The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.

1. Introduction

Stability analysis of stochastic delay equations is quite popular due to its numerous applications. It is therefore impossible to give a more or less extensive overview of the topic. Some of the results are summarized in the monograph [1]; others can be found in more recent works of the author of this monograph as well as in many other publications. Indisputably, the main method of stability analysis is based on Lyapunov functions and their generalizations. However, this method may not be applicable in certain situations or may give too restrictive conditions, both in the deterministic and stochastic cases. It is particularly challengeable to use the Lyapunov framework in the case of complicated delays or/and random coefficients and initial conditions (see, e.g., discussion in the recent paper [2]). Yet, many applications require such kinds of models, e.g., in mathematical finance, especially when modeling stock prices, interest rates, or volatilities (see, e.g., [3] and the references therein), stochastic control theory [46], semi-Markov systems [7], epidemic models [8], and many others.

Stability analysis used in this paper is based an alternative approach utilizing a regularization technique. This approach has been successfully used by several authors in the theory and applications of deterministic and stochastic delay equations. Its theoretical foundations are presented in the monograph [9], and since then numerous applications of the method have been investigated in a number of papers. Among the recent ones are the publications [10] (the Mackey–Glass equation), [11, 12] (control theory), [13] (neural networks), [14] (hyperjerk systems), and [15] (the stochastic pantograph equation).

The regularization method has its roots in the theory of the input-to-state stability, which usually implies the Lyapunov stability (see, e.g., [16]). As soon as this relationship between the two types of stability is established, the algorithm starts with choosing a simpler equation (called a reference equation) that is assumed to already have the required stability properties. Resolved and substituted into the original equation, the reference equation produces a new integral equation of the form . If the latter equation is solvable (for instance, if ), then stability of the original equation is proven. Thus, the method is similar to Lyapunov’s direct method, but instead of seeking a Lyapunov function(al), one aims to first find a suitable reference equation possessing the prescribed asymptotic property. The reference equation is then used to regularize the original equation. The method proven to be particularly efficient when constructing Lyapunov function(al) may be technically difficult (random coefficients and delays, distributed delays, unbounded delays, complicated noises, and so on).

In [13, 14, 17], the regularization method was combined with the estimation technique based on positive invertible matrices. This led to new, verifiable stability conditions in the case of differential equations with variable, in particular, distributed delays. In general, the approach based on –matrices gives better stability results as the technique utilizing the norm estimation (see, e.g., [15, 17]). In the above publications, verifiable stability conditions were formulated in terms of the coefficients of the systems in question.

The present paper is a continuation of the authors’ work started in [16, 17]. Minding future applications of the method (see Section 8), we offer a general framework combining the regularization approach from [9] with the theory of positive invertible matrices. After that, we demonstrate how this scheme can be applied to (rather complicated) systems of Itô equations with variable delays and random coefficients, i.e., in the situations where the Lyapunov-like functionals may be difficult to construct. A motivation to study such systems goes back to several sophisticated stochastic models used, for instance, in population dynamics. Thus, equations for the aggregated state variables derived from the McKendrick–von Foerster equation for structured populations always contain random, distributed delays, random coefficients, and random initial conditions [14], so that the Lyapunov framework may be difficult to apply.

For the sake of simplicity, we chose to describe the framework for the case of linear equations and exponential stability, as it was done in the monograph [9]. Even if the most realistic models are nonlinear, we stress that the methods of the present paper, primarily developed for the linear case, can be directly used to study local stability of stochastic nonlinear equations. Moreover, the regularization technique can be applied to global stability of nonlinear deterministic and stochastic delay equations. This has been clearly shown in many publications based on the regularization method (see, e.g., the monograph [9] and the references therein as well as the recent publications [10, 11, 13, 14, 17]).

The presentation of the regularization method will be incomplete without mentioning the recent publications [1820], where many profound stability results were obtained. The paper [20] deserves a special attention, as the authors regularize, in our terminology, nonlinear stochastic equations by means of nonlinear reference equations with the known stability properties. This approach is close to the one used in the present paper, but does not exploit –matrices. It seems to be a fruitful idea to combine both techniques to obtain more general stability results and to cover more applications.

