{\displaystyle \sum _{k=1}^n}\underset{\tau \in N_{+}}{sup}\left.\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}}{\hat{a}}_{lk}^i(\tau ,j)\right]{\parallel \phi \parallel }_{l_{2p}}\right](l=1,\dots ,n).\end{array}$$

(12)

As $\underset{s\in N}{sup}\left[{\displaystyle \sum _{\tau =0}^{s-1}}{\lambda }_l^{s-\tau -1}\right]=\underset{s\in N}{sup}\left[1+{\lambda }_l+{\lambda }_l^2+\dots +{\lambda }_l^{s-1}\right]={\displaystyle \sum _{\tau =0}^{\infty }}{\lambda }_l^s={\displaystyle \frac{1}{1-{\lambda }_l}}$ ($l=1,2,\dots ,n$), the set of inequalities (12) can be rewritten as

$$\overline{E}\overline{x}\le C\overline{x}+\overline{c}{\parallel b\parallel }_{k_{2p}^n}\overline{e}+n\hat{c}{\parallel \phi \parallel }_{l_{2p}}\overline{e},$$

where C is an $n\times n$-matrix, the entries of which are given by Condition 3) of Theorem 5, $\overline{c}=max\{{\overline{c}}_1,\dots ,{\overline{c}}_n\}$, ${\overline{c}}_l,l=1,\dots ,n$ are also defined by 3), while $\hat{c}=max\{{\hat{c}}_{lk},l,k=1,\dots ,m\}$, ${\hat{c}}_{lk},l,k=1,\dots ,m$ are described in Condition 4). Theorem 2 guarantees the $m_{2p}$-stability of Equation (1). □

Let us now turn to exponential p-stability of the zero solution of Equation (1). Again, we use Definition 4 in order to conveniently formulate this property in terms of the $m_p^{\gamma }$-stability of this equation. Recall that in this case we exploit the weight function $\gamma \left(s\right)=exp\left\{\beta s\right\}$$(s\in N_{+})$ for some $\beta >0$.

We start with some important remarks, which explain some assumptions put on Equation (1); see the authors’ paper [6] for more details:

• Exponential p-stability can only be achieved in the case of bounded delays;
• The stochastic terms based on the Wiener processes cannot give exponential p-stability if the unperturbed (i.e., deterministic) equation does not possess this property.

Theorem 6.

Assume that Equation (1) satisfies the following conditions:

(1) The entries of the matrix $A_i(s,j)$ in (1) are equal to 0 for $s\in N_{+}$, $j\le s-d_i-1$, where $d_i$ are some natural numbers ($i=1,\dots ,m$), while the entries of this matrix are arbitrary real numbers for $s\in N_{+}$, $j=s-d_1,\dots ,s$;

(2) The entries of the matrix $B(s,j)$ in Equation (3) are equal to 0 if $0\le s\le d_1$ ($0\le j\le s$) and if $s>d_1$ ($0\le j\le s-d_1$);

(3) The fundamental matrix $X(s,\tau )={\left(x_{lk}(s,\tau )\right)}_{l,k=1}^n(s,\tau \in N_{+},0\le \tau \le s)$ of Equation (3) with $B(s,j)=A_1(s,j)$ ($s\in N_{+}$, $0\le j\le s$) admits the estimates $|x_{lk}(s,\tau )|\le {\overline{c}}_{lk}{\lambda }_l^{s-\tau }$ ($s\in N_{+}$, $0\le \tau \le s$), where ${\overline{c}}_{lk},0<{\lambda }_l<1$ ($l,k=1,\dots ,n$) are some constants;

