Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes

Abstract

This paper concerns Razumikhin-type theorems on exponential stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain involves small parameters. The smaller the parameter is, the rapider switching the system will experience. In order to reduce the complexity, we will “replace” the original systems by limit systems with a simple structure. Under Razumikhin-type conditions, we establish theorems that if the limit systems are 𝑝th-moment exponentially stable; then, the original systems are 𝑝th-moment exponentially stable in an appropriate sense.

1. Introduction

The stability of time delay systems is a field of intense research [1, 2]. In [2], the global uniform exponential stability independent of time delay linear and time invariant systems subjected to point and distributed delays was studied. Moreover, noise and time delay are often the sources of instability, and they may destabilize the systems if they exceed their limits [3].

Hybrid delay systems driven by continuous-time Markov chains have been used to model many practical systems in which abrupt changes may be experienced in the structure and parameters caused by phenomena such as component failures or repairs. An area of particular interest has been the automatic control of the underlying systems, with consequent emphasis on the analysis of stability of the stochastic models. For systems with time delay, there are two approaches to proving stability that correspond to the conventional Lyapunov stability theory. The first is based on Lyapunov-Krasovski functionals, the second on Lyapunov-Razumikhin functions. The latter one originated with Razumikhin [4] for the ordinary differential delay equation which is called Razumikhin-type theorem and was developed by several people [5]. In his paper, Mao [6] was the first who established a Razumikhin-type theorem for stochastic functional differential equations (SFDEs). Roughly speaking, a Razumikhin-type theorem states that if the derivative of a Lyapunov function along trajectories is negative whenever the current value of the function dominates other values over the interval of time delay; then, the Lyapunov function along trajectories will converge to zero. The Razumikhin methods have been widely used in the study of stability for functional and differential-delay systems. In this work, we shall investigate stochastic differential delay equations with Markovian switching (SDDEwMSs). The switching we shall use will be a finite-state Markov chain, which incorporates various considerations into the models and often results in the underlying Markov chain having a large state space. To overcome the difficulties and to reduce the computational complexity, much effort has been devoted to the modeling and analysis of such systems, in which one of the main ideas is to split a large-scale system into several classes and lumping the states in each class into one state; see [7–9]. Starting from the work [10], by introducing a small parameter 𝜀>0, a number of asymptotic properties of the Markov chain 𝑟^𝜀(⋅) have been established. One of the main results in [9] is that a complicated system can be replaced by the corresponding limit system having a much simpler structure. In [11, 12], long-term behavior of SDEwMSs and SDDEwMSs was investigated, respectively, while in [13, 14] the stability of random delay system with two-time-scale Markovian switching was studied. Using the stability of the limit system as a bridge, the desired asymptotic properties of the original system is obtained using perturbed Lyapunov function methods. In this work, we shall establish a Razumikhin-type theorem for SDDEwMSs.

The remainder of this work is organised as follows: in the next section, we shall begin with the formulation of the problem. Section 3 investigates the Razumikhin-type theorem for SDDEs driven by Brownian motion. The exponential stability for SDDEs driven by pure jumps is discussed in Section 4.

2. Preliminaries

Let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e. it is increasing and right continuous, and ℱ₀ contains all ℙ-null sets). Throughout the paper, we let 𝐵(𝑡)=(𝐵₁(𝑡),…,𝐵_𝑚(𝑡))^𝑇 be an 𝑚-dimensional Brownian motion defined on the probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,ℙ). If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. Let |⋅| denote the Euclidean norm in ℝ^𝑛 as well as the trace norm of a matrix. For 𝜏>0, 𝐶([−𝜏,0];ℝ^𝑛) denotes the family of continuous functions from [−𝜏,0] to ℝ^𝑛 with the norm ‖𝜑‖=sup_{−𝜏≤𝜃≤0}|𝜑(𝜃)|. Denote by 𝐶^𝑏_ℱ([−𝜏,0];ℝ^𝑛) the family of all ℱ measurable and bounded 𝐶([−𝜏,0];ℝ^𝑛)-valued random variable. We will denote the indicator function of a set 𝐺 by 𝐼_𝐺.

Let 𝑟(𝑡)(𝑡≥0) be a right-continuous Markov chain on (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,ℙ) taking values in a finite state space 𝕊={1,2,…,𝑁} with the generator Γ=(𝛾_𝑖𝑗)_𝑁×𝑁 given by ℙ𝑟()(𝑡)=𝛾𝑡+𝛿=𝑗∣𝑟=𝑖_𝑖𝑗𝛿+∘(𝛿),if𝑖≠𝑗,1+𝛾_𝑖𝑖𝛿+∘(𝛿),if𝑖=𝑗, (1) where 𝛿>0 and 𝛾_𝑖𝑗 is the transition rate from 𝑖 to 𝑗 satisfying 𝛾_𝑖𝑗>0 if 𝑖≠𝑗 and 𝛾_𝑖𝑖=−∑_𝑖≠𝑗𝛾_𝑖𝑗. We assume the Markov 𝑟(⋅) is independent of the Brownian motion 𝐵(⋅). It is well known that almost every sample path 𝑟(⋅) is a right-continuous step function with finite number of simple jumps in any finite subinterval of ℝ₊∶=[0,∞). As a standing hypothesis, we assume that the Markov chain is irreducible. This is equivalent to the condition that for any 𝑖,𝑗∈𝕊, we can find 𝑖₁,𝑖₂,…,𝑖_𝑘∈𝕊 such that 𝛾_𝑖,𝑖1𝛾_𝑖1,𝑖2…𝛾_{𝑖𝑘,𝑗}>0. (2) Thus, Γ always has an eigenvalue 0. The algebraic interpretation of irreducibility is rank(Γ)=𝑁−1. Under this condition, the Markov chain has a unique stationary (probability) distribution 𝜋Γ=0, subject to ∑^𝑁_𝑗=1𝜋_𝑗=1 and 𝜋_𝑗>0 for all 𝑗∈𝕊. For a real valued function 𝜎(⋅) defined on 𝕊, we define Γ𝜎(⋅)(𝜅)∶=_ℓ∈𝕊𝛾_𝜅ℓ𝜎(ℓ)=_ℓ≠𝜅𝛾_𝜅ℓ(𝜎(ℓ)−𝜎(𝜅)), (3) for each 𝜅∈𝕊.

