Input-to-State Stability of Linear Stochastic Functional Differential Equations

The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “theW-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.


Introduction
This review paper is aimed at describing a general framework for analysis of asymptotic properties of linear stochastic functional differential equation driven by a semimartingale.The core idea of the method is an alternative description of asymptotic properties in terms of solvability of the equation in certain pairs of spaces of stochastic processes on the semiaxis.Similar to the deterministic case, this property can be called input-to-state stability (ISS) [1] or, alternatively, admissibility of the pairs of spaces for the equation in question [2,3].
As long as the relationship between a desired asymptotic property and ISS with respect to a certain pair of spaces is established, one applies a special regularization method to verify ISS.Usually, such a regularization starts with choosing a simpler equation (called a reference equation), which is already ISS with respect to this pair of spaces.Resolved and substituted into the original equation, the reference equation produces a new, integral equation of the form  − Θ = .If the latter is solvable (e.g., if ‖Θ‖ < 1) in a suitable space, then ISS and hence the related asymptotic property are proved.
This framework was proposed by Azbelev (who also gave the name the -method to this framework) and his students in 1986 for stability analysis of deterministic functional differential equations.The -method was meant to serve as an alternative to the Lyapunov direct method for linear delay equations (see [4]).Later on the method was generalized in [2,[5][6][7] (see also the references therein) and applied to other classes of equations, for example, in [8][9][10] and in many other papers.In [3], the method was for the first time applied to linear stochastic functional differential equations and developed further in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].In the recent paper [26] the idea of input-to-state stability was applied to nonlinear stochastic equations describing neural networks.
In some sense, the -method is similar to Lyapunov's direct method.But instead of seeking a Lyapunov function(al) one aims to find a suitable reference equation which possesses the prescribed ISS property and which then is used to regularize the original equation.Like Lyapunov's method, the -method also provides necessary and sufficient stability conditions (currently for linear equations, only).
The present review paper offers a short yet consistent description of the results which have been published by the authors since 1992.The material is organized as follows.Notation and a short introduction to the concept of a linear stochastic functional differential equation are given in Section 2. In Section 3 we introduce (stochastic) input-to-state 2 Journal of Function Spaces stability, describe its connections to various types of stochastic Lyapunov stability, and outline two regularization methods (left and right -transforms).In Section 4 we provide conditions which guarantee ISS in the weighted spaces (Bohl-Perron type theorems).These results are used to deduce asymptotic Lyapunov stability from simple stability.Applications to stochastic functional differential and difference equations are discussed in Sections 5 and 6, respectively.These two chapters contain several concrete examples of stochastic delay equations, which demonstrate efficiency of our method.Finally, in Section 7 we offer a short summary and mention several generalizations of the results presented.
Most proofs, many of which are rather technical, are omitted.In such a case, the papers are cited right before the corresponding theorems, where detailed proofs are available.