(4) $c_{lk}\stackrel{def}{=}{\displaystyle \frac{{\overline{c}}_l}{1-{\lambda }_l}}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{j={\nu }_1\left(\tau \right)}^{\tau }}|a_{lk}^1(\tau ,j)-b_{lk}(\tau ,j)|h+c_p\sqrt{h}{\displaystyle \sum _{i=2}^m\sum _{j={\nu }_i\left(\tau \right)}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)\right]<\infty$, where ${\overline{c}}_l\stackrel{def}{=}{\displaystyle \sum _{k=1}^n}{\overline{c}}_{lk}$, ${\nu }_i\left(\tau \right)=0$ if $0\le \tau \le d_i$) and ${\nu }_i\left(\tau \right)=\tau -d_i$ if $\tau >d_i$ ($i=1,\dots ,m$) for all $k,l=1,\dots ,n$;

(5) ${\hat{c}}_{lk}\stackrel{def}{=}{\displaystyle \frac{{\overline{c}}_l}{1-{\lambda }_l}}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{j=-d_1}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^1(\tau ,j)h+{\displaystyle \sum _{i=2}^m\sum _{j=-d_i}^{-1}}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}\right]<\infty ,$

$k,l=1,\dots ,n$, where ${\chi }_i\left(\tau \right)$ is the indicator of the set $0\le \tau \le d_i$ ($\tau \in N_{+}$) ($i=1,\dots ,m$);

(6) The inverse ${(\overline{E}-C)}^{-1}$, where $C={\left(c_{lk}\right)}_{l,k=1}^n$, exists and all its entries are positive.

Then, Equation (1) is $m_{2p}^{\gamma }$-stable with $\gamma \left(s\right)=exp\left\{\beta s\right\}$$(s\in N_{+})$ for some $\beta >0$.

Proof. 

As before, Theorem 2 plays a central role in the proof. First of all, we put $\lambda :=max\{{\lambda }_1,\dots ,{\lambda }_n\}$, pick an arbitrary $\beta$ satisfying $0<\beta <-ln\lambda$, and define with this $\beta$ the weight $\gamma \left(s\right)=exp\left\{\beta s\right\}$$(s\in N_{+})$ and the constants ${\hat{\lambda }}_l:={\lambda }_{l}exp\left\{\beta \right\}$ ($l=1,\dots ,n$). Now, observe that the fundamental matrix

$$X(s,\tau )={\left(x_{lk}(s,\tau )\right)}_{l,k=1}^n(s,\tau \in N_{+},0\le \tau \le s)$$

of Equation (3) admits, according to Condition (3) of Theorem 6, the estimate $|x_{lk}(s,0)\gamma \left(s\right)|\le {\hat{c}}_1$ ($s\in N_{+}$, $l,k=1,\dots ,n$).

Then, Equation (6) becomes

$$\begin{array}{c}{x}_l\left(s\right)={\displaystyle \sum _{k=1}^n}{x}_{lk}(s,0)b_k+{\displaystyle \sum _{\tau =0}^{s-1}\sum _{\sigma =1}^n}{x}_{l\sigma }(s,\tau +1)\times \\ \left[{\displaystyle \sum _{j={\nu }_1\left(\tau \right)}^{\tau }\sum _{k=1}^n}(a_{lk}^1(\tau ,j)-b_{lk}(\tau ,j))x_k\left(j\right)h+\right.\\ \left.{\displaystyle \sum _{i=2}^m\sum _{j={\nu }_i\left(\tau \right)}^{\tau }\sum _{k=1}^n}{a}_{lk}^i(\tau ,j)x_k\left(j\right)\underset{\tau h}{\overset{(\tau +1)h}{\int }}d{\mathcal{B}}_i\left(\zeta \right)\right]+\\ {\displaystyle \sum _{\tau =0}^{s-1}\sum _{\sigma =1}^n}{x}_{l\sigma }(s,\tau +1)\left[{\displaystyle \sum _{j=-d_1}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right)a_{lk}^1(\tau ,j){\phi }_k\left(j\right)h+\right.\\ \left.{\displaystyle \sum _{i=2}^m\sum _{j=-d_i}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right)a_{lk}^i(\tau ,j){\phi }_k\left(j\right){\displaystyle \underset{\tau h}{\overset{(\tau +1)h}{\int }}}d{\mathcal{B}}_i\left(\zeta \right)\right],s\in N_{+},l=1,\dots ,n,\end{array}$$

(13)

where ${\nu }_i\left(\tau \right)$ and ${\chi }_i\left(\tau \right)$ are defined in Conditions (4) and (5), respectively.

