# Lyapunov Stability of the Generalized Stochastic Pantograph Equation

Authors: Ramazan Kadiev; Arcady Ponosov
Journal: Journal of Mathematics
 (2018)

---

## Abstract

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.


---

## Body

## 1. Introduction

In this paper we study Lyapunov stability of the stochastic pantograph equation (see, e.g., [1–3]):(1)dxt=Axt+∑j=0m0Aj0xgj0tdt+∑l=1m-1∑j=0mlAjlxgjltdBlt,where 0<gjl<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., [4–11] 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 byinput-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-to-state 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 M-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 techniquetheW-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 W-method also provides necessary and sufficient stability conditions. The W-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 W-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 x˙+αx=g, where α>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 Section2 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 M-stability for different spaces of stochastic processes. The central result of Section 3 describes relationship between stochastic Lyapunov stability and M-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 Section3 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].Section4 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 p-stability for p≠2, stability with probability 1, and stability conditions in the vector case.Finally, Section5 contains a short overview of the main results of the paper as well as some suggestions on further applications of the developed method.


## 2. Notation and Preliminaries

Let(Ω,F,(Ft)t≥0,P) be a stochastic basis (see, e.g., [19]), where Ω is a set of elementary probability events, F is a σ-algebra of all events on Ω, (Ft)t≥0 is a right continuous family of σ-subalgebras of F, and P is a probability measure on F; all the above σ-algebras are assumed to be complete with respect to (w.r.t. in what follows) the measure P, i.e., they contain all subsets of zero measure; the symbol E stands for the expectation related to the probability measure P.In the sequel, we use an arbitrary yet fixed norm. in Rn, the real-valued index p satisfying the assumption 0≤p≤∞, and a continuous positive function γ(t) defined for all t≥0.ByZ=(z1,…,zm)T we denote an m-dimensional semimartingale (see, e.g., [19]), while B=(B1,…,Bm)T stands for the standard Brownian motion (the Wiener process), the particular case of Z.The general linear stochastic functional differential equation is defined as follows (see, e.g., [16]):(2)dxt=VxtdZtt≥0,and the initial condition reads in this case as(3)x0=x0∈Rn.Here V is a k-linear Volterra operator (see below), which is defined in certain linear spaces of vector-valued stochastic processes.By thek-linearity of the operator V we mean the property (4)Vα1x1+α2x2=α1Vx1+α2Vx2,which holds for all F0-measurable, bounded, and scalar random values α1,α2 and all stochastic processes x1,x2 belonging to the domain of the operator V.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).