Preliminaries and the Concept of a Stochastic Functional Differential Equation
Let (Ω, F, (F) ≥0 , ) be a stochastic basis (see, e.g., [27]), where Ω is a set of elementary probability events, F is a algebra of all events on Ω, (F) ≥0 is a right continuous family of -subalgebras of F, and  is a probability measure on F; all the above -algebras are assumed to be complete with respect to the measure , that is, containing all subsets of zero measure; the symbol  stands below for the expectation related to the probability measure .
In the sequel, we will use an arbitrary yet fixed norm | ⋅ | in   , ‖ ⋅ ‖ being the associated matrix norm.
stands for the identity matrix (as long as its size is defined).
The space   consists of all -dimensional, F 0 -measurable random variables, and  =  1 is a commutative ring of all scalar F 0 -measurable random variables.
We consider a homogeneous stochastic hereditary equation equipped with two extra conditions Here  ℎ is a -linear Volterra operator (see the following), which is defined in certain linear spaces of vector stochastic processes,  is an F 0 -measurable stochastic process, and  0 ∈   .By -linearity of the operator  ℎ we mean the following property: holding for all F 0 -measurable, bounded, and scalar random values  1 ,  2 and all stochastic processes  1 ,  2 belonging to the domain of the operator  ℎ .
The following kinds of stochastic Lyapunov stability will be discussed in this paper.
In the sequel the following linear spaces of stochastic processes will be used: (i)   () consists of all predictable ×-matrix stochastic processes on [0, +∞), the rows of which are locally integrable with respect to the semimartingale  (see, e.g., [29] or [27]).
A solution of the initial value problem (4) and (5a) is a stochastic process from the space   satisfying the equation where ()() = ∫  0 [()() + ()]() is a -linear Volterra operator in the space   and the integral is understood as a stochastic one with respect to the semimartingale  (see, e.g., [27] or [29]) and  0 ∈   .Equation (4) will be referred to as a linear functional differential equation with respect to a semimartingale.For () = col(, B 2 (), . . ., B  ()) we obtain a linear functional differential equation of Itô type which is a particular case of the general equation (4).
According to [3] (see also the habilitation thesis [13]) the following classes of linear stochastic equations also are particular cases of (4): (a) Systems of linear ordinary (i.e., nondelay) stochastic differential equations driven by an arbitrary semimartingale (in particular, systems of ordinary Itô equations).
(b) Systems of linear stochastic differential equations with discrete delays driven by a semimartingale (in particular, systems of Itô equations with discrete delays).(c) Systems of linear stochastic differential equations with distributed delays driven by a semimartingale (in particular, systems of Itô equations with distributed delays).(d) Systems of linear stochastic integrodifferential equations driven by a semimartingale (in particular, systems of Itô integrodifferential equations).(e) Systems of linear stochastic functional difference equations driven by a semimartingale (in particular, systems of Itô functional difference equations).
For instance, a general linear stochastic differential equation with distributed delay where and R  are vector functions defined on {(, ):  ∈ [0, ∞), −∞ <  ≤ } for  = 1, . . ., , which is equipped with the prehistory condition can be, under natural assumptions on R  and  (see, e.g., [3,13]), rewritten as the functional differential equation ( 4) with It is also worth mentioning that our concept of a stochastic functional differential equation covers as well the case of functional differential equations with respect to random Borel measures.In this case, we can simply put () = () in ( 4), where () is a random Borel measure of bounded variation, so that the space   () contains all -dimensional stochastic processes on [0, +∞) with the trajectories that are a.s.locally integrable with respect to .Thus, (4) contains linear random differential equations including no delay, discrete and distributed delays, and linear random integrodifferential equations and linear random functional difference equations.Finally, if () is nonrandom (or equivalently if Ω contains only one point), then we obtain the deterministic versions of all the above classes of equations.
Representation of solutions of the deterministic functional differential equations (the generalized Cauchy formula) plays an important role in the stability analysis and in the theory of quasilinear equations.The following lemma gives the representation of the solutions of the initial value problem ( 4) and (5a).4) and (5a) have a unique (up to the -equivalence) solution   (,  0 ) for any  ∈   () and  0 ∈   .Then one has the following representation:
Proof.Using the -linearity of the operator  it is easy to see that () 0 is a solution of the homogeneous equation corresponding to (4).Now consider Due to the assumptions of the lemma, the initial value problem ( 4) and (13b) has a unique solution for any  ∈   ().Thus, this problem gives rise to an operator from   () to   .Denote this operator by .Clearly, ()(0) = 0.The -linearity of  follows directly from the -linearity of the operator  and the unique solvability of the initial value problem ( 4) and (13b).Therefore, the stochastic process on the right-hand side of ( 13) satisfies ( 4) and (5a).
Remark 4. For some classes of (4), an explicit integral formula for the operator  in the representation ( 13) can be given.The corresponding results can be found in [13].The problems of existence and uniqueness of solutions of the initial value problem ( 4) and (5a) are considered in [13,28].
This definition says that the solutions belong to    whenever  ∈   and  0 ∈    and that they continuously depend on  and  0 in the appropriate topologies.The choice of the spaces is closely related to the kind of stability we are interested in.Formally speaking, one has to mention the space of initial values    in this definition [25].However, this space is kept fixed in the present paper, so that we skip it.
Below we demonstrate how this result leads to efficient algorithms in analysis of Lyapunov stability of linear stochastic functional differential equations.The main idea, which is described in [3,[19][20][21], is to convert the given property of Lyapunov stability, via the property of ISS, into the property of invertibility of a certain regularized operator in a suitable functional space.This operator can be constructed with the help of an auxiliary equation.
The description of this algorithm applied to (4) starts from choosing an auxiliary equation, which we call a reference equation.The latter is similar to (4), but it is "simpler," so that the required ISS property is already established for this equation: where :   →   () is a -linear Volterra operator and  ∈   ().For (15) the existence and uniqueness property is always assumed, that is, that for any (0) ∈   there is the only (up to a -equivalence) solution () satisfying (15).According to the lemma, for the solutions of ( 15) we have the following representation: where () is the fundamental matrix of the associated homogeneous equation and  is the corresponding Cauchy operator for (15).As in the deterministic case, we have two versions of the regularization: the right and the left one.They stem formally from the same reference equation but produce different integral equations.Also the applicability conditions are different.
We start with the right regularization.Inserting (16a) into (4) yields Denoting ( − ) = Θ  , we obtain the operator equation ( 1) The regularization is called "right" as the operator  is placed to the right of the operator  in (4).The letter "" in the operator Θ  is due to the word "right." The next result of this section lists the assumptions on the reference equation, under which the right regularization may be applied.
Theorem 7. Given a weight  (i.e., a positive continuous function defined for  ≥ 0), let one assume that (4) and the reference equation ( 15) satisfy the following conditions: (1) The operators  and  act continuously from    to   .
If now the operator −Θ  :   →   has a bounded inverse, then ( 4) is ISS with respect to the pair (   ,   ).
Proof.Under the assumptions of the theorem we have for arbitrary  0 ∈    ,  ∈   .The ISS property of the reference equation implies which holds for all  0 ∈    ,  ∈   .Here c is some positive number.Taking now the norms we arrive at the inequality for some  > 0. This implies that (4) is ISS with respect to the pair (   ,   ).
Consider now the case of the left regularization rewriting (4) as follows: or alternatively as Denoting (−) = Θ  , we obtain the operator equation ( 2) Theorem 8. Given a weight  (i.e., a positive continuous function defined for  ≥ 0), let one assume that ( 4) and the reference equation ( 15) satisfy the following conditions: ( Proof.Under the above assumptions we have that (⋅) 0 ∈    whenever  0 ∈    and also that for arbitrary  0 ∈    ,  ∈   .Taking the norms and using the assumptions put on the reference equation, we, as in the previous theorem, obtain the inequality where  0 ∈    ,  ∈   .Thus, ( 4) is ISS with respect to the pair (   ,   ).
The left and right regularization give usually different stability results, in both the deterministic and stochastic theory.In the stochastic case, the left regularization appears to be more efficient.
Finally, we remark that the choice of the space  and the weight  depends on the asymptotic property we are studying.In the next section we describe typical examples which are general enough to cover most interesting cases of stochastic stability and, on the other hand, specific enough to ensure applicability of the important Bohl-Perron property.