The estimates $\parallel b_l{\parallel }_{k_{2p}^1}\le {\parallel b\parallel }_{k_{2p}^n}$ and $\underset{j<0}{sup}(E|{\phi }_l{(j)|}^{2p}{)}^{1/2p}\le {\parallel \phi \parallel }_{l_{2p}}$, together with the inequality (8), produce the following estimates for Equation (13):

$$\begin{array}{c}{\overline{x}}_l^{\gamma }\le {\overline{c}}_l{\parallel b\parallel }_{k_{2p}^n}+\underset{s\in N}{sup}\left[exp\left\{\beta s\right\}{\displaystyle \sum _{\tau =0}^{s-1}}{\overline{c}}_l{\hat{\lambda }}_l^{s-\tau -1}\left({\displaystyle \sum _{j={\nu }_1\left(\tau \right)}^{\tau }\sum _{k=1}^n}|a_{lk}^1(\tau ,j)-b_{lk}(\tau ,j)|\times \right.\right.\\ hexp\{-\beta j\}{\overline{x}}_k^{\gamma }+\left.\left.{\displaystyle \sum _{i=2}^m\sum _{j={\nu }_i\left(\tau \right)}^{\tau }\sum _{k=1}^n}{\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}exp\{-\beta j\}{\overline{x}}_k^{\gamma }\right)\right]+\\ \underset{s\in N}{sup}\left[exp\left\{\beta s\right\}{\displaystyle \sum _{\tau =0}^{s-1}}{\overline{c}}_l{\hat{\lambda }}_l^{s-\tau -1}\left({\displaystyle \sum _{j=-d_1}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^1(\tau ,j)h{\parallel \phi \parallel }_{l_{2p}}+\right.\right.\\ \left.\left.{\displaystyle \sum _{i=2}^m\sum _{j=-d_i}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}{\parallel \phi \parallel }_{l_{2p}}\right)\right]\le \\ {\overline{c}}_l\underset{s\in N}{sup}\left({\displaystyle \sum _{\tau =0}^{s-1}}{\hat{\lambda }}_l^{s-\tau -1}\right)\left[{\displaystyle \sum _{k=1}^n}\underset{\tau \in N_{+}}{sup}\left({\displaystyle \sum _{j={\nu }_1\left(\tau \right)}^{\tau }}|a_{lk}^1(\tau ,j)-b_{lk}(\tau ,j)|\times \right.\right.\\ \left.hexp\left\{\beta (\tau +1-j)\right\}+c_p\sqrt{h}{\displaystyle \sum _{i=2}^m\sum _{j={\nu }_i\left(\tau \right)}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)exp\left\{\beta (\tau +1-j)\right\}\right){\overline{x}}_k^{\gamma }+\\ {\displaystyle \sum _{k=1}^n}\underset{\tau \in N_{+}}{sup}\left({\displaystyle \sum _{j=-d_1}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^1(\tau ,j)hexp\left\{\beta (\tau +1)\right\}+\right.\\ \left.\left.{\displaystyle \sum _{i=2}^m\sum _{j=-d_i}^{-1}}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}exp\left\{\beta (\tau +1)\right\}\right){\parallel \phi \parallel }_{l_{2p}}\right],l=1,\dots ,n.\end{array}$$

(14)