## 3. Lyapunov Stability andM-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.Definition 1.
The zero solution of (2) is called (1)
weakly stable in probability if for anyϵ>0,δ>0 there is η(ϵ,δ)>0 such that P{ω∈Ω:xt,x0>ϵ}<δ for all x0<η and t≥0;(2)
asymptotically weakly stable in probability if it is weakly stable in probability and if, in addition, for anyϵ>0 and all x0∈Rn one has P{ω∈Ω:xt,x0>ϵ}→0 as t→+∞;(3)
stable in probability if for anyϵ,δ>0 there is η(ϵ,δ)>0 such that P{ω∈Ω:supt≥0xt,x0>ϵ}<δ for all x0<η;(4)
asymptotically stable in probability if it is stable in probability and if, in addition, for anyε>0 and all x0∈Rn one has P{ω∈Ω:xt,x0>ϵ}→0 as t→+∞;(5)
p-stable if for any ϵ>0 there is η(ϵ)>0 such that x0<η implies E|x(t,x0)|p≤ϵ for all t≥0;(6)
asymptoticallyp-stable if it is p-stable and, in addition, limt→+∞⁡Ext,x0p=0 for all x0∈Rn;(7)
exponentiallyp-stable if there exist positive constants K, β such that the inequality Ext,x0|p≤Kx0|pexp⁡{-βt} holds true for all t≥0 and all x0∈Rn;(8)
stable with probability 1 ifsupt≥0|x(t,xν)|→0 with probability 1 whenever xν→0 as ν→+∞;(9)
asymptotically stable with probability 1 if it is stable with probability 1 and if, in addition,xt,x0→0 as t→+∞ for all x0∈Rn;(10)
strongly stable with probability 1 if for anyϵ>0 there exists η(ϵ)>0 such that P{ω∈Ω:supt≥0|x(t,x0)|≤ϵ}=1 whenever |x0|<η;(11)
strongly asymptotically stable with probability 1 if it is strongly stable with probability 1 and if, in addition, for anyϵ>0x(t,x0) it tends to 0 with probability 1 as t→+∞ for all x0∈Rn.Remark 2.
The initial conditionx0 can also be random. In this case the norm of x0 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)Ln(Z) consists of all predictable n×m-matrix stochastic processes on [0,+∞), the rows of which are locally integrable w.r.t. the semimartingale Z (see, e.g., [19]).(ii)Dn consists of all n-dimensional stochastic processes on [0,+∞), which can be represented as (5)xt=x0+∫0tHsdZs,where x(0)∈Rn, H∈Ln(Z).The spaces below ("M-spaces") are crucial for studying the stochastic Lyapunov stabilities listed above.M 0 γ = { x : x ∈ D n such that for any δ>0 there is K>0, for which supt≥0P{ω:ω∈Ω,|γ(t)x(t)|>K}<δ}.M ^ 0 γ = { x : x ∈ D n such that for any δ>0 there is K>0, for which P{ω:ω∈Ω,supt≥0|γ(t)x(t)|>K}<δ}.M p γ = { x : x ∈ D n , s u p t ≥ 0  E | γ ( t ) x ( t ) | p < ∞ } (0<p<∞).M ^ p γ = { x : x ∈ D n , E s u p t ≥ 0 | γ ( t ) x ( t ) | p < ∞ } (0<p<∞).M ∞ γ = M ^ ∞ γ = { x : x ∈ D n , e s s  s u p ( t , ω ) ∈ [ 0 , + ∞ [ × Ω | γ ( t ) x ( t ) | < ∞ }.Forγ(t)=1 (t≥0) we also put Mp1=Mp and M^p1=M^p (0≤p≤∞).LetB be a linear subspace of the space Ln(Z) equipped with some norm .B. For a given positive and continuous function γ(t)(t∈[0,∞)) we define Bγ={f:f∈B,γf∈B}. The latter space becomes a linear normed space if we put fBγ≔γfB. By this, the linear spaces Mpγ, M^pγ become normed spaces if 1≤p≤∞.Remark 3.
The above spaces can also be described as follows. LetL∞(X) be the space consisting of all essentially bounded functions g:[0,∞)→X, while Lp(Y) is the space of measurable (p=0), p-integrable (0<p<∞), essentially bounded (p=∞) functions h:Ω→Y, where X and Y are arbitrary separable Banach spaces. Then it is easy to see that Mpγ=L∞(Lp(Rn)) and M^pγ=Lp(L∞(Rn)) for all 0≤p≤∞ and an arbitrary positive and continuous function γ:[0,∞)→R. This means that the above list of the M-spaces covers all possible combinations of Lebesgue spaces with respect to the variable ω∈Ω and spaces of essentially bounded functions with respect to the variable t∈[0,∞). As we will see, this list covers also all types of stochastic Lyapunov stability described in Definition 1.Below we use the following assumptions on a continuous positive functionγ(t), t∈[0,∞).Propertyγ1: the function γ satisfies the conditions γ(t)≥σ (t∈[0,+∞)), σ>0, and limt→+∞γ(t)=+∞.Propertyγ2: γ(t)=exp{βt} for some β>0.Definition 4.