ISS with respect to Weighted Spaces and Bohl-Perron Type Theorems
By Bohl-Perron type theorems one means results ensuring equivalence between ISS with respect to the spaces with and without weights.This allows for deducing asymptotic (exponential) stability from the simple stability which is much easier to handle.
For technical reasons we restrict ourselves to the so-called special semimartingales in this section.In this case we will be able to give a more explicit description of the spaces   and   ().
Clearly, we can always do it by adding, if necessary, new, ( + 1)th component to the -dimensional semimartingale ().
It is known [29] that for special semimartingales the space   () consists of all predictable  × -matrices () = [  ()], for which for any  ≥ 0.Here Under the above assumptions we can also write Finally, the space   consists of all -dimensional F  -adapted stochastic processes on [0, +∞), which are right continuous and have left-hand side limits at all points.
The following particular cases of the general space  are crucial for our further considerations: The following parameters are used in the above definitions: (i) The function  is a nonnegative measurable function defined for  ≥ 0.
(iv)  + , (v)    is the linear space of scalar functions defined on [0, +∞), -integrable (1 ≤  < ∞) with respect to the increasing function , and essentially (with respect to the measure generated by ) bounded if  = ∞.
Notice that the spaces Λ  , (,  1 ,  2 ) consist of the same stochastic processes; that is, they in fact are independent of the choice of the positive numbers  1 ,  2 , the only difference being the norms in these spaces, which is introduced by technical reasons.The same applies to the spaces Λ + , (,  1 ,  2 ).We remark that all of these spaces are of importance for analysis of ISS of (4).
Finally, we let   stand for the space   in the scalar case.Below we assume, for the sake of simplicity, that the operator  in the reference equation ( 15) is deterministic.More specifically, it means the following.
Remark 15.In the last two theorems it is required that the operator  satisfies the Δ-condition.This is always true if this operator comes from an ordinary stochastic differential equation.In the case of delay equations, the operator  typically satisfies the Δ-condition if the delay is bounded, but the latter assumption can be essentially generalized.This problem is studied in more detail in [3, 11-14, 30, 31] and in some other papers.

Lyapunov Stability of Linear Hereditary Itô Equations
Stability analysis of stochastic differential equations including delays and impulses, which is based on the Lyapunov second (direct) method and its generalizations, is technically difficult and, in some cases, probably, even impossible.In this and in the next section we concentrate on such examples of stochastic hereditary equations, where, as we show, our technique based on ISS and subsequent regularization is directly applicable.We restrict ourselves to the case of Itô equations, which are driven by the standard Brownian motion [27], although the results presented below can be generalized to hereditary equations driven by semimartingales.
Given a number 1 ≤  < ∞, the following inequality [27, Ineq.(3.1)] is essential for estimates of the operator norms in the spaces of stochastic processes: where B is a scalar standard Brownian motion, ,  are arbitrary positive numbers,  is an arbitrary measurable and (F  ) ≥0 -adapted (i.e., () is (F  )-measurable for any  ≥ 0) scalar stochastic process on [, ], and   is a universal constant which only depends on .Note that  1 = 1, while for  > 1 the estimates for   can be found in the literature (see, e.g., [27] or [32, page 40]), where  should be replaced with 2.Similar estimates for general semimartingales (the Burkholder-Davis-Gundy inequalities) are also available and can be used to estimate operator norms in the -method, but we do not use these norms in the present paper referring the reader to the cited publications.
Our first example is a scalar Itô equation with infinite delay.The proof of this result can be found in [15].
Notice that system (38), (39a), and (39b) is a particular case of the general system (1) and (2a) considered in Section 1. Stability analysis can be therefore performed by our method with a proper choice of the reference equation.
In the paper [22] several corollaries of this theorem are listed, which can be used to obtain more explicit conditions of the exponential 2-stability (1 ≤  < ∞) of ( 38) and (39b) with respect to the initial data.
The following theorem was proved in [23].
In [23] several corollaries of this theorem are given, which provide more explicit conditions of the asymptotic 2stability (1 ≤  < ∞) of the impulsive equations of the form ( 38) and (39b).