Since $\underset{s\in N}{sup}\left({\displaystyle \sum _{\tau =0}^{s-1}}{\hat{\lambda }}_l^{s-\tau -1}\right)=\underset{s\in N}{sup}\left(1+{\hat{\lambda }}_l+{\hat{\lambda }}_l^2+\dots +{\hat{\lambda }}_l^{s-1}\right)={\displaystyle \sum _{\tau =0}^{\infty }}{\hat{\lambda }}_l^s={\displaystyle \frac{1}{1-{\hat{\lambda }}_l}}$ for any $l=1,2,\dots ,n$, we can rewrite (14) as

$$\overline{E}{\overline{x}}^{\gamma }\le C\left(\beta \right){\overline{x}}^{\gamma }+\overline{c}{\parallel b\parallel }_{k_{2p}^n}\overline{e}+n\hat{c}\left(\beta \right){\parallel \phi \parallel }_{l_{2p}}\overline{e},$$

where $C\left(\beta \right)$ is an $n\times n$-matrix, the entries of which are given by

$$\begin{array}{c}{c}_{lk}\left(\beta \right):={\displaystyle \frac{{\overline{c}}_l}{1-{\hat{\lambda }}_l}}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{j={\nu }_1\left(\tau \right)}^{\tau }}|a_{lk}^1(\tau ,j)-b_{lk}(\tau ,j)|hexp\left\{\beta (\tau +1-j)\right\}+\right.\\ \left.c_p\sqrt{h}{\displaystyle \sum _{i=2}^m\sum _{j={\nu }_i\left(\tau \right)}^{\tau }}{\hat{a}}_{lk}^i(\tau ,j)exp\left\{\beta (\tau +1-j)\right\}\right],l,k=1,\dots ,n,\end{array}$$

the constant $\overline{c}=max\{{\overline{c}}_1,\dots ,{\overline{c}}_n\}$ is defined via ${\overline{c}}_l(l=1,\dots ,n$) described in Condition (3) of Theorem 6, while

$$\hat{c}\left(\beta \right)=max\{{\hat{c}}_{lk}\left(\beta \right),l,k=1,\dots ,m\},$$

where

$$\begin{array}{c}{\hat{c}}_{lk}\left(\beta \right):={\displaystyle \frac{{\overline{c}}_l}{1-{\hat{\lambda }}_l}}\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{j=-d_1}^{-1}\sum _{k=1}^n}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^1(\tau ,j)hexp\left\{\beta (\tau +1)\right\}+\right.\\ \left.{\displaystyle \sum _{i=2}^m\sum _{j=-d_i}^{-1}}{\chi }_1\left(\tau \right){\hat{a}}_{lk}^i(\tau ,j)c_p\sqrt{h}exp\left\{\beta (\tau +1)\right\}\right],l,k=1,\dots ,n,\end{array}$$

are continuous functions of the parameter $\beta$.

Observe that by construction $c_{lk}\left(0\right)=c_{lk},{\hat{c}}_{lk}\left(0\right)={\hat{c}}_{lk}(l,k=1,\dots ,n);$ see Condition (5), and recall that $\beta$ was defined to be an arbitrary number satisfying the estimates $0<\beta <-ln\lambda$. By assumption (6), the entries of the matrix ${(\overline{E}-C\left(0\right))}^{-1}={(\overline{E}-C)}^{-1}$ are positive. Therefore, for a sufficiently small positive ${\beta }_0<-ln\lambda$, the matrix $\overline{E}-C\left({\beta }_0\right)$ will still be invertible, and the entries of ${(\overline{E}-C\left({\beta }_0\right))}^{-1}$ will be positive as well. Thus, $\overline{E}-C\left({\beta }_0\right)$ will be inverse-positive. Applying Theorem 2, we see that Equation (1) is $m_{2p}^{\gamma }$-stable, where $\gamma \left(s\right)=exp\left\{{\beta }_{0}s\right\}$$(s\in N_{+})$ with ${\beta }_0>0$. □

Remark 1.

Let us notice that in Theorem 6, unlike Theorems 3–5, we require that the inverse ${(\overline{E}-C)}^{-1}$ only contains positive entries, which is a stronger assumption, which is stable under small perturbations. This stability property is not necessarily valid for inverse-positive matrices.

5. Some Examples

Here, we apply Theorems 3–6 to some specific examples of stochastic difference systems with or without delay. To illustrate the results of Section 4, we chose to study systems of two scalar equations. In this case, the sufficient conditions for $\overline{E}-C$, where $C={\left(c_{lk}\right)}_{l,k=1}^2$, to be inverse-positive read as follows:

$$c_{11}<1,(1-c_{11})(1-c_{22})>c_{12}{c}_{21},$$

(15)

which means that the leading principal minors of the $2\times 2$-matrix $\overline{E}-C$ are positive, while the off-diagonal entries $-c_{12}$ and $-c_{21}$ of the matrix $\overline{E}-C$ are by construction non-positive.

Example 1.

We start with a non-delay system of two scalar difference Itô equations

$$\begin{array}{c}{x}_1(s+1)=x_1\left(s\right)+(a_{11}^1\left(s\right)x_1\left(s\right)+a_{12}^1\left(s\right)x_2\left(s\right))h+\\ {\displaystyle \sum _{i=2}^m}(a_{11}^i\left(s\right)x_1\left(s\right)+a_{12}^i\left(s\right)x_2\left(s\right))({\mathcal{B}}_i\left((s+1)h\right)-{\mathcal{B}}_i\left(sh\right))(s\in N_{+}),\\ x_2(s+1)=x_2\left(s\right)+(a_{21}^1\left(s\right)x_1\left(s\right)+a_{22}^1\left(s\right)x_2\left(s\right))h+\\ {\displaystyle \sum _{i=2}^m}(a_{21}^i\left(s\right)x_1\left(s\right)+a_{22}^i\left(s\right)x_2\left(s\right))({\mathcal{B}}_i\left((s+1)h\right)-{\mathcal{B}}_i\left(sh\right))(s\in N_{+}),\end{array},$$

(16)

where $h>0$ and $a_{lk}^i\left(s\right)$ ($s\in N_{+}$) are some random, ${\mathcal{F}}_s$-measurable variables for all $i=1,\dots ,m$, $l,k=1,2$.

From Theorem 3, we immediately obtain

Proposition 1.

Assume that

(1) $a_{11}^1\left(s\right)={\hat{a}}_{11}^1$ and $a_{22}^1\left(s\right)={\hat{a}}_{22}^1$ are real constants, and $|a_{lk}^i\left(s\right)|\le {\hat{a}}_{lk}^i\left(s\right)$ almost surely ($s\in N_{+}$) for all other $i=1,\dots ,m,l,k=1,2$, where ${\hat{a}}_{lk}^i\left(s\right)$ are non-random reals;

(2) The system (16) satisfies the inequalities in (15) with $c_{lk}$ given by

$$c_{lk}\stackrel{def}{=}{\displaystyle \sum _{s=0}^{\infty }}\left[|{\hat{a}}_{lk}^1\left(s\right)|h+c_p\sum _{i=2}^m|{\hat{a}}_{lk}^i\left(s\right)|\sqrt{h}\right](l,k=1,2).$$

Then, the zero solution of the system (16) is $2p$-stable.

The exponential stability of the system (16) can be examined if one applies Theorem 6. In this case, we choose the following auxiliary system (3):

$$\begin{array}{c}{x}_1(s+1)=x_1\left(s\right)+{\hat{a}}_{11}^1{x}_1\left(s\right)h+f_1\left(s\right)h(s\in N_{+}),\\ x_2(s+1)=x_2\left(s\right)+{\hat{a}}_{22}^1{x}_2\left(s\right)h+f_2\left(s\right)h(s\in N_{+}),\end{array}$$

(17)

where ${\hat{a}}_{11}^1$ and ${\hat{a}}_{22}^1$ are real constants. It is easy to see the assumptions of Theorem 6 are fulfilled.

In addition, we need slightly stronger assumptions on the matrix C:

$$c_{12}>0,c_{21}>0,c_{11}<1,(1-c_{11})(1-c_{22})>c_{12}{c}_{21}.$$

(18)

Then, the entries of the matrix ${(\overline{E}-C)}^{-1}$ are all positive. By making use of the auxiliary system (17), we obtain the following result:

Proposition 2.

Assume that the system (16) satisfies the following conditions:

(1) $a_{11}^1\left(s\right)={\hat{a}}_{11}^1$ and $a_{22}^1\left(s\right)={\hat{a}}_{22}^1$ are real constants, and $|a_{lk}^i\left(s\right)|\le {\hat{a}}_{lk}^i$ almost surely ($s\in N_{+}$) for all other $i=1,\dots ,m,l,k=1,2$, where ${\hat{a}}_{lk}^i$ are some reals;

(2) $0<{\lambda }_l<1$, where ${\lambda }_l\stackrel{def}{=}1+{\hat{a}}_{ll}^{1}h$ ($l=1,2$);

(3) The inequalities in (15) are fulfilled with $c_{lk}$ ($l,k=1,2$) defined as

$$c_{ll}\stackrel{def}{=}{\displaystyle \frac{1}{1-{\lambda }_l}}\left[c_p\sqrt{h}\sum _{i=2}^m\left|{\hat{a}}_{ll}^i\right|\right](l=1,2),c_{lk}\stackrel{def}{=}{\displaystyle \frac{1}{1-{\lambda }_l}}\left[|{\hat{a}}_{lk}^1|h+c_p\sqrt{h}\sum _{i=2}^m|{\hat{a}}_{lk}^i|\right](l\ne k).$$

Then, the zero solution of the system (16) is exponentially $2p$-stable.

Example 2.

Consider the system of two scalar difference Itô equations with infinite delay and random coefficients defined as

$$\begin{array}{c}{x}_1(s+1)=x_1\left(s\right)+{\displaystyle \sum _{j=-\infty }^s}[a_{11}^1(s,j)x_1\left(j\right)+a_{12}^1(s,j)x_2\left(j\right)]h+\\ {\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^s}[a_{11}^i(s,j)x_1\left(j\right)+a_{12}^i(s,j)x_2\left(j\right)]({\mathcal{B}}_i\left((s+1)h\right)-{\mathcal{B}}_i\left(sh\right))(s\in N_{+}),\\ x_2(s+1)=x_2\left(s\right)+{\displaystyle \sum _{j=-\infty }^s}[a_{21}^1(s,j)x_1\left(j\right)+a_{22}^1(s,j)x_2\left(j\right)]h+\\ {\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^s}[a_{21}^i(s,j)x_1\left(j\right)+a_{22}^i(s,j)x_2\left(j\right)]({\mathcal{B}}_i\left((s+1)h\right)-{\mathcal{B}}_i\left(sh\right))(s\in N_{+}),\\ x_1\left(j\right)={\phi }_1\left(j\right),x_2\left(j\right)={\phi }_2\left(j\right)(j<0),\end{array}$$

(19)

where $h>0$, $a_{kl}^i(s,j)$ ($s\in N_{+}$) are some random, ${\mathcal{F}}_s$-measurable variables ($i=1,\dots ,m$, $k,l=1,2$, $j\le s$), and ${\phi }_i\left(j\right)$ ($j<0$) are ${\mathcal{F}}_0$-measurable scalar random variables for $i=1,2$.

From Theorem 3, we deduce the following:

Proposition 3.