The paper is organized as follows. We start with some preliminary information and the notation (Section 2). Then, we introduce the system of stochastic delay equations we intend to study as well as the main assumptions on it (Section 3). In Section 4, we offer a brief description of the regularization method and its adjustment based on the input-to-state stability and the theory of positive invertible matrices. We remark that the framework described in this section goes far beyond the tasks of the present paper. It can be also used in the case of arbitrary semi-martingales, unbounded delays, nonlinear systems, etc. We chose this level of generality because we intend to apply the framework to many other equations and models in the future. Section 5 contains formulation and the proof of the main result of the paper. It provides sufficient conditions (in terms of the coefficients) of exponential -stability of the system introduced in Section 3. In Section 6, the main result is specified for some particular cases of the general system. We believe that even in the case of deterministic equations, we produced some new results. Section 7 contains some examples illustrating the stability conditions offered in the previous section. Finally, in Section 8, we provide a short summary and describe some open problems.

2. Preliminaries

Let be a stochastic basis, where is set of elementary probability events, is a -algebra of all events on , is a right continuous family of -subalgebras of , and is a probability measure on ; all the above -algebras are assumed to be complete w.r.t. , i.e., containing all subsets of zero measure.

The following notational agreements are used throughout the paper:(i) is the set of all real numbers.(ii) is a column vector.(iii) is an arbitrary yet fixed norm in , being the associated matrix norm.(iv) is the -identity matrix.(v) is the Lebesgue measure on .(vi) is the norm in a normed space .(vii) is an arbitrary real number satisfying .(viii) is the -algebra of all Borel subsets of an interval .(ix) is the standard -dimensional Brownian motion (i.e., the scalar Brownian motions are independent).(x) (only used in Section 4).(xi)The expectation (the integral w.r.t. the measure ) is denoted by .(xii) is the linear space of all –dimensional, –measurable random values.(xiii).(xiv) is the constant from inequality (1) used in the stability conditions.

Recall that a –measurable stochastic process , , is called –adapted if is -measurable for all .

A -matrix is said to be nonnegative, resp. positive if resp. for all .

The following definition is crucial for what follows.

Definition 1. A matrix is called a (nonsingular) –matrix if for , , and all the principal minors of the matrix are positive.
According to [21], a matrix is a nonsingular –matrix if for all , , and there exist positive numbers , , such that one of the following conditions is fulfilled:(1), .(2), .In particular, if , , in the first of the above conditions, then we obtain the class of strictly diagonally dominant matrices. In [21], one can find plenty of characterizations of nonsingular –matrices.
One of the most important properties of nonsingular –matrices states the following.
The inverse of a nonsingular –matrix is positive. In what follows we will always silently assume that any –matrix is nonsingular.
The next two lemmas contain inequalities to be used in this paper.

Lemma 1. for any -adapted stochastic process , any , and any component of the Brownian motion .

Proof. Inequality (1) follows directly from inequality (5) in [22, p.65], where one can find explicit formulas for .

Lemma 2. Let be a scalar function that is square integrable on and be an -adapted stochastic process satisfying . Then,and

Proof. We only prove inequality (2), as inequality (3) can be justified similarly.

3. The Main System and Formulation of the Problem

We will study exponential stability of the following system of Itô delay differential equationsequipped with the initial conditions

Remark 1. According to [9], we separate the initial conditions for and , as we do not require the continuity of the function . This function is assumed to be only bounded and measurable, so that changing its value at countably many points does not change the solution of equation (5). On the other hand, it can be easily checked by examples that changing the value of usually changes the solution of equation (5). That is, the space of initial functions and the space of initial values have different topologies.
The following assumptions are put on (5)–(7) throughout the paper:(1) is an -dimensional –measurable random variable belonging to the space .(2) is an -dimensional –measurable stochastic process with essentially bounded trajectories, defined on the interval , where .(3) is an unknown -dimensional stochastic process defined for ; it is -adapted for and -measurable for .(4) are –matrices for , , and the entries of the matrices , , are scalar, -adapted stochastic processes with almost surely Lebesgue integrable trajectories, while the entries of the matrices , , , are scalar, -adapted stochastic processes with almost surely square integrable trajectories.(5), , , are Borel measurable scalar functions defined on satisfying the inequalities –everywhere for some constants , , .

