Lyapunov Stability of the Generalized Stochastic Pantograph Equation

The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions.


Introduction
In this paper we study Lyapunov stability of the stochastic pantograph equation (see, e.g., [1][2][3]): where 0 <   < 1, and its generalizations (see (21) in Section 4).A very good and comprehensive description of the role of the classical pantograph equation and its stochastic counterpart, including historical comments, can be found in the paper [2].Let us only mention that generalizations of the pantograph equations have also attracted attention of many mathematicians; see, e.g., [4][5][6][7][8][9][10][11] and the references therein.Stability analysis of ( 1) and ( 21) has a special feature: the delay is unbounded, so that many methods, including those based on Lyapunov-Krasovskii functionals, are inapplicable.One uses therefore various special techniques, which can, e.g., be found in the papers [12,13] (the stochastic case) and [5] (the deterministic case).These techniques help to produce verifiable stability criteria, mostly in the case of the classic pantograph equation (1).
Our approach goes back to the framework developed in the monographs [14] (for linear differential equations in Banach spaces) and [15] (for linear deterministic functional differential equations), where Lyapunov stability is replaced by input-to-state stability, i.e., the property of the equation where its solutions belong to certain linear topological spaces and continuously depend (in the corresponding topology) on the initial data.In the stochastic case this approach is outlined in [16].On the other hand, (1) and ( 21) possess a very specific property: their initial conditions are finite dimensional, i.e., identical to the ones for ordinary differential equations.This considerably simplifies the analysis of the input-tostate stability, as all linear finite dimensional operators are bounded, and we only need to prove that all solutions of the equation belong to a certain topological space.For brevity, we will call this property -stability keeping in mind that this is, in fact, a particular case of the input-to-state stability for linear equations with finite dimensional spaces of initial data.
The idea of how to verify the property of input-to-state stability for linear deterministic functional differential equations goes back to the papers of N.V.Azbelev and his students (see [15] and the references therein) who call their technique the -method.It is somewhat 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 asymptotic property and which then is used to regularize the original equation.Like Lyapunov's method, the -method also provides necessary and sufficient stability conditions.The -method proven to be rather efficient for many classes of delay equations, especially those where searching for Lyapunov functionals seems to be difficult.Equations with infinite delays can serve as a prominent example of such a class.
In [17], the method was for the first time applied to linear stochastic functional differential equations and developed further by the authors in the series of publications (see the review article [16]).The first efficient stability conditions for stochastic differential equations with unbounded delays, obtained by the -method, were presented in the paper [7].
In the present paper we develop this approach further by concentrating on specific stochastic equations with unbounded delays and finite dimensional initial conditions: the pantograph equation (1) and its generalization (21).In the examples below (see Section 4) we only use the simplest reference equation ẋ +  = , where  > 0 is a parameter, variation of which ensures best possible stability conditions.More sophisticated reference equations (e.g., those including delays) can be found in other publications of the authors.
The paper is organized as follows.
In Section 2 we introduce some notation and define the general linear stochastic functional differential equation, which is used in Section 3 in the definitions of different kinds of stochastic stability.In this section we also offer a precise definition of -stability for different spaces of stochastic processes.The central result of Section 3 describes relationship between stochastic Lyapunov stability and stability, where we relate specially defined spaces of stochastic processes to different kinds of stochastic Lyapunov stability.
Let us remark that the role of the definitions and results presented in Section 3 goes far beyond the applications to the stability analysis of the generalized stochastic pantograph equation.Bearing in mind these future applications, we chose to formulate and prove the results of Section 3 for the case of the general functional differential equation (2).In addition to (21), (2) covers integrodifferential equations arising, e.g., in electrical circuit analysis [18].
Section 4 contains applications to the stability analysis of the generalized stochastic pantograph equation ( 21), but we stress that most results are also new for the stochastic pantograph equation (1).This includes, e.g., conditions of stability for  ̸ = 2, stability with probability 1, and stability conditions in the vector case.
Finally, Section 5 contains a short overview of the main results of the paper as well as some suggestions on further applications of the developed method.