Consider the following stochastic delay system with Markovian swtching: (𝑡)(𝑥(𝑡)()(𝑡(𝑥(𝑡)()(𝑡(𝑡),𝑥𝑑𝑥=𝑓,𝑥𝑡−𝜏,𝑟))𝑑𝑡+𝑔,𝑥𝑡−𝜏,𝑟))𝑑𝐵₀=𝜉∈𝐶[−𝜏,0];ℝ^𝑛,𝑟(0)∈𝕊, (4) where 𝑓∶ℝ^𝑛×ℝ^𝑛×𝕊→ℝ^𝑛 and 𝑔∶ℝ^𝑛×ℝ^𝑛×𝕊→ℝ^𝑛×𝑚.

To highlight the fast and slow motions, we introduce a parameter 𝜀>0 and rewrite the Markov chain 𝑟(𝑡) as 𝑟^𝜀(𝑡) and the generator Γ as Γ^𝜀. Γ^𝜀 is given by Γ^𝜀=1𝜀Γ+Γ, (5) where Γ/𝜀 represents the fast varying motions, and Γ represents the slowly changing dynamics. We denote Γ^𝜀=(𝛾^𝜀_𝑖𝑗)_𝑁×𝑁, Γ=(𝛾_𝑖𝑗)_𝑁×𝑁, and Γ=(𝛾_𝑖𝑗)_𝑁×𝑁. To the reduction of complexity, Γ needs to have a certain structure. Suppose that 𝕊=𝕊¹∪𝕊²∪⋯∪𝕊^𝑙 (6) with 𝕊^𝑖={𝑠_𝑖1,…,𝑠_𝑖𝑁𝑖} and 𝑁=𝑁₁+𝑁₂+⋯+𝑁_𝑙, and that Γ=diagΓ¹,…,Γ^𝑙, (7) where for each 𝑘∈{1,…,𝑙} and Γ^𝑘 is a generator of a Markov chain taking values in 𝕊^𝑘. We impose the following hypothesis:

(H1) For each 𝑘∈{1,…,𝑙}, Γ^𝑘 is irreducible.

To highlight the effect of the fast switching, we rewrite the system (4) as 𝑑𝑥^𝜀(𝑡)=𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡𝑥)𝑑𝑡+𝑔^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)(𝑡),𝑥𝑑𝑤_𝜀0=𝜉∈𝐶[−𝜏,0];ℝ^𝑛,𝑟^𝜀=𝑟₀. (8) To assure the existence and uniqueness of the solution, we give the following standard assumptions.

(H2) For any integer 𝑅, there is a constant ℎ_𝑅>0, such that 𝑓𝑥,𝑦,𝜅−𝑓𝑥₁,𝑦₁,𝜅∨𝑔𝑥,𝑦,𝜅−𝑔𝑥₁,𝑦₁,𝜅≤ℎ_𝑅𝑥−𝑥₁+𝑦−𝑦₁ (9)for all 𝜅∈𝕊 and those 𝑥,𝑥₁,𝑦,𝑦₁∈ℝ^𝑛 with |𝑥|∨|𝑥₁|∨|𝑦|∨|𝑦₁|≤𝑅.

(H3) There is an ℎ>0, such that for any 𝑥,𝑦∈ℝ^𝑛,𝜅∈𝕊, 𝑓∨𝑔|𝑥|+𝑦,𝑓()()𝑥,𝑦,𝜅𝑥,𝑦,𝜅≤ℎ1+0,0,𝜅≡0,𝑔0,0,𝜅≡0. (10)

Under the assumptions (H2) and (H3), system (8) has a unique solution denoted by 𝑥^{𝜀,𝜉,ℓ}(𝑡) on 𝑡≥−𝜏, where the notation 𝑥^{𝜀,𝜉,ℓ} emphasizes the dependence on the initial data (𝜉,ℓ). Moreover, for every 𝑝>0 and any compact subset 𝐾 of 𝐶([−𝜏,0];ℝ^𝑛), there exists a positive constant 𝐻 which is independent of 𝜀 such that sup_{()𝜉,ℓ∈𝐾×𝕊}𝐸sup_{−𝜏≤𝑠≤𝑡}𝑥^{𝜀,𝜉,ℓ}(𝑠)^𝑝≤𝐻,on𝑡≥0. (11)

We will consider the stability of system (8), but the state space of the Markov chain is large, and it is difficult to handle (8) directly. So we will consider the average system of (8). To proceed, lump the states in each 𝕊^𝑘 into a single state and define an aggregated process 𝑟^𝜀(⋅) as 𝑟^𝜀(𝑡)=𝑘,if𝑟^𝜀(𝑡)∈𝕊^𝑘. (12) Denote the state space of  𝑟^𝜀(𝑡) by 𝕊={1,…,𝑙}, the stationary distribution Γ^𝑘 by 𝜇^𝑘=(𝜇^𝑘₁,…,𝜇^𝑘_𝑁𝑘)∈ℝ^1×𝑁𝑘 and 𝜇=diag(𝜇¹,…,𝜇^𝑙)∈ℝ^𝑙×𝑁. Define Γ=𝛾_{𝑖𝑗𝑙×𝑙}=𝜇Γ𝟏 (13) with 𝟏=diag(𝟏_𝑁1,…,𝟏_𝑁𝑙) and 𝟏_𝑁𝑘=(1,…,1)^𝑇∈ℝ^𝑁𝑘×1,𝑘=1,…,𝑙. It has been known that 𝑟^𝜀(⋅) converges weakly to 𝑟(⋅) as 𝜀→0, where 𝑟(⋅) is a continuous-time Markov chain with generator Γ and state space 𝕊 (cf. [9]).

Define 𝑓𝑥,𝑦,𝑖=^𝑁𝑖_𝑗=1𝜇^𝑖_𝑗𝑓𝑥,𝑦,𝑠_𝑖𝑗,𝑔𝑥,𝑦,𝑖𝑔^𝑇𝑥,𝑦,𝑖=^𝑁𝑖_𝑗=1𝜇^𝑖_𝑗𝑔𝑥,𝑦,𝑠_𝑖𝑗𝑔^𝑇𝑥,𝑦,𝑠_𝑖𝑗 (14)(15) for each 𝑠_𝑖𝑗∈𝕊^𝑖 with 𝑖∈{1,…,𝑙} and 𝑗∈{1,…,𝑁_𝑖}. It is easily seen that 𝑓(𝑥,𝑦,𝑖) and 𝑔(𝑥,𝑦,𝑖) are the averages with respect to the stationary distribution of the Markov chain. Note that for any (𝑥,𝑦)≠(0,0),𝑔(𝑥,𝑦,𝑠_𝑖𝑗)𝑔^𝑇(𝑥,𝑦,𝑠_𝑖𝑗) are nonnegative definite matrices, so we find its “square root” of (15), which is denoted by 𝑔(𝑥,𝑦,𝑖). For degenerate diffusions, we can see the argument in [15].

The averaged system of (8) is defined as follows: 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑥(𝑡−𝜏),𝑟(𝑡+))𝑑𝑡𝑔(𝑥(𝑡),𝑥(𝑡−𝜏),𝑟(𝑡))𝑑𝑤(𝑡),𝑥₀=𝜉,𝑟=𝑟₀. (16)

3. Moment Exponential Stability

In this section, we shall establish the Razumikhin-type theorem on the exponential stability for (8).

Let 𝐶^𝑝(ℝ^𝑛×𝕊;ℝ₊) be the class of nonnegative real-valued functions defined on ℝ^𝑛×𝕊 that are 𝑝-times continuously differentiable with respect to 𝑥. We give the following assumption about 𝑉(𝑥,𝑖)∈𝐶^𝑝(ℝ^𝑛×𝕊;ℝ₊) for some 𝑝≥4.

(H4) For each 𝑖∈𝕊, 𝑉(𝑥,𝑖)→∞ as |𝑥|→∞. Moreover, 𝜕^𝑝𝑉(𝑥,𝑖)=𝑂(1),𝜕^ℓ𝑉(𝑥,𝑖)(|𝑥|^ℓ+|𝑦|^ℓ)≤𝐾(|𝑥|^𝑝+|𝑦|^𝑝+1) for 1≤ℓ≤𝑝−1, where 𝜕^ℓ𝑉(𝑥,𝑖) denotes the ℓth derivative of 𝑉(𝑥,𝑖) with respect to 𝑥 and 𝑂(𝑦) denotes the function of 𝑦 satisfying sup_𝑦|𝑂(𝑦)|/𝑦<∞.

Theorem 1. Let (H1)–(H3) hold; there is a function 𝑉(𝑥,𝑖)∈𝐶^𝑝(ℝ^𝑛×𝕊;ℝ₊) satisfying (H4), and there are positive constants 𝜆,𝑐₁,𝑐₂, and 𝑞>1 such that(i)𝑐₁|𝑥|^𝑝≤𝑉(𝑥,𝑖)≤𝑐₂|𝑥|^𝑝, (ii)𝔼[max_𝑖∈𝕊ℒ𝑉(𝑥(𝑡),𝑥(𝑡−𝜏),𝑖)]≤−𝜆𝔼[max_𝑖∈𝕊𝑉(𝑥(𝑡),𝑖)] provided 𝔼[min_𝑖∈𝕊𝑉(𝑥(𝑡+𝜃),𝑖)]<𝑞𝔼[max_𝑖∈𝕊𝑉(𝑥(𝑡),𝑖)],−𝜏≤𝜃≤0,
where ℒ𝑉𝑥,𝑦,𝑖=𝑉_𝑥(𝑥,𝑖)𝑓+1𝑥,𝑦,𝑖2trace𝑉_𝑥𝑥(𝑥,𝑖)𝑔𝑥,𝑦,𝑖𝑔^𝑇+𝑥,𝑦,𝑖_𝑙𝑗=1𝛾_𝑖𝑗𝑉𝑥,𝑗. (17) Then, for all 𝜉∈𝐶([−𝜏,0];ℝ^𝑛), limsup_𝜀→0𝔼𝑥^𝜀(𝑡)^𝑝≤𝜈₂𝑒^−𝜈1𝑡, (18) where 𝜈₁=min𝜆,log𝑞𝜏;𝜈₂𝜈𝑖𝑠𝑎𝑓𝑖𝑥𝑒𝑑𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡₂=𝑐₂𝑐₁sup_{−𝜏≤𝜃≤0}𝔼𝜉^𝑝. (19)

Remark 2. Note that the conditions of Theorem 1 are sufficient conditions for the average system (16) 𝑥(𝑡) (or the limit process 𝑥(𝑡)). However the conclusion of Theorem 1 is about the process 𝑥^𝜀(𝑡). Since the structure of the the average system (16) is much simpler than that of 𝑥^𝜀(𝑡), this theorem has reduced the computational complexity for the system (8).

Remark 3. limsup_𝜀→0𝔼|𝑥^𝜀(𝑡)|^𝑝 does exist by (11).

Proof of Theorem 1. Define 𝑉(𝑥,𝜁)=^𝑙_𝑖=1𝑉(𝑥,𝑖)𝐼_{𝜁∈𝕊^𝑖_}=𝑉(𝑥,𝑖),if𝜁∈𝕊^𝑖. (20) Note that 𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)=𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡),_𝑁𝜅=1𝛾_𝑙𝜅𝑉(𝑥,𝜅)=^𝑁_𝜅=1𝛾^𝑙_{𝑙𝜅𝑖=1}𝑉(𝑥,𝑖)𝐼_{𝜅∈𝕊^𝑖_}=0. (21) We extend 𝑟(𝑡) to [−𝜏,0] by setting 𝑟(𝑡)=𝑟(0); then, 𝔼𝑉(𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)) is right continuous on 𝑡≥−𝜏.
Let 𝜈∈(0,𝜈₁) be arbitrary, and define 𝑈(𝑡)∶=𝑒^𝜈𝑡limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)=𝑒^𝜈𝑡limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡). (22) If we can show that 𝑈(𝑡)≤𝑐₁𝜈₂, then the proof is completed.
If 𝑡∈[−𝜏,0], by condition (i), 𝑈(𝑡)≤lim_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)=𝔼𝑉(𝜉,0)≤𝑐₂𝔼𝜉(0)^𝑝≤𝑐₂sup_{−𝜏≤𝜃≤0}𝔼𝜉(𝜃)^𝑝=𝑐₁𝜈₂. (23)
If 𝑡≥0, we will prove that 𝑈(𝑡)≤𝑐₁𝜈₂. Otherwise, there exists the smallest 𝜌∈(0,∞) such that all 𝑡∈[−𝜏,𝜌),𝑈(𝑡)≤𝑐₁𝜈₂ and 𝑈(𝜌)≥𝑐₁𝜈₂ as well as 𝑈(𝜌+𝛿)>𝑈(𝜌) for all suffieciently small 𝛿.
For 𝑡∈[𝜌−𝜏,𝜌), limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)=𝑒^−𝜈𝑡𝑈(𝑡)≤𝑒^−𝜈𝑡𝑈𝜌=𝑒^−𝜈𝑡𝑒^𝜈𝜌limsup_𝜀→0𝔼𝑉𝑥^𝜀𝜌,𝑟^𝜀𝜌≤𝑒^𝜈𝜏limsup_𝜀→0𝔼𝑉𝑥^𝜀𝜌,𝑟^𝜀𝜌. (24) If limsup_𝜀→0𝔼𝑉(𝑥^𝜀(𝜌),𝑟^𝜀(𝜌))=0, then limsup_𝜀→0𝔼𝑉(𝑥^𝜀(𝑡),𝑟^𝜀(𝑡))=0, 𝑡∈[𝜌−𝜏,𝜌).
Since (𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)) converges to (𝑥(𝑡),𝑟(𝑡)) with probability one (see Lemma  2.3 in [12]), by condition (i), we can derive 𝑥(𝑡)=0,𝑡∈𝜌−𝜏,𝜌. (25) Recalling the fact 𝑓(0,0,𝑖)≡0, 𝑔(0,0,𝑖)≡0 and using the uniqueness of the equation, we then have 𝑥(𝑡)=0, a.e. 𝑡>0. Therefore we have limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)=0,𝑡>0. (26) Then 𝑈(𝜌)=0, which is a contradiction. Hence we see that lim_𝜀→0𝔼𝑉(𝑥^𝜀(𝜌),𝑟^𝜀(𝜌))≠0. For 𝑡∈[𝜌−𝜏,𝜌), there exists a 𝑞>1 such that limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)≤𝑞limsup_𝜀→0𝔼𝑉𝑥^𝜀𝜌,𝑟^𝜀𝜌,𝜈<log𝑞𝜏. (27) Consequently, there exists a sufficiently small 𝜀₀>0, such that, for any 𝜀∈(0,𝜀₀), 𝔼min_𝑖∈𝕊𝑉𝑥^𝜀𝜌+𝜃,𝑖≤𝑞𝔼max_𝑖∈𝕊𝑉𝑥^𝜀𝜌,𝑖,𝜃∈[−𝜏,0]. (28) By condition (ii), 𝔼max_𝑖∈𝕊ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑖≤−𝜆𝔼max_𝑖∈𝕊𝑉𝑥^𝜀(𝑡),𝑖; (29) then, 𝔼ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)≤−𝜆𝔼𝑉𝑥^𝜀(𝑡),𝑟(𝑡). (30) Noting that 𝜈<𝜈≤𝜆, we have 𝔼ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)≤−𝜈𝔼𝑉𝑥^𝜀(𝑡),𝑟(𝑡). (31) We now consider 𝑈𝜌+𝛿−𝑈𝜌=limsup_𝜀→0𝑒^{𝜈(𝜌+𝛿)}𝔼𝑉𝑥^𝜀𝜌+𝛿,𝑟^𝜀𝜌+𝛿−𝑒^𝜈𝜌𝔼𝑉𝑥^𝜀𝜌,𝑟^𝜀𝜌=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡+)𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡. (32) By the definition of operator ℒ, we have ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)=𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+12trace𝑉_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡𝑥)×𝑔^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑁_𝜅=1𝛾^𝜀_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡)=,𝜅𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+12trace𝑉_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑁_𝜅=1𝛾_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡),𝜅=𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+12trace𝑉_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑙_𝑗=1𝛾^𝜀_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗+𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡+1)2trace𝑉_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑁_𝜅=1𝛾_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡)−,𝜅_𝑙𝑗=1𝛾_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡)𝑥,𝑗=ℒ𝑉^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑓𝑥^𝜀(𝑡),𝑥^𝜀((𝑡,𝑟𝑡−𝜏))_𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏(𝑡)),𝑟^𝜀(𝑡)+12trace𝑉_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑁_𝜅=1𝛾_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡)−,𝜅_𝑙𝑗=1𝛾_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗. (33)
So 𝑈𝜌+𝛿−𝑈𝜌=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡+limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+1𝑑𝑡2limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡×𝑉trace_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡−)𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏(𝑡)),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)𝑑𝑡+limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑁_𝜅=1𝛾^𝜀_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡)−,𝜅_𝑙𝑗=1𝛾^𝜀_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗𝑑𝑡=∶𝐼₁+𝐼₂+𝐼₃+𝐼₄. (34) By the definition of 𝑓, 𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡=_{𝑙𝑁𝑖=1𝑖𝑗=1})𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑠_𝑖𝑗×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}. (35) This, together with assumption (H2), implies lim_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)𝑑𝑡≤lim_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡×𝑓𝑥_𝜀)(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)𝑑𝑡^21/2=lim_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×^𝑙𝑁_𝑖=1^𝑖_𝑗=1𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑠_𝑖𝑗×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}𝑑𝑡^21/2≤lim_𝜀→0𝔼^𝜌+𝛿_𝜌^𝑙𝑁_𝑖=1^𝑖_𝑗=1𝑒^𝜈𝑡ℎ1+𝑥^𝜀(𝑡)^𝑝+𝑥^𝜀(𝑡−𝜏)^𝑝×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}𝑑𝑡^21/2. (36) By the argument of Lemma  7.14 in [9], the right side of above inequality is equivalent to to 0;thatis,𝐼₂=0. Similarly, we can show 𝐼₃=12limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡trace×𝑉_𝑥𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑔𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)×𝑔^𝑇𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)𝑑𝑡=0. (37) By the definition of Γ and Γ, we have ^𝑁_𝜅=1𝛾_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡),𝜅=Γ𝑉𝑥^𝜀(𝑡),⋅𝑟^𝜀(𝑡),_𝑙𝑗=1𝛾_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗=Γ𝑉𝑥^𝜀(𝑡),⋅𝑟^𝜀(𝑡), (38) hence 𝐼₄=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑁_𝜅=1𝛾^𝜀_{𝑟𝜀(𝑡)𝜅}𝑉𝑥^𝜀(𝑡)−,𝜅_𝑙𝑗=1𝛾^𝜀_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗𝑑𝑡=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡Γ𝑉𝑥^𝜀(𝑡),⋅𝑟^𝜀(𝑡)−Γ𝑉𝑥^𝜀(𝑡),⋅𝑟^𝜀(𝑡)𝑑𝑡=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^{𝑙𝜈𝑡𝑁}_𝑖=1^𝑖_𝑗=1Γ𝑉𝑥^𝜀(𝑡),⋅𝑠_𝑖𝑗×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}𝑑𝑡≤limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^{𝑙𝜈𝑡𝑁}_𝑖=1^𝑖_𝑗=1Γ𝑉𝑥^𝜀(𝑡),⋅𝑠_𝑖𝑗×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}^21/2. (39) By assumption (H4) and the argument of Lemma  7.14 in [9], we have the right side of above inequality is equivalent to 0,thatis,𝐼₄=0.
Therefore by the condition (ii) 𝑈𝜌+𝛿−𝑈𝜌=lim_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡≤0; (40) this is 𝑈𝜌+𝛿≤𝑈𝜌. (41) This contradicts the definition of 𝜌. The proof is now completed.

Example 4. Let 𝑟^𝜀(⋅) be a Markov chain generated by Γ^𝜀 given in (5) with ,.Γ=−220002−200000−303001−100010−1Γ=−101000−101000−101010−101000−1 (42) The generator Γ consists of two irreducible blocks. The stationary distributions are 𝜇¹=(0.5,0.5), 𝜇²=(1/7,2/7,4/7), and 6Γ=−117−67. (43) Consider a one-dimensional equation 𝑑𝑥^𝜀(𝑡)=𝑓𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡+𝑔𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)𝑑𝑤(𝑡) (44) with 𝑓𝑥,𝑠₁₁=𝑥8,𝑓𝑥,𝑠₁₂=𝑥8,𝑔𝑥,𝑠₁₁=𝑥cos𝑥8√2,𝑔𝑥,𝑠₁₂=𝑥sin𝑥8√2,𝑓𝑥,𝑠₂₁(),𝑓=−28𝑥+sin𝑥𝑥,𝑠₂₂=7𝑥+14sin𝑥,𝑓𝑥,𝑠₂₃7=−4𝑔𝑥,𝑥,𝑠₂₁=√74𝑔𝑥sin𝑥,𝑥,𝑠₂₂√=−74𝑥cos𝑥,𝑔𝑥,𝑠₂₃=√78𝑥. (45) Then the limit equation is 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑟(𝑡))𝑑𝑡+𝑔(𝑥(𝑡−𝜏),𝑟(𝑡))𝑑𝑤(𝑡), (46) where 𝑟 is the Markov chain generated by Γ and 𝑓()=𝑥𝑥,18,𝑓(𝑥,2)=−3𝑥,𝑔()=𝑥𝑥,1,16𝑔()=𝑥𝑥,24. (47) Let 𝑉(𝑥,1)=2𝑥²,𝑉(𝑥,2)=𝑥²; then, 1ℒ𝑉𝑥,𝑦,1≤−2|𝑥|²+𝑦²,128ℒ𝑉𝑥,𝑦,2≤−367|𝑥|²+𝑦²,16 (48) Consequently max_𝑖=1,21ℒ𝑉𝑥,𝑦,𝑖≤−2|𝑥|²+1𝑦16²1=−4max_𝑖=1,2𝑉(+1𝑥,𝑖)16min_𝑖=1,2𝑉𝑦,𝑖. (49)(50) It is easy to see that we can find a 𝑞>1 such that (1/4)−(𝑞/16)>0. Therefore, for any 𝜙∈𝐿²_ℱ𝑡([−𝜏,0];ℝ^𝑛) satisfying 𝔼[min_𝑖∈𝕊𝜙(𝜃)]≤𝑞𝔼[max_𝑖∈𝕊𝜙(0)] on −𝜏≤𝜃≤0, (49) yields 𝔼max_𝑖∈𝕊1ℒ𝑉𝑥,𝑦,𝑖≤−4−𝑞𝔼16max_𝑖=1,2𝑉(𝑥,𝑖). (51) Hence, by Theorem 1, the solution 𝑥^𝜀(𝑡) is mean square stable when 𝜀 is sufficient small.

4. Stochastic Delay System with Pure Jumps

In this section we discuss the stability of the following stochastic delay system with pure jumps: 𝑑𝑥^𝜀(𝑡)=𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝑑𝑡_ℝ𝑚𝑏𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡)𝑁(),𝑥,𝑧𝑑𝑡,𝑑𝑧₀=𝜉∈𝐶[−𝜏,0];ℝ^𝑛,𝑟(0)∈𝕊, (52) where 𝑥^𝜀(𝑡−)=lim_𝑠↑𝑡𝑥^𝜀(𝑠),  𝑏∶ℝ^𝑛×ℝ^𝑛×𝕊×ℝ^𝑚→ℝ^𝑛×𝑚. We assume that the each column 𝑏^(𝑙) of the 𝑛×𝑚 matrix 𝑏=[𝑏_𝑖𝑗] depends on 𝑧 only through the 𝑙th coordinate 𝑧_𝑙; that is, 𝑏^(𝑘)𝑥,𝑦,𝜅,𝑧=𝑏^(𝑘)𝑥,𝑦,𝜅,𝑧_𝑘;𝑧=𝑧₁,…,𝑧_𝑚∈ℝ^𝑚,𝜅∈𝕊. (53)𝑁(𝑡,𝑧) is a 𝑚-dimensional Poisson process, and the compensated Poisson, process is defined by 𝑁(𝑑𝑡,𝑑𝑧)=𝑁₁𝑑𝑡,𝑑𝑧₁,…,𝑁_𝑑𝑑𝑡,𝑑𝑧_𝑚=𝑁₁𝑑𝑡,𝑑𝑧₁−𝜆₁𝑑𝑧₁𝑁𝑑𝑡,…,_𝑚𝑑𝑡,𝑑𝑧_𝑚−𝜆_𝑚𝑑𝑧_𝑚𝑑𝑡, (54) where {𝑁_𝑗,𝑗=1,…,𝑚} are independent one-dimensional Poisson random measures with characteristic measure {𝜆_𝑗,𝑗=1,…,𝑚} coming from 𝑚 independent one-dimensional Poisson point processes.

The averaged system of (18) is defined as follows: 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑥(𝑡−𝜏),𝑟(𝑡+))𝑑𝑡_ℝ𝑚𝑏(𝑥(𝑡−),𝑥((𝑡−𝜏)−),𝑟(𝑡))×𝑁(),,𝑧𝑑𝑡,𝑑𝑧𝑥₀=𝜉∈𝐶[−𝜏,0];ℝ^𝑛,𝑟(0)∈𝕊, (55) where 𝑥(𝑡−)=lim_𝑠↑𝑡𝑥(𝑠), 𝑏∶ℝ^𝑛×ℝ^𝑛×𝕊×ℝ^𝑚→ℝ^𝑛×𝑚. Similar to the definition of 𝑓, we define 𝑏𝑥,𝑦,𝑖,𝑧=^𝑁𝑖_𝑗=1𝜇^𝑖_𝑗𝑏𝑥,𝑦,𝑠_𝑖𝑗,𝑧. (56) For each 𝑠_𝑖𝑗∈𝕊^𝑖 with 𝑖∈{1,…,𝑙} and 𝑗∈{1,…,𝑁_𝑖}.

To assure the existence and uniqueness of the solution of (52), we also give the following standard assumptions.

(H2′) For any integer 𝑅, there is a constant ℎ_𝑅>0, such that 𝑓𝑥,𝑦,𝑖−𝑓𝑥₁,𝑦₁∨,𝑖_{𝑚𝑘=1ℝ}𝑏^(𝑘)𝑥₂,𝑦₂,𝜅,𝑧_𝑘−𝑏^(𝑘)𝑥₁,𝑦₁,𝜅,𝑧_𝑘𝜆_𝑘𝑑𝑧_𝑘≤ℎ_𝑅𝑥₂−𝑥₁+𝑦₂−𝑦₁ (57) for all 𝑖∈𝕊 and those 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ^𝑛 with |𝑥₁|∨|𝑥₂|∨|𝑦₁|∨|𝑦₂|≤𝑅.

(H3′) There is an ℎ>0, such that for any 𝑥,𝑦∈ℝ^𝑛, 𝑖∈𝕊, 𝑓𝑥,𝑦,𝑖∨^𝑚_𝑘=1ℝ𝑏^(𝑘)𝑥,𝑦,𝜅,𝑧_𝑘𝜆_𝑘𝑑𝑧_𝑘≤ℎ1+|𝑥|+𝑦,𝑓(0,0,𝜅)≡0,𝑏(0,0,𝜅,𝑧)≡0. (58) Given 𝑉∈𝐶^𝑝(ℝ^𝑛×𝕊;ℝ₊), we define the operator 𝕃𝑉 by 𝕃𝑉𝑥,𝑦,𝑖=𝑉_𝑥(𝑥,𝑖)𝑓𝑥,𝑦,𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑉+𝑥,𝑗_{ℝ𝑚𝑘=1}𝑉𝑥+𝑏^(𝑘)𝑥,𝑦,𝜅,𝑧_𝑘,𝜅−𝑉(𝑥,𝑖)−𝑉_𝑥(𝑥,𝑖)𝑏^(𝑘)𝑥,𝑦,𝜅,𝑧_𝑘𝜆_𝑘𝑑𝑧_𝑘, (59) where 𝑉_𝑥()=()𝑥,𝑖𝜕𝑉𝑥,𝑖𝜕𝑥₁(),…,𝜕𝑉𝑥,𝑖𝜕𝑥_𝑚. (60) We need the following lemma, for details see [16].

Lemma 5. Let (H1) and (H2′), (H3′) hold, as 𝜀→0; then, (𝑥^𝜀(⋅),𝑟^𝜀(⋅)) converges weakly to (𝑥(⋅),𝑟(⋅)) in 𝐷([0,∞),ℝ^𝑛×𝕊), where 𝐷([0,∞),ℝ^𝑛×𝕊) is the space of functions defined on [0,∞) that are right continuous and have left limits taking values in ℝ^𝑛×𝕊 and endowed with the Skorohod topology.

We now state our main result in this section.

Theorem 6. Let (H1) and (H2′), (H3′) hold; there is a function 𝑉(𝑥,𝑖)∈𝐶^𝑝(ℝ^𝑛×𝕊;ℝ₊) satisfying (H4), and there are positive constants 𝜆,𝑐₁,𝑐₂, and 𝑞>1 such that(i)𝑐₁|𝑥|^𝑝≤𝑉(𝑥,𝑖)≤𝑐₂|𝑥|^𝑝, (ii)𝔼[max_𝑖∈𝕊𝕃𝑉(𝑥(𝑡),𝑥(𝑡−𝜏),𝑖)]≤−𝜆𝔼[max_𝑖∈𝕊𝑉(𝑥(𝑡),𝑖)] provided 𝔼[min_𝑖∈𝕊𝑉(𝑥(𝑡+𝜃),𝑖)]<𝑞𝔼[max_𝑖∈𝕊𝑉(𝑥(𝑡),𝑖)],−𝜏≤𝜃≤0,Then, for all 𝜉∈𝐶([−𝜏,0];ℝ^𝑛), limsup_𝜀→0𝔼𝑥^𝜀(𝑡)^𝑝≤𝜈₄𝑒^−𝜈3𝑡, (61) where 𝜈₃=min𝜆,log𝑞𝜏𝜈,𝑎𝑛𝑑₄𝜈𝑖𝑠𝑎𝑓𝑖𝑥𝑒𝑑𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡₄=𝑐₂𝑐₁sup_{−𝜏≤𝜃≤0}𝐸𝜉^𝑝. (62)

Proof. As the proof of Theorem 1, define 𝑉(𝑥,𝜁)=^𝑙_𝑖=1𝑉(𝑥,𝑖)𝐼_{𝜁∈𝕊^𝑖_}=𝑉(𝑥,𝑖)if𝜁∈𝕊^𝑖. (63) We extend 𝑟(𝑡) to [−𝜏,0] by setting 𝑟(𝑡)=𝑟(0). Then, 𝔼𝑉(𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)) is right continuous on 𝑡≥−𝜏.
Let 𝜈∈(0,𝜈₃) be arbitrary, and define 𝑈(𝑡)∶=𝑒^𝜈𝑡limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)=𝑒^𝜈𝑡limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡). (64) If we can show that 𝑈(𝑡)≤𝑐₁𝜈₄, then the proof is completed.
If 𝑡∈[−𝜏,0], by condition (i), is the same as the proof of Theorem 1, we have 𝑈(𝑡)≤𝑐₁𝜈₄.
In the following we shall prove that 𝑈(𝑡)≤𝑐₁𝜈₄ if 𝑡≥0. Otherwise, there exists the smallest 𝜌∈(0,∞) such that all 𝑡∈[−𝜏,𝜌),𝑈(𝑡)≤𝑐₁𝜈₄, and 𝑈(𝜌)≥𝑐₁𝜈₄ as well as 𝑈(𝜌+𝛿)>𝑈(𝜌) for all suffieciently small 𝛿.
As the same in the proof of Theorem 1 we can have that lim_𝜀→0𝔼𝑉(𝑥^𝜀(𝜌),𝑟^𝜀(𝜌))≠0. Hence for 𝑡∈[𝜌−𝜏,𝜌), there exists a 𝑞 such that limsup_𝜀→0𝔼𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)<𝑞limsup_𝜀→0𝔼𝑉𝑥^𝜀𝜌,𝑟^𝜀𝜌,𝜈<log𝑞𝜏. (65) Consequently, there exists a sufficiently small 𝜀₀>0, such that for any 𝜀∈(0,𝜀₀), 𝔼min_𝑖∈𝕊𝑉𝑥^𝜀𝜌+𝜃,𝑖≤𝑞𝔼max_𝑖∈𝕊𝑉𝑥^𝜀𝜌,[].,𝑖𝜃∈−𝜏,0 (66)
By condition (ii), 𝔼max_𝑖∈𝕊𝕃𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑖≤−𝜆𝔼max_𝑖∈𝕊𝑉𝑥^𝜀(𝑡),𝑖, (67) we then have for 𝜈<𝜈≤𝜆, 𝔼𝕃𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)≤−𝜈𝔼𝑉𝑥^𝜀(𝑡),𝑟(𝑡). (68)
We now consider 𝑈𝜌+𝛿−𝑈𝜌=limsup_𝜀→0𝑒^{𝜈(𝜌+𝛿)}𝔼𝑉𝑥^𝜀𝜌+𝛿,𝑟^𝜀𝜌+𝛿−𝑒^𝜈𝜌𝔼𝑉𝑥^𝜀𝜌,𝑟^𝜀𝜌=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝕃𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡+)𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝕃𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡. (69) By the definition of the operator 𝕃 similar to that of the proof of Theorem 1, we have 𝕃𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)=𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑚_𝑘=1ℝ𝑉𝑥^𝜀(𝑡)+𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀)−),𝑟((𝑡−𝜏_𝜀(𝑡),𝑧_𝑘,𝑟^𝜀(𝑡−)𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)−𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑏^(𝑘)×𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘×𝜆_𝑘𝑑𝑧_𝑘+^𝑁_𝑗=1𝛾^𝜀_{𝑟𝜀(𝑡)𝑗}𝑉𝑥^𝜀(𝑡)𝑥,𝑗=𝕃𝑉^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡−)𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+^𝑚_𝑘=1ℝ𝑉𝑥^𝜀(𝑡)+𝑏^(𝑘)×𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘,𝑟^𝜀(𝑡)−𝑉𝑥^𝜀(𝑡)+b^(𝑘)×𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘,𝑟^𝜀(𝑡)𝜆_𝑘𝑑𝑧_𝑘−^𝑚_𝑘=1ℝ𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘−𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘𝜆_𝑘𝑑𝑧_𝑘+^𝑁_𝑗=1𝛾_{𝑟𝜀(𝑡)𝑗}𝑉𝑥^𝜀(𝑡),𝑗−^𝑙_𝑗=1𝛾_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗. (70) This implies 𝑈𝜌+𝛿−𝑈𝜌=limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡ℒ𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡+limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)−𝑓𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)𝑑𝑡+limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡×^𝑚_𝑘=1ℝ𝑉𝑥^𝜀(𝑡)+𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀)−),𝑟((𝑡−𝜏_𝜀(𝑡),𝑧_𝑘,𝑟^𝜀(𝑡)−𝑉𝑥^𝜀(𝑡)+𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑙,𝑟^𝜀(𝑡)𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡−limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡×^𝑚_𝑘=1ℝ𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×𝑏^(𝑘)𝑥^𝜀(),𝑥𝑡−_𝜀)−),𝑟((𝑡−𝜏_𝜀(𝑡),𝑧_𝑘−𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘×𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡+limsup_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑁_𝑗=1𝛾_{𝑟𝜀(𝑡)𝑗}𝑉𝑥^𝜀(𝑡)−,𝑗_𝑙𝑗=1𝛾_𝑟^𝜀_(𝑡)𝑗𝑉𝑥^𝜀(𝑡),𝑗𝑑𝑡=∶𝐽₁+𝐽₂+𝐽₃+𝐽₄+𝐽₅. (71) By the definition of 𝑏, 𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘−𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘=^𝑙𝑁_𝑖=1^𝑖_𝑗=1𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑠_𝑖𝑗,𝑧_𝑘×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}. (72) By assumption (H2′), we have 𝐽₄=limsup^𝑚_{𝜀→0𝑘=1}𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×_ℝ𝑏^(𝑘)𝑥^𝜀(),𝑥𝑡−_𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘−𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀)−),((𝑡−𝜏𝑟^𝜀(𝑡),𝑧_𝑘×𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡=limsup^𝑚_𝜀→0^𝑙_𝑘=1^𝑁_𝑖=1^𝑖_𝑗=1𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×_ℝ𝑏^(𝑘)×𝑥^𝜀(),𝑥𝑡−_𝜀((𝑡−𝜏)−),𝑠_𝑖𝑗,𝑧_𝑘×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}×𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡≤limsup^𝑚_𝜀→0^𝑙_𝑘=1^𝑁_𝑖=1^𝑖_𝑗=1𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥×𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)×_ℝ𝑏^(𝑘)×𝑥^𝜀(),𝑥𝑡−_𝜀((𝑡−𝜏)−),𝑠_𝑖𝑗,𝑧_𝑘×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}×𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡^21/2. (73) By the argument of Lemma  7.14 in [9], the right side of the inequality above is equivalent to 0,thatis,𝐽₄=0. Similarly, by mean-value theorem, we can show that there exists 𝜂^(𝑘)(𝑡) which is between 𝑥^𝜀(𝑡)+𝑏^(𝑘)(𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘) and 𝑥^𝜀(𝑡)+𝑏^(𝑘)(𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘) such that 𝐽₃=lim^𝑚_{𝜀→0𝑘=1}𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡×_ℝ𝑉_𝑥𝜂(𝑡)×𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘−𝑏^(𝑘)𝑥^𝜀(𝑡−),𝑥^𝜀((𝑡−𝜏)−),𝑟^𝜀(𝑡),𝑧_𝑘𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡=lim^𝑚_𝜀→0^𝑙_𝑘=1^𝑁_𝑖=1^𝑖_𝑗=1𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡×𝑉_𝑥𝜂(𝑡)_ℝ𝑏^(𝑘)𝑥^𝜀(),𝑥𝑡−_𝜀((𝑡−𝜏)−),𝑠_𝑖𝑗,𝑧_𝑘×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡≤lim^𝑚_𝜀→0^𝑙_𝑘=1^𝑁_𝑖=1^𝑖_𝑗=1𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝑉_𝑥𝜂(𝑡)×_ℝ𝑏^(𝑘)𝑥^𝜀(),𝑥𝑡−_𝜀((𝑡−𝜏)−),𝑠_𝑖𝑗,𝑧_𝑘×𝐼_{{𝑟𝜀(𝑡)=𝑠𝑖𝑗}}−𝜇^𝑖_𝑗𝐼_{𝑟^𝜀_(𝑡)=𝑖}×𝜆_𝑘𝑑𝑧_𝑘𝑑𝑡^21/2. (74) By the argument of Lemma  7.14 in [9], we have 𝐽₃=0. Similar to the proof of Theorem 1, we can derive 𝐽₂=0,𝐽₅=0.
Therefore we arrive at 𝑈𝜌+𝛿−𝑈𝜌=lim_𝜀→0𝔼^𝜌+𝛿_𝜌𝑒^𝜈𝑡𝕃𝑉𝑥^𝜀(𝑡),𝑥^𝜀(𝑡−𝜏),𝑟^𝜀(𝑡)+𝜈𝑉𝑥^𝜀(𝑡),𝑟^𝜀(𝑡)𝑑𝑡≤0; (75) then, 𝑈𝜌+𝛿≤𝑈𝜌. (76) This contradicts the definition of 𝜌. The proof is therefore completed.

We shall give an example to illustrate our theory:

Example 7. Let 𝑟^𝜀(⋅) be a Markov chain generated by Γ^𝜀=1𝜀1Γ+Γ=𝜀1−101021−1020102−20212−1; (77) here we set Γ=0. The stationary distribution is 𝜇=(4/19,8/19,3/19,4/19). Consider a one-dimensional equation 𝑑𝑥^𝜀(𝑡)=𝑓𝑥^𝜀(𝑡),𝑟^𝜀(𝑡+)𝑑𝑡_∞0𝜎𝑟^𝜀(𝑡),𝑧𝑥^𝜀((𝑡−𝜏)−)𝑁(𝑑𝑡,𝑑𝑧) (78) with 𝑓(𝑥,1)=2sin𝑥,𝑓(𝑥,2)=−198𝑓()𝑥,𝑥,3=−196𝑥,𝑓(𝑥,4)=−2sin𝑥. (79) Let 𝛽(𝑧)=4𝜎()+8191,𝑧𝜎()+3192,𝑧𝜎()+4193,𝑧𝜎(),194,𝑧_∞0𝛽²(𝑧)𝜆(𝑑𝑧)<2. (80) Then the limit equation is 𝑑𝑥(𝑡)3=−2𝑥(𝑡)𝑑𝑡+^∞₀𝛽(𝑧)𝑥((𝑡−𝜏)−)𝑁(𝑑𝑡,𝑑𝑧). (81) Let 𝑉(𝑥)=𝑥²; then, 𝕃𝑉𝑥,𝑦≤−3|𝑥|²+^∞₀𝛽²(𝑧)𝜆(𝑑𝑧)𝑦². (82) We can find a 𝑞>1 such that 3−2𝑞>0. Therefore, for any 𝜙∈𝐿²_ℱ𝑡([−𝜏,0];ℝ^𝑛) satisfying 𝔼[min_𝑖∈𝕊𝜙(𝜃)]≤𝑞𝔼[max_𝑖∈𝕊𝜙(0)] on −𝜏≤𝜃≤0, (49) yields 𝔼max_𝑖∈𝕊ℒ𝑉𝑥,𝑦,𝑖≤−3−2𝑞𝔼max_𝑖=1,2𝑉(𝑥,𝑖). (83) Hence, by Theorem 6, the solution 𝑥^𝜀(𝑡) is mean square stable.

Acknowledgment

This paper was supported by the National Science Foundation of China with Grant no. 60904005.