Itô Type Linear Functional Difference Equations
Stochastic difference equations were truly defined in [33].Stochastic functional difference equations were introduced in [34] and studied further in [35].Analysis of Lyapunov stability for stochastic difference equations is a challenging mathematical problem which has attracted attention of many researchers but has not yet been comprehensively studied.Some theorems on stability of ordinary stochastic difference equations can be, for example, found in [36][37][38][39], while results on stability of stochastic functional difference equations are summarized in the monograph [40].As in the case of differential equations, stochastic versions of Lyapunov's classical methods can be applied to difference equations, too.These methods work in many situations, yet some important classes of equations seem to be insufficiently studied, which may be due to several technical restrictions one has to put on Lyapunov functions and especially functionals, in the stochastic case.In this section we apply our method instead and show that it is efficient in many situations.
In the sequel  is the set of all natural numbers,  + = {0} ∪ , and the variable  is always assumed to belong to  + ; that is,  = 0, 1, 2, . ... Below we consider the following stochastic difference equations: (2)   is the linear space of all sequences of ×-matrices ()( ∈  + ), with the entries being F  -measurable random variables.
It is easy to see that  is a linear operator from   to   .The normed subspace  can be now chosen in the way described in Section 4. We omit here the technicalities referring the reader to [24,25].

Conclusion
In the present paper we considered the property of -stability (1 ≤  < ∞) for various linear stochastic equations.We showed how to obtain efficient stability results formulated in terms of the parameters of the equations in question.However, let us stress that the central results of the paper are also valid for other kinds of stochastic stability, for example, stability in probability and almost sure stability (see, e.g., [13]).The major technical problem arising in analysis of these types of stability is mostly due to the fact that the corresponding spaces of stochastic processes do not admit norms, which makes the analysis of invertibility of the associated linear operators more difficult and less efficient.
Moreover, the method can also be applied to analysis of other asymptotic properties of stochastic equations and not only to the Lyapunov stability.For instance, in [13,14,16,41] the problem of partial Lyapunov stability was addressed.
Unlike the Lyapunov-Krasovskii-Razumikhin method, which is based on the existence of a suitable Lyapunov function (or the Lyapunov-Krasovskii functional in the delay case), our method is based on the ISS property of the equation and requires a suitable auxiliary (reference) equation which is used to regularize the equation under consideration.The crucial point is then to check solvability of a regularized equation in a carefully chosen space of stochastic processes.This yields ISS and thus the desirable asymptotic property.
Several techniques of constructing reference equations have been worked out in the papers cited above.Normally, such an equation is chosen to be dependent on a parameter, which ensures a certain flexibility.This approach has led to several new and interesting results, many of which seem to be difficult (impossible?) to obtain by other methods.One of such examples, equations with random delays, is studied in [17].It is also worth mentioning that our techniques allow for studying Lyapunov stability with respect to random initial data (see Definition 1), which is rarely possible when using Lyapunov function(al)s.
Another crucial idea, which is of great importance in the stability theory, is the Bohl-Perron principle, which states that asymptotic and exponential stability in rather general cases can be deduced from invertibility of the operators in nonweighted spaces.Technically, it is much easier to deal with the latter spaces rather than the spaces with a nontrivial weight.It turned out to be possible to formulate the Bohl-Perron principle in terms of the verifiable Δ-condition.Roughly speaking, this condition says that the delays in the system should not be unbounded, although many delay equations with unbounded delays also satisfy the Δ-condition.
Finally, we remark that although the method was originally designed for stochastic linear equations, several classes of nonlinear stochastic functional differential equations can also be investigated similarly; see [12,13,16] for some results.In [26] the mean square input-to-state stability was introduced to study neural networks.The technique is rather promising and might be particularly efficient in several important applied problems, for example, in the synchronization theory for networks with random switching [42].