Notation and Preliminaries
Let (Ω, F, (F  ) ≥0 , ) be a stochastic basis (see, e.g., [19]), 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 (w.r.t. in what follows) the measure , i.e., they contain all subsets of zero measure; the symbol  stands for the expectation related to the probability measure .
In the sequel, we use an arbitrary yet fixed norm |.| in   , the real-valued index  satisfying the assumption 0 ≤  ≤ ∞, and a continuous positive function () defined for all  ≥ 0.
The general linear stochastic functional differential equation is defined as follows (see, e.g., [16]): and the initial condition reads in this case as Here  is a -linear Volterra operator (see below), which is defined in certain linear spaces of vector-valued stochastic processes.
By the -linearity of the operator  we mean the property which holds 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 .
According to [17] the following classes of linear stochastic equations can be represented as (2): (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).
Of course, the stochastic pantograph equation (1) and its generalization (21) are also particular cases of the general functional differential equation (2).

Lyapunov Stability and 𝑀-Stability
In this section we study different kinds of stochastic Lyapunov stability of the zero solution of the linear equation ( 2) with respect to the initial data (3).Let us start with the precise definitions.
Remark 2. The initial condition  0 can also be random.In this case the norm of  0 should be adjusted accordingly.
For brevity, we will also write " (2) is stable" in a certain sense instead of "the zero solution of ( 2) is stable" in this sense.
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 w.r.t. the semimartingale  (see, e.g., [19]).
For () = 1 ( ≥ 0) we also put Let  be a linear subspace of the space   () equipped with some norm ‖.‖  .For a given positive and continuous function () ( ∈ [0, ∞)) we define   = { :  ∈ ,  ∈ }.The latter space becomes a linear normed space if we put ‖‖   fl ‖‖  .By this, the linear spaces Remark 3. The above spaces can also be described as follows.Let  ∞ () be the space consisting of all essentially bounded functions  : [0, ∞) → , while L  () is the space of measurable ( = 0), -integrable (0 <  < ∞), essentially bounded ( = ∞) functions ℎ : Ω → , where  and  are arbitrary separable Banach spaces.Then it is easy to see that )) for all 0 ≤  ≤ ∞ and an arbitrary positive and continuous function  : [0, ∞) → .This means that the above list of the spaces covers all possible combinations of Lebesgue spaces with respect to the variable  ∈ Ω and spaces of essentially bounded functions with respect to the variable  ∈ [0, ∞).As we will see, this list covers also all types of stochastic Lyapunov stability described in Definition 1.
Our first theorem describes relationships between the different kinds of the stochastic Lyapunov stability and the associated S-stabilities for (2).Theorem 5.The following statements describe relationship between stochastic stability properties of ( 2) and the -spaces: (1) weak stability in probability is equivalent to the  0stability; (2) weak asymptotic stability in probability is equivalent to the   0 -stability for some  satisfying Property 1; (3) stability in probability is equivalent to the M0 -stability; (4) if 0 <  < ∞, then -stability is equivalent to the  stability; (5) if 0 <  < ∞, then asymptotic -stability is equivalent to the    -stability for some  satisfying Property 1; (6) if 0 <  < ∞, then exponential -stability is equivalent to the    -stability for some  satisfying Property 2; (7) stability with probability 1 is equivalent to the M0stability; (8) strong stability with probability 1 is equivalent to the  ∞ -stability.
Proof.We consider all cases separately.
Proof.The first and the second part follow directly from the first statement of Lemma 7 combined with statements ( 4)-( 6) and ( 1)-( 2) and ( 4)-( 5) of Theorem 5, respectively.The third part follows from the second statement of Lemma 7 combined with statements (1)-( 4) of Theorem 5. To prove the last part first we observe that statements (3) and (8) of Theorem 5 contain the same space   0 , so that stability in probability and stability with probability 1 are equivalent for (2).
In the next theorem, we describe more relations between different kinds of the stochastic Lyapunov stability.Some of these results are used in the examples below.(2) M  -stability with  satisfying Property 1 implies asymptotic stability with probability 1.
(2).Due to the first statement of the theorem and the inclusion M  ⊂ M 0 we obtain the property of stability with probability 1.To prove asymptotic stability with probability 1, we proceed as in the proof of statement (7) of Theorem 5 adding () to (,  0 ) and (), which gives almost everywhere boundedness of the random function  = () = sup ≥0 ()‖()‖.Therefore so that almost everywhere as  → ∞.

Delay-Independent Stability Conditions for the Generalized Pantograph Equation Driven by the Brownian Motion
Delay-independent stability is also called absolute stability (see, e.g., [20]).This property is important if the delay functions are unknown or difficult to determine.It is well-known (see, e.g., [3]) that using Itô's formula usually results in a delay-dependent stability condition for stochastic pantograph equations.Below we present an alternative method which is based on the theory developed in Section 3.
We start with the analysis of -stability, where we use the integral form of the Marcinkiewicz-Zygmund inequality which holds true for any predictable stochastic process () (0 ≤  ≤ ), any  > 0, and the scalar standard Brownian motion B().The constant   depends on the number  (0 <  < ∞), only.In 1988 D.L. Burkholder proved (see, for example, [21,22]) that the constant   = 2 − 1 in the Marcinkiewicz-Zygmund inequality ( 20) is best possible for all for  ≥ 1.
Now we briefly describe explicit stability conditions for one particular case of the vector equation (21).
Corollary 12. Let the matrix  have only real eigenvalues   (1 ≤  ≤ ) and   be scalar matrices: , where  0 is the only positive root of the quadratic polynomial and Proof.Let us choose  =  2 0 .To verify condition (22) of Theorem 10 we first observe that for any positive  > 0 there exists a basis in   , in which the Euclidean matrix norm ‖.‖ satisfies ‖+  ‖ < |+|+, where  is the largest eigenvalue of the matrix .At the same time, the Euclidean norms of the scalar matrices   (0 ≤  ≤  − 1; 1 ≤  ≤   ) are independent of the choice of the basis.Denoting for all  = 0, . . .,  − 1 we obtain, exactly in the same manner as in the proof of Corollary 11, that for sufficiently small . In where   is the  ×  identity matrix.Then ( 21) is stable with probability 1.
Proof.We want to apply the first statement from Theorem 9 by checking that the solutions () of ( 21) belong to the space M2 for all (0) ∈   .This is done exactly in the same way as in the proof of Theorem 10 provided that the norm (sup ≥0 |()| 2 ) 1/2 is replaced by the norm ( sup ≥0 |()| 2 ) 1/2 and the Doob inequality is used instead of the Marcinkiewicz-Zygmund inequality.

Conclusions and Outlook
In the paper we described and justified a new framework for stability analysis of stochastic functional differential equations in the case when initial data are finite dimensional.
In particular, this framework covers the generalized stochastic pantograph equation.The main feature of our analysis consists in replacing Lyapunov stabilities with input-tostate stabilities by choosing appropriate spaces of stochastic processes.It is shown that this approach is applicable to all known kinds of stochastic Lyapunov stability.We demonstrated the efficiency of this idea by applying it to the generalized stochastic pantograph equation, where the emphasis was put on delay-independent stability conditions.In particular, we studied 2-stability ( ≥ 1) and stability with probability 1.
The future development of the suggested framework may include the following topics: (1) Stability analysis of the stochastic pantograph equations may be extended to the case of nonconstant coefficients.In this case, Theorem 5 and other results of Section 3 could still be applied.
(2) More attention should be paid to the vector case.In particular, coefficient-based stability conditions may be derived from Theorems 10 and 13 to get more general stability tests than those offered in Section 4.
(3) Asymptotic -stability and asymptotic stability with probability 1 are only described in terms of stability, but not in the form of specific stability conditions, so that further analysis of these asymptotic properties should be continued.
(4) The stochastic pantograph equation driven by an arbitrary semimartingale can also be studied using the techniques developed in Section 3.
(5) The main results of Section 3 stretch are far beyond the generalized pantograph equation.As it is mentioned in Section 2, wide classes of stochastic functional differential equations can be represented as (2).The technique developed in this paper can be, R  are vector functions defined on {(, ) :  ∈ [0, ∞), 0 <  ≤ } for  = 1, . . ., .But this analysis is beyond the scope of the present paper.
(6) Stochastic hereditary equations with more general initial conditions () = (),  ≤ 0, can easily be transformed into (2) as well (see, e.g., [16]).However, a possible generalization of the main results of Section 3 will not be straightforward, as in this case, one will, in addition, need to prove boundedness of the infinite dimensional linear operators assigning  to the associated solution ().
.Proof.The first statement simply follows from the standard relationship between Lebesgue spaces generated by finite measure sets.The second and the third statements are a direct consequence of the definitions of the spaces −(−)    (ℎ  ()) B  () the next theorem, where we study stability with probability 1, we use the Doob inequality where B is the scalar standard Brownian motion and  is an arbitrary predictable stochastic process on [0, ∞).