Remark 2. It can be proven that under the assumptions 1–5, the initial value problem ((5)–(7)) has a unique (up to the natural equivalence of stochastic processes) continuous, -adapted solution .

Definition 2. We say that equation (5) is exponentially –stable with respect to the initial data, i.e., the initial value and the “prehistory” function   , if there are positive numbers such that all solutions of the initial value problem ((5)–(7)) satisfyIn the next section, we describe the method of our analysis. We remark that this description is very general and goes far beyond the particular case of equation (5).

4. Regularization Method in Stability Analysis

In this section, we briefly describe a framework which we use to study stability properties of stochastic functional differential equations. The main idea of this approach is to convert the property of Lyapunov stability to the property of invertibility of certain operators in suitable functional spaces.

In what follows, we put . We remark, however, that the framework described below is valid for any -dimensional semi-martingale (see, e.g., [16]).

We consider a general linear homogeneous stochastic hereditary equationequipped with two initial conditions

Here is a -linear Volterra operator (see below) defined in certain linear spaces of vector stochastic processes, is an -measurable stochastic process, and . By -linearity of the operator we mean the following property:holding for all -measurable, bounded, and scalar random values and all stochastic processes belonging to the domain of the operator .

It is easy to see that equation (5) is a particular case of equation (9).

The solution of the initial value problem ((9)–(11)) will be denoted by , . Below the solution is always assumed to exist and to be unique for an appropriate choice of .

We also need an adaptation of Definition 1 to the case of equation (9).

Definition 3. For a given real number , we call the zero solution of equation (9)(i)-stable (with respect to the initial data and ) if for any there is such that implies for all and all -measurable , .(ii)Exponentially -stable if there exist positive constants , such that the inequalityholds true for all and all -measurable , .
Introducing the -norm in the space of the prehistory functions byand minding the norm in the space of the initial values defined in the previous section yield a shorter version of Definition 3, where the expression is replaced by .
To describe the regularization method, we need to represent (9) and (10) in a slightly different form. Let be a stochastic process and be a stochastic process on coinciding with for and equalling 0 for, while let be a stochastic process on coinciding with for and equalling 0 for. Then, the stochastic process, defined for , will be a solution of the (9)–(11) if satisfies the initial value problemwhere , for . Indeed, by -linearity we have that yielding (15). Note that is uniquely defined by the stochastic process , “the prehistory function.” Let us also observe that the initial value problem ((15) and (16)) is equivalent to the initial value problem ((9)–(11)) only for admitting the representation , where is an arbitrary extension of the function to the real line .
Let be a linear subspace of the space of -adapted stochastic processes with trajectories that are almost surely essentially bounded on . The norm in this space is defined byAs we assume the existence and uniqueness property for equation (21) for all and , we can denote the corresponding solution by . Let stand for the space of all solutions of equation (15), and we define its linear subspace byThe construction above produces two linear operators:The next theorem is crucial for the framework (see, e.g., [12]). It says that the stochastic Lyapunov stability follows from the input-to-state stability (“the stochastic Bohl–Perron theorem”).

Theorem 1. Assume that the linear operators and , defined by (17) and (18), respectively, are bounded. Then, the zero solution of equation (9) is -stable in the sense of Definition 3.

We remark that the operator is, as a rule, bounded, so that the only challenge in application of Theorem 1 is to prove boundedness of the operator . This can be done by the regularization method, also known as “the W-method” [9, 16]. The regularization is usually constructed with the help of an auxiliary or reference equationwhere is again a -linear Volterra operator. The reference equation is, thus, similar to equation (15), but it is supposed to be “simpler” in the sense that the required stability property for this equation is already proven (see condition (2) in Theorem 2 below).

Assuming the existence and uniqueness property for equation (21), we get the following “variation-of-constants” formula for its solutions:where is the fundamental matrix of the associated homogeneous equation and is the corresponding Cauchy operator.

Using representation (22), we can regularize equation (15) in two ways: on the right and on the left. In this paper, we only use the left regularization to be described below. The algorithm based on the right regularization is presented in [16].

Using equation (21), we rewrite equation (15) as follows:or, if we use the representation (22), as

Denoting , we obtain the operator equation

Theorem 2. Assume that equation (15) and reference equation (21) satisfy the following conditions:(1)The linear operators , act continuously from to .(2)The Cauchy operator in (22) constructed for reference equation (21) is bounded as an operator from to .(3)The operator has a bounded inverse.