Assume that the system (19) satisfies the following conditions:

(1) $a_{11}^1(s,j)={\hat{a}}_{11}^1$ and $a_{22}^1(s,j)={\hat{a}}_{22}^1$ are real constants, and $|a_{lk}^i(s,j)|\le {\hat{a}}_{lk}^i(s,j)$ almost surely ($s\in N_{+}$) for all other $i=1,\dots ,m,l,k=1,2$, where ${\hat{a}}_{lk}^i(s,j)$ are some reals;

(2) ${\displaystyle \sum _{\tau =0}^{\infty }}\left[{\displaystyle \sum _{j=-\infty }^{-1}}|a_{lk}^1(\tau ,j)|h+c_p{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}}\left|a_{lk}^i(\tau ,j)\right|\sqrt{h}\right]<\infty ,almostsurely(l,k=1,2);$

(3) The inequalities in (15) are fulfilled with $c_{lk}$ defined as

$$c_{lk}\stackrel{def}{=}{\displaystyle \sum _{\tau =0}^{\infty }}\left[{\displaystyle \sum _{j=0}^{\tau }}{\hat{a}}_{lk}^1(\tau ,j)|h+c_p{\displaystyle \sum _{i=2}^m\sum _{j=0}^{\tau }}\left|{\hat{a}}_{lk}^i(\tau ,j)\right|\sqrt{h}\right](l,k=1,2).$$

Then, the zero solution of the system (19) is $2p$-stable.

Applying Theorems 5 and 6 to the system (19) in the next two propositions, we again use the auxiliary system (17).

Proposition 4.

Assume that the system (19) satisfies the following conditions:

(1) $a_{lk}^1(s,j)={\hat{a}}_{lk}^1(s,j)=0$ for all $s\in N_{+}$, $0\le j\le s$, $l,k=1,2$, except for $a_{ll}^1(s,j)={\hat{a}}_{ll}^1$ ($l=1,2$), which are non-zero, real constants;

(2) $|a_{lk}^i(s,j)|\le {\hat{a}}_{lk}^i(s,j)$ almost surely ($s\in N_{+}$) for all other $i=1,\dots ,m,l,k=1,2$, where ${\hat{a}}_{lk}^i(s,j)$ are some reals;

(3) $0<{\lambda }_l<1$, where ${\lambda }_l\stackrel{def}{=}1+{\hat{a}}_{ll}^{1}h$ ($l=1,2$);

(4) $\underset{\tau \in N_{+}}{sup}\left[{\displaystyle \sum _{i=2}^m\sum _{j=-\infty }^{-1}}\left|a_{lk}^i(\tau ,j)\right|\right]<\infty ,almost surely(l,k=1,2)$;

(5) The inequalities in (15) are fulfilled with $c_{lk}$ given by

$$c_{lk}\stackrel{def}{=}{\displaystyle \frac{c_p\sqrt{h}}{1-{\lambda }_l}}\underset{\tau \in N_{+}}{sup}\left[\sum _{i=2}^m\sum _{j=0}^{\tau }\left|{\hat{a}}_{lk}^i(\tau ,j)\right|\right](l,k=1,2).$$

Then, the zero solution of the system (19) is $2p$-stable.

Proposition 5.

Assume that the system (19) satisfies the following conditions:

(1) $a_{ll}^1(s,j)={\hat{a}}_{ll}^1$ ($l=1,2$) are real constants and $|a_{lk}^i(s,j)|\le {\hat{a}}_{lk}^i(s,j)$ almost surely ($s\in N_{+}$) for all other $i=1,\dots ,m,k,l=1,2$, where $a_{lk}^i(s,j)$ are some reals;

(2) $d_1=0$ and $d_i$ ( $i=2,\dots ,m$) are some natural numbers;

(3) ${\hat{a}}_{kl}^i(s,j)=0$ ($s\in N_{+}$, $j\le s-d_i-1$) for $i=1,\dots ,m$, $k,l=1,2$, and ${\hat{a}}_{kl}^1(s,s)={\hat{a}}_{kl}^1$ ($s\in N_{+}$), ${\hat{a}}_{kl}^i(s,j)={\hat{a}}_{kl}^i$ ($s\in N_{+}$, $j=s-d_i,\dots ,s$) are arbitrary reals for all $i=1,\dots ,m,k,l=1,2$, except for ${\hat{a}}_{12}^1$${\hat{a}}_{21}^1$, which is equal to 0;

(4) $0<{\lambda }_l<1$, where ${\lambda }_l\stackrel{def}{=}1+{\hat{a}}_{ll}^1$ ($l=1,2$);

(5) The inequalities in (18) are fulfilled with $c_{lk}$ defined as

$$c_{lk}\stackrel{def}{=}{\displaystyle \frac{c_p\sqrt{h}}{1-{\lambda }_l}}\left[\sum _{i=2}^m{d}_i\left|{\hat{a}}_{lk}^i\right|\right](k,l=1,2).$$

Then, the zero solution of the system (19) is exponentially $2p$-stable.

In conclusion, we remark that all stability criteria in this section are new, in particular, because they cover systems with random coefficients.

6. A Numerical Example

In this section, we provide a numerical illustration of one of the theoretical stability results; see Proposition 2 of the previous section. First of all, let us notice that if $a_{12}^1\left(s\right)=a_{21}^1\left(s\right)=0$ and stochastic perturbations are absent, then the second condition of Proposition 2 is necessary and sufficient for the exponential stability of the deterministic counterpart of the system (16). The other conditions of Proposition 2 help us to examine the influence of the coefficients $a_{12}^1\left(s\right)$, $a_{21}^1\left(s\right)$ and stochastic terms on the stability of (16). Note that the system contains many parameters. So, to be able to describe the obtained stability conditions geometrically, we introduce the aggregated variables $u={\hat{a}}_{12}^{1}h$, $v={\hat{a}}_{21}^{1}h$ and put ${\theta }_1\stackrel{def}{=}(1-c_{11})(1-c_{22}){\hat{a}}_{11}^1{\hat{a}}_{22}^1{h}^2,{\theta }_2\stackrel{def}{=}{c}_p\sqrt{h}{\displaystyle \sum _{i=2}^m}|{\hat{a}}_{12}^i|,{\theta }_3\stackrel{def}{=}{c}_p\sqrt{h}{\displaystyle \sum _{i=2}^m}\left|{\hat{a}}_{21}^i\right|.$ Then, we can reformulate the $2p$-stability condition from Proposition 2 as $\left(\right|u|+{\theta }_2\left)\right(\left|v\right|+{\theta }_3)<{\theta }_1.$ Using the Python language library, we drew the $2p$-stability region; see Figure 1, minding that stability can also occur outside this region, as Proposition 2 only gives sufficient conditions.

7. Conclusions and Outlook

The main results of the paper can be summarized as follows:

(1)

A novel framework to examine the p-stability, asymptotic p-stability, and exponential p-stability of stochastic difference equations with and without aftereffect is suggested and justified. The framework combines the method of auxiliary equations and the theory of inverse-positive matrices and is applicable to both linear and nonlinear systems.

(2)

Several stability criteria for linear stochastic difference systems with random coefficients and bounded and unbounded delays are obtained on the basis of this method. The criteria cover p-stability and exponential p-stability and are conveniently formulated in terms of the systems’ parameters. The choice of different auxiliary systems leads to different criteria.

(3)

The examples based on systems of two scalar equations show the efficiency of the framework.

(4)

The study findings can be important for many theoretical and computational models based on stochastic differential and difference equations, especially in cases where the method of Lyapunov functionals only give implicitly formulated stability criteria or may even be difficult to apply.

In the future, the authors plan to extend the stability analysis suggested in the paper to a broad class of nonlinear stochastic difference equations, especially those with random coefficients and delays. A particular emphasis will be put on non-exponential asymptotic p-stability, which is typical for systems with unbounded delays.