Letx(.,x0) be the solution of the initial value problem (2)-(3) defined on [0,∞) and S be a certain subspace of the space Dn. We say that (2) is S-stable if x(.,x0)∈S for any x0∈Rn.Our first theorem describes relationships between the different kinds of the stochastic Lyapunov stability and the associatedS-stabilities for (2).Theorem 5.
The following statements describe relationship between stochastic stability properties of (2) and the M-spaces:(1)
weak stability in probability is equivalent to theM0-stability;(2)
weak asymptotic stability in probability is equivalent to theM0γ-stability for some γ satisfying Property γ1;(3)
stability in probability is equivalent to theM^0-stability;(4)
if0<p<∞, then p-stability is equivalent to the Mp-stability;(5)
if0<p<∞, then asymptotic p-stability is equivalent to the Mpγ-stability for some γ satisfying Property γ1;(6)
if0<p<∞, then exponential p-stability is equivalent to the Mpγ-stability for some γ satisfying Property γ2;(7)
stability with probability 1 is equivalent to theM^0-stability;(8)
strong stability with probability 1 is equivalent to theM∞-stability.Proof.
We consider all cases separately.
(1)⇐). Choosing arbitrary positive numbers ϵ and δ and minding that x(t,x0)=X(t)x0 for all x0∈Rn and all t≥0, we first find K(δ)>0 such that supt≥0P{ω∈Ω:X(t)>K(δ)}<δ. Letting η=ϵ/K(δ) we observe that |x0|<η implies supt≥0P{ω∈Ω:|x(t,x0)|>ϵ}<δ for all x0, |x0|<η, which is equivalent to weak stability in probability.
⇒ ) . Assume that the zero solution of (2) is weakly stable in probability. Using the notation from the previous part of the proof, we assume, in addition, that there exists x0∈Rn such that x(.,x0) does not belong to M0. Hence there is δ0>0 such that for any ϵ>0(6)Pω∈Ω:xtϵ,x0>ϵ>δ0for at least one t(ϵ)>0. In such a case, the solution x^(t)≡η′x(t,x0)/|x0|, where 0<η′<η) satisfies the estimate P{ω∈Ω:|x(t(ϵ))|>ϵ}>δ0, although |x0|<η, which contradicts the assumption.
(2)⇐). The weak stability in probability follows from the observation that M0γ is topologically imbedded into M0. Let us now take arbitrary x0∈Rn, ϵ>0, and δ>0. As x(.,x0)∈M0γ, there exists K>0, for which (7)supt≥0⁡Pω∈Ω:γtxt,x0>K<δ.Now, choosing T>0 satisfying Kγ-1(t)<ϵ for all t≥T we obtain (8)supt≥0⁡Pω∈Ω:xt,x0>ϵ≤supt≥0⁡Pω∈Ω:γtxt,x0>K<δ,which implies weak asymptotic stability.
⇒ ) . From the weak asymptotic stability in probability and the representation x(t,x0)=X(t)x0 for all x0∈Rn, we deduce that the function γ¯(t)=E(X(t)∧1)→0 as t→∞, as E(|x-y|∧1) is the metric in the space L0. Then the function γ(t)=1/γ¯(t) satisfies Property γ1 with σ=1. Now, we take an arbitrary x0∈Rn, |x0|≤1, and define K=2Cδ-1 where C=supt≥0γ¯(t). Let also choose T>0 satisfying the estimate P{ω∈Ω:X(t)>1}<δ/2 for all t≥T. Then we obtain (9)Pω∈Ω:xt,x0>K≤Pω∈Ω:Xt>K≤Pω∈Ω:Xt>1+Pω∈Ω:Xt∧1>K≤δ2+δ2CEXt∧1=δ2+δ2Cγ¯t<δfor all t≥T. This yields supt≥TP{ω:ω∈Ω,|γ(t)x(t)|>K}<δ}, which is sufficient to conclude that x(.,x0)∈M0γ, as x(.,x0) is continuous with probability 1 and hence bounded in probability on the interval [0,T].
(3)⇐). This proof is similar to the one for case (1), where we only need to replace supt≥0P{ω∈Ω:X(t)>K(δ)} and supt≥0P{ω∈Ω:|x(t,x0)|>ϵ} by P{ω∈Ω:supt≥0X(t)>K(δ)} and P{ω∈Ω:supt≥0|x(t,x0)|>ϵ}, respectively.
⇒ ) . Assume that there exists x0∈Rn such that x(.,x0) does not belong to M^0. Hence there is δ0>0 such that for any natural number ν>0(10)Pω∈Ω:supt≥0⁡xt,x0>ν>δ0.Then the solutions xν(t)=x(t,x0ν-1) satisfy P{ω∈Ω:supt≥0|xν(t)|>1}>δ0, although the sequence xnu(0)=x0ν-1 tends to 0. This contradicts the property of stability in probability.
(4)⇐). Let ek, k=1,…,n, be the standard basis in Rn, the latter being equipped with the norm |b|=∑k=1n|bk| (all norms are equivalent). Due to p-stability we have supt≥0E|x(t,ek)|p≤K for some K>0 and any k=1,…,n. Using the associated matrix norm [a1,…,an]=max1≤k≤n|ak|, where ak∈Rn, yields supt≥0EX(t)p≤K. For any ϵ>0 we now define η=(ϵ/K)1/p obtaining(11)supt≥0⁡Ext,x0p≤supt≥0⁡EXtpx0p≤Kηp=ϵfor any |x0|<η, which completes the proof of p-stability.
⇒ ) . Assume, on the contrary, that for some x0∈Rn the solution x(.,x0) does not belong to the space Mp. Hence for any natural number ν there exists tν such that E|x(tν)|p>νp. As xν(0)=x0ν-1 can be arbitrarily small in the norm, while E|xν(tν)|p>1, this contradicts the property of p-stability.
(5)⇐). As the space Mpγ is imbedded in Mp, the p-stability is trivial. On the other hand, for any x0∈Rn there exists a positive number K such that supt≥0Eγ(t)|x(t,x0)|p≤K. Therefore supt≥0E|x(t,x0)|p≤γ-1(t)→0 as t→∞ by Property γ1.
⇒ ) . Using the same norm as in part (5), we obtain EX(t)p→0 as t→∞, so that supt≥TEX(t)p<1 for some T>0. Choosing γ¯(t)=E(X(t)p∧1) and putting γ(t)=1/γ¯(t) (which satisfies Property γ1 with σ=1), we obtain (12)supt≥T⁡Eγtxt,x0p≤supt≥T⁡EγtXtpx0p=x0p.As sup0≤t≤TE(γ(t)|x(t,x0)|p) is bounded due to continuity of the solutions, we deduce that supt≥TE(γ(t)|x(t,x0)|p)<∞, so that x(.,x0)∈Mpγ for any x0∈Rn.
(6)⇐). Assume that γ(t)=exp⁡{βt}. Using the norm from (5), we conclude that supt≥0γ(t)E|x(t,ek)|p≤K for all standard basis vectors ek. This yields supt≥0γ(t)EX(t)p≤K and thus (13)Ext,x0p≤EXtpx0p≤Kx0pexp-βt.⇒). If E|x(t,x0)|p≤K|x0|pexp{-βt} for all t>0 and x0∈Rn, then supt≥0γ(t)E|x(t,x0)|p≤K|x0|p is finite, so that x(.,x0)∈Mpγ for any x0∈Rn.
(7)⇐). We use again the norm from (5). According to the assumptions, for any δ>0 there is K=K(δ)>0, for which P{ω∈Ω:supt≥0|x(t,ek)|>K}<δ/n for all standard basis vectors ek. Let Ωk(δ) contain all ω∈Ω where supt≥0|x(t,ek)|≤K and Ω(δ)=⋂1≤k≤nΩk(δ). Evidently, PΩ(δ)≥1-δ. As X(t)=xt,ek for some ek (dependent on t and ω), we have that supt≥0X(t)≤K(δ) for all ω∈Ω(δ). This means that supt≥0X(t) is almost surely finite.
Taking now an arbitrary sequencexν∈Rn going to 0 as ν→∞, we conclude that (14)supt≥0⁡xt,xν≤supt≥0⁡Xtxνtends to 0 almost surely. This completes the proof of this part.
⇒ ) . Given x0∈Rn, the stability with probability 1 implies that the sequence supt≥0|x(t,x0n-1) converges to zero almost surely as n→∞. Thus, this sequence converges in probability, and therefore, for any δ>0 there exists a number N=N(δ) such that P{ω∈Ω:supt≥0|x(t,x0/n)≥1}<δ for all n≥N. Choosing K(δ)=N(δ) we see that P{ω∈Ω:supt≥0|x(t,x0)≥K}<δ, so that x(.,x0)∈M^0.
(8)⇐). Utilizing the norm from part (5) and the definition of the space M∞, we can write that, for some sufficiently large K, the estimates supt≥0|x(t,ek)|≤K} hold for all standard basis vectors ek, k=1,…,n, and all ω∈Ωk, where PΩk=1. Therefore, on the set Ω^=⋂1≤k≤nΩk of full measure we have (15)supt≥0⁡Xt≤supt≥0⁡max1≤k≤n⁡xt,ek=max1≤k≤n⁡supt≥0⁡xt,ek≤K.Hence, given δ>0 we set η=δ/K and obtain (16)Pω:ω∈Ω,supt≥0⁡xt,x0≤Pω:ω∈Ω,supt≥0⁡Xtx0≤δas long as x0∈Rn,|x0|<η. By this, we verified strong stability with probability 1.
⇒ ) . Assume that there exists a solution x(.,x0) that does not belong to the space M∞. This means that there exists δ0>0 such that for any natural number ν  P{ω:ω∈Ω,supt≥0|x(t,x0)|≤ν}≤1-ϵ0. Defining xν(t)=x(t,x0ν-1) we see that P{ω:ω∈Ω,supt≥0|xnu(t)|≤1}≤1-ϵ0, although xν(0)=x0ν-1 goes to 0 as ν→∞. This contradicts the property of strong stability with probability 1.Remark 6.
Note that theM-stabilities listed in Theorem 5 imply the respective stability properties not only w.r.t. the initial values x0∈Rn, but also w.r.t. the random initial values, provided that the increments of the semimartingale Z are independent of the initial σ-algebra F0. This is, e.g., the case if the family (Ft)t≥0 is generated by the Brownian motion. This comment applies to all results proven below.Lemma 7.
Letp,q∈[0,∞] and γ,κ be continuous functions defined on [0,∞) and satisfying Property γ1. Then the following inclusions are topological embedding: (1)
Mpγ⊂Mqγ and M^pγ⊂M^qγ if p≥q;(2)
Mpγ⊂Mpκ and M^pγ⊂M^pκ if γ(t)≥κ(t) for t≥t0 (t0≥0 is some real number);(3)
M^pγ⊂Mpγ.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 spacesMpγ and M^pγ and the inequality supt≥0E|y(t)|≤Esupt≥0|y(t)|, which holds for any measurable stochastic process y(t).Theorem 8.
The following is valid for the stochastic functional differential equation (2): (1)
if0<q<p<∞, then p-stability (resp. asymptotic, exponential p-stability) implies q-stability (resp., asymptotic, exponential q-stability);(2)
if0<p<∞, then p-stability (resp., asymptotic p-stability) implies weak stability in probability (resp., weak asymptotic stability in probability);(3)
stability in probability (resp., asymptotic stability in probability) implies weak stability with probability 1 (resp., weak asymptotic stability with probability 1).(4)
stability in probability is equivalent to stability with probability 1.Proof.
The first and the second part follow directly from the first statement of Lemma7 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 M0γ, 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.Theorem 9.
Letp∈[0,∞]. Then(1)
M^p-stability implies stability with probability 1.(2)
M^pγ-stability with γ satisfying Property γ1 implies asymptotic stability with probability 1.(3)
M^∞γ-stability with γ satisfying Property γ1 implies strong asymptotic stability with probability 1.Proof.
(1). This follows from statement (7) of Theorem5 and the inclusion M^p⊂M^0.
(2). Due to the first statement of the theorem and the inclusionM^pγ⊂M^0γ 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 γ(t) to x(t,x0) and X(t), which gives almost everywhere boundedness of the random function ξ=ξ(ω)=supt≥0γ(t)X(t). Therefore (17)γtxt,x0≤supt≥0⁡γtXtx0=ξx0,so that (18)xt,x0≤ξx0γt⟶0almost  everywhere  ast⟶∞.
(3). AsM^∞γ⊂M^∞, we readily obtain strong asymptotic stability with probability 1 from the last statement of Theorem 5. Adding γ(t) to x(t,x0) and X(t) and following the proof of this statement yield the estimate supt≥0γ(t)X(t)≤K for some (nonrandom) constant K. Hence (19)γtxt,x0≤supt≥0⁡γtXtx0≤Kx0,and xt,x0≤K|x0|/γ(t)→0 with probability 1 as t→∞.


## 4. 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 ofp-stability, where we use the integral form of the Marcinkiewicz-Zygmund inequality(20)E∫0tζsdBs2p≤ρp2pE∫0tζs2dsp,which holds true for any predictable stochastic process ζ(s) (0≤s≤t), any t>0, and the scalar standard Brownian motion B(s). The constant ρp depends on the number p (0<p<∞), only. In 1988 D.L. Burkholder proved (see, for example, [21, 22]) that the constant ρp=2p-1 in the Marcinkiewicz-Zygmund inequality (20) is best possible for all for p≥1.Consider the following equation:(21)dxt=Axt+∑j=0m0Aj0xhj0tdt+∑l=1m-1∑j=0mlAjlxhjltdBltt≥0.For all 0≤l≤m-1, 0≤j≤ml, the n×n-matrices A,Ajl contain constant, real entries, the standard Wiener processes Bl(t) are independent, and the functions hjl are Borel measurable functions satisfying 0≤hjl(t)≤t on [0,∞).In the next theorem we putAl=∑j=0mlAjl for all l=0,…,m-1, where . is the matrix norm induced by the chosen vector norm ..Theorem 10.
Letp satisfy the inequality 1≤p<∞. Assume that there exists a positive number α such that(22)A+αIn+A0+ρpα2∑l=1m-1Al<α,where In is the n×n identity matrix. Then (21) is 2p-stable.Proof.
We apply the forth statement from Theorem5. To show that the solutions x(t) of (21) belong to the space M2p for all x(0)∈Rn, we replace this equation by an equivalent integral equation(23)xt=e-αtx0+∫0te-αt-sA+αInxs+∑j=0m0Aj0xhj0sds+∑l=1m-1∑j=0ml∫0te-αt-sAjlxhjlsdBlst≥0.Using the Marcinkiewicz-Zygmund inequality (20) we obtain (24)supt≥0⁡Ext2p1/2p≤x0+supt≥0⁡E∫0te-αt-sA+αInxsds2p1/2p+∑j=0m0supt≥0⁡E∫0te-αt-sAj0xhj0sds2p1/2p+∑l=1m-1∑j=0mlsupt≥0⁡E∫0te-αt-sAjlxhjlsdBls2p1/2p≤x0+α-1A+αInsupt≥0⁡Ext2p1/2p+α-1∑j=0j=m0Aj0supt≥0⁡Exhj0t2p1/2p+ρp∑l=1m-1∑j=0j=mlsupt≥0⁡E∫0te-2αt-sAjl2xhjls2dsp1/2p≤x0+Ksupt≥0⁡Ext2p1/2p,where K=α-1A+αIn+α-1A0+ρp(2α)-1/2∑l=1m-1Al. In the estimates above we used the assumptions 0≤hjl(t)≤t for all t≥0, 0≤j≤ml, 0≤l≤m-1. Due to (22), the constant K is less than 1, so that (25)supt≥0⁡Ext2p1/2p≤1-K-1x0.Therefore x∈M2p for all x(0)∈Rn.Consider the generalized scalar pantograph equation(26)dx=axt+bxhtdt+cxt+dxgtdBtt≥0,where a,b,c,d are real constants, h(t) and g(t) (t≥0) are Borel measurable functions satisfying 0≤h(t)≤t, 0≤g(t)≤t (t≥0), and B(t) is the standard scalar Brownian motion.Corollary 11.
Letξ0 be the (only) positive root of the quadratic polynomial P(ξ)=ξ2-Kξ-|b|, where K=ρp(c+d/2). If a<-ξ02, then (26) is 2p-stable.Proof.
Inequality (22) reads in this case as follows: (27)a+α+b+Kα<α.Assuming that a<-ξ02 and putting α=ξ02>-a>0 we observe that (28)α-a+α-b-Kα=ξ2-b-Kξ>0,as ξ>-a>ξ0.Let us compare Corollary11 with the following well-known result (see [1] or [2]): (26) with the linear delays h(t)=g(t)=kt (0<k<1) is 2-stable if 2a+2b+cd+c2+d2<0.In the case whenb=0, this result and Corollary 11 give the same condition: a<-(|c|+|d|)2/2.Ifc=0, then the result from [1] gives, in fact, a better stability condition in the case of linear delays: a<-d2/2-|b| versus a<-d2/4-|b|-(d/2)-d2/8+|b|, but our result covers more general delays and the case of p-stability for p≠2.Now we briefly describe explicit stability conditions for one particular case of the vector equation (21).Corollary 12.
Let the matrixA have only real eigenvalues λi(1≤i≤n) and Ajl be scalar matrices: (29)Ajl=diagajl,…,ajl0≤l≤m-1;1≤j≤ml.Then (21) is 2p-stable if λi<-ξ02(1≤i≤n), where ξ0 is the only positive root of the quadratic polynomial (30)Piξ=ξ2-Kξ-a0and (31)K=ρp2∑l=1m-1∑j=0mlajl.Proof.
Let us chooseα=ξ02. To verify condition (22) of Theorem 10 we first observe that for any positive ϵ>0 there exists a basis in Rn, in which the Euclidean matrix norm . satisfies A+αIn<|λ+α|+ϵ, where λ is the largest eigenvalue of the matrix A. At the same time, the Euclidean norms of the scalar matrices Ajl (0≤l≤m-1;1≤j≤ml) are independent of the choice of the basis. Denoting (32)Al=∑j=0mlajlfor all l=0,…,m-1 we obtain, exactly in the same manner as in the proof of Corollary 11, that (33)A+αIn+A0+ρpα2∑l=1m-1Al<λ+α+ϵ+A0+Kα<αfor sufficiently small ϵ.In the next theorem, where we study stability with probability 1, we use the Doob inequality(34)Esupt≥0⁡∫0tζsdBs2≤4Esupt≥0⁡∫0tζs2dswhere B is the scalar standard Brownian motion and ζ is an arbitrary predictable stochastic process on [0,∞).Theorem 13.
Assume that there exists a positive numberα such that(35)A+αIn+A0+2α∑l=1m-1Al<α,where In is the n×n identity matrix. Then (21) is stable with probability 1.Proof.
We want to apply the first statement from Theorem9 by checking that the solutions x(t) of (21) belong to the space M^2 for all x(0)∈Rn. This is done exactly in the same way as in the proof of Theorem 10 provided that the norm supt≥0E|x(t)|2p1/2p is replaced by the norm Esupt≥0|x(t)|2p1/2p and the Doob inequality is used instead of the Marcinkiewicz-Zygmund inequality.


## 5. 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-to-state 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 studied2p-stability (p≥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, Theorem5 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 Theorems10 and 13 to get more general stability tests than those offered in Section 4.(3)
Asymptoticp-stability and asymptotic stability with probability 1 are only described in terms of M-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 Section3.(5)
The main results of Section3 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, e.g., directly applied to integrodifferential equations, where (36)Vxt=∫0,tdsR1t,sxs,…,∫0,tdsRmt,sxs,Ri are vector functions defined on {(t,s):t∈[0,∞),0<s≤t} for i=1,…,m. But this analysis is beyond the scope of the present paper.(6)
Stochastic hereditary equations with more general initial conditionsx(s)=φ(s), s≤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 φ to the associated solution x(t).



---

*Source: 7490936-2018-06-19.xml*