Then, the operator in (20) is bounded.

Theorems 1 and 2 justify the regularization framework in the analysis of Lyapunov stability for stochastic linear functional differential equations. In fact, all the conditions except condition (3) in Theorem 2 are usually fulfilled, and this is easy to check. The only real challenge is therefore the invertibility of the operator . In [16] (see also the references therein), the invertibility of this operator is verified by estimating the norm of the integral operator : if in the inequalitythen equation (6) is -stable due to Theorem 1. If one also proves that the equation remains -stable after the substitution for some positive , then equation (9) becomes exponentially -stable.

However, calculation of the norm might be challengeable, especially in the vector case and in the case of random coefficients. In [13], and later in [14, 17] for the stochastic case, it was suggested to use the properties of monotone operators. The main idea was to perform all the estimates componentwise, which is much simpler, and then to check the positive invertibility of a certain matrix. Below we offer a generic description of this method.


Suppose that after estimating each component of vector equation (23), we arrived at the vector inequalitywhere is an –matrix and , are some column -vectors with nonnegative components. Typically, , where is the identity matrix, while and replace and in the scalar inequality (26), respectively. Then, we have the following.

Theorem 3. If is an –matrix in the sense of Definition 1, then the operator in (20) is bounded.

Proof. As is an –matrix, the matrix is positive, and we can rewrite (28) asTherefore,where . As , we conclude from (30) that and for some positive . Thus, the operator is bounded.
In the next section, the scheme just outlined is applied to equation (5), i.e., for the system of Itô equations with variable delays and random coefficients. Using the boundedness of all delay functions in equation (5), the substitution for some positive , and Theorems 1 and 3 for -stability of the equation for , we prove exponential -stability of equation (5) for , .
We also remark that the second Lyapunov method may be difficult to use in this case.

5. The Main Result

The following conditions are supposed to be valid throughout this section:(C1) There exist positive numbers , so that the coefficients of equation (5) satisfy the inequalities–almost everywhere.(C2) For each , there is a nonempty subset and positive number , so that

–almost everywhere.

The –matrix is defined as follows:

Theorem 4. If conditions (C1) and (C2) are fulfilled and if is an –matrix, then equation (5) is exponentially –stable in the sense of Definition 2.

Proof. We rewrite equation (5) and the initial condition (6) as a single system:where , , are scalar, -adapted stochastic processes defined on satisfying the condition for , while , , are scalar, –measurable stochastic processes defined on satisfying the conditions for and for . Due to the assumed property of the existence and uniqueness of the solutions of the initial value problem ((34) and (7)), we can use the notation for its solution. It is straightforward to check that for , where is the solution of (5)–(7).
The next step consists in introducing the new variables , where is defined on and the number satisfies the inequalities for all . Clearly, for , and we obtain the following system:Denoting for and minding , transform (35) to the systemSubstituting the expression for from the -th equation in (35) into the -th equation in (36), where , leads toFinally, setting , and remembering initial condition (7), carry (37) over to the systemTo obtain estimates for the solutions of (38), we will adopt the following notation:(i), .(ii), .(iii).In addition, we will use the following inequalities:(i)    for , , .(ii).(iii).(iv)Now, from (38) and inequalities (1)–(3), we obtainFrom the definition, we have that , . Hence, (39) yieldswherePut now , , and and let be an –matrix with the entries given byFrom (40), we then deduce the following estimate:where . Evidently, . According to the assumptions of the theorem, is an –matrix, so that is also an –matrix for small . Thus, from (43) and Theorem 3, we obtainfor some constant . Combining the substitution with the inequalities and (44), we get the estimateHence, equation (5) is exponentially –stable in the sense of Definition 2.
The theorem is proven.

Remark 3. Whether or not the matrix is an –matrix can be verified by calculation of its principal minors: if all of them are positive, then is an –matrix. Otherwise, one can use the sufficient conditions described right after Definition 1.

6. Some Corollaries

In this section, we produce some sufficient conditions for exponential stability of (5).

Let us start with the following particular case of equation (5):

We define the –matrix as follows: