DSpace at VNU: On stability in distribution of stochastic differential delay equations with Markovian switching

Contents lists available atScienceDirect Systems & Control Letters journal homepage:www.elsevier.com/locate/sysconle On stability in distribution of stochastic differential delay equatio

Contents lists available atScienceDirect Systems & Control Letters

journal homepage:www.elsevier.com/locate/sysconle

On stability in distribution of stochastic differential delay equations

with Markovian switching

aFaculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

bDepartment of Mathematics, Wayne State University, Detroit, MI 48202, USA

cDepartment of Mathematics,Vinh University, 182 Le Dan, Vinh, Nghe An, Viet Nam

a r t i c l e i n f o

Article history:

Received 11 December 2012

Received in revised form

13 December 2013

Accepted 17 December 2013

Available online 23 January 2014

Keywords:

Stochastic differential delay equations

Stability in distribution

Itô’s formula

Markov switching

a b s t r a c t

This paper provides a new sufficient condition for stability in distribution of stochastic differential delay equations with Markovian switching (SDDEs) It can be considered as an improvement to the result given

by Yuan C et al in [6]

© 2013 Elsevier B.V All rights reserved

1 SDDEs with Markovian switching

Following the development of stochastic differential equations

(SDEs), stochastic differential equations with Markovian switching

have recently become an active area of stochastic analysis

Tak-ing a further step, it is realized that, in real systems, the future

state is usually dependent not only on the present state but also

on the past states, so they should be modeled by differential

equa-tions with time delays For this reason, stochastic differential delay

equations with Markovian switching and their stability have been

studied carefully in many papers; especially, for a systematic and

detailed review, the book [1] should be referred to Since there are

four types of convergence in probability theory, at least four

differ-ent types of stochastic stability should be studied, namely, stability

in distribution, stability in probability as well as moment

stabil-ity and almost sure stabilstabil-ity For the past decade, almost sure

sta-bility and moment stasta-bility have received enormous attention [2

while to the best of our knowledge, very few studies on the stability

in distribution of stochastic differential equations with Markovian

switching can be found Among them, [3] is a noticeable

contribu-tion to the stability in distribucontribu-tion In [3], the authors consider a

∗Corresponding author Fax: +84 4 8588817.

E-mail addresses:nhdu2001@yahoo.com, dunh@vnu.edu.vn (N.H Du),

dangnh.maths@gmail.com (N.H Dang), dieunguyen2008@gmail.com (N.T Dieu).

homogeneous SDDE with Markovian switching

dX(t) =fX(t),X(t− τ),r(t)dt

+g

X(t),X(t− τ),r(t)dB(t) (1.1)

on a probability space(Ω,F, (Ft)t≥ 0,P)satisfying the usual

con-ditions, where f : Rn×Rn ×S → Rn, g : Rn ×Rn×S →

Rn×Rm,B(t) = (B1(t), ,B m(t))T is an m-dimensional Brow-nian motion, r(t)is a Markov chain taking values in a finite state

space S = {1,2, ,N}with the generatorΓ = (γij)N×N with

γij >0 if i̸=j (see [4 ]) and r(·)is independent of B(·) Moreover,

B(t)and r(t)areFt-adapted

In order to prevent possible confusion, the notations in this pa-per are the same as in [3] which will be reintroduced below for

convenience Denote by C([−τ;0];Rn)the family of continuous functionsϕ(·)from[− τ;0]to Rnwith norm∥ ϕ∥ = sup−τ≤θ≤ 0

| ϕ(θ)|, by C2(Rn × S;R+) the family of nonnegative functions

V(x,i)on Rn×S which are twice continuously differential in x For

a continuous Rn -valued stochastic process X(t)on t ∈ [− τ; ∞),

denote X t = {X(t+ θ) : −τ ≤ θ ≤0}for t≥0 so X tis a stochastic

process with the state space C([−τ;0];Rn) Moreover, for a subset

A⊂Ω, we denote by 1A the indicator function of A, i.e 1 A =1 if

ω ∈A and 1 A=0 otherwise For V∈C2(Rn×S;R+), we define

LV(x,y,i) =

n



j= 1

γij V(x,j) +V x(x,i)f(x,y,i)

+1

2trace[g

T(x,y,i)V xx(x,i)g(x,y,i)],

0167-6911/$ – see front matter © 2013 Elsevier B.V All rights reserved.

http://dx.doi.org/10.1016/j.sysconle.2013.12.006

LV(x,y,z1,z2,i) = n

j= 1

γij V(x−y,j) +V x(x−y,i)

× 

f(x,z1,i) −f(y,z2,i)

+1

2trace





g T(x,z1,i) −g T(y,z2,i)

×V xx(x−y,i)g(x,z1,i) −g(y,z2,i),

where

V x(x,i) =

 ∂V(x,i)

∂x1 , , ∂V∂ (x,i)

x n

 ,

V xx(x,i) =

 ∂2V(x,i)

∂x k∂x j



n×n

.

For any two stopping times 0≤ τ1≤ τ2< ∞, it follows from the

generalized Itô formula that

EV(X(τ2),r(τ2)) = EV(X(τ1),r(τ1))

+E

 τ 2

τ 1

LV(X(s),X(s− τ),r(s))ds

provided that the integrations involved exist and are finite

It is well-known that(X t,r(t))is a homogeneous Markov

pro-cess with transition probability p(t, ξ,i,dζ × {j} )

Definition 1 (See [ 3 ]) Eq.(1.1)is said to be stable in distribution if

there exists a probability measureπ(· × ·)on C([−τ,0];Rn) ×S

such that the transition probability p(t, ξ,i,dζ × {j} )of(X t,r(t))

converges weakly toπ(dζ × {j} )as t → ∞for every(ξ,i) ∈

C([−τ,0];Rn) ×S.

In [3], the authors provided a criterion for stability in

distribu-tion of Eq.(1.1)which is cited as the following theorem

Theorem 1.1 Assume that the coefficients of Eq.(1.1)satisfy the local

Lipschitz condition and the linear growth condition, that is, there exist

h;h k>0(k∈N)such that for all x,y,x,y∈Rn,i∈S,

|f(x,y,i)| + |g(x,y,i)| ≤h(1+ |x| + |y| ),

and for all|x| ∨ |y| ∨ |x| ∨ |y| ≤k, i∈S,

|f(x,y,i) −f(x,y,i)|2+ |g(x,y,i) −g(x,y,i)|2

≤h k(|x−x|2+ |y−y|2).

Moreover, the three following hypotheses are also satisfied.

(H1) There isα >0 such that

2(x−x)T

f(x,y,i) −f(x,y,i)

+17|g(x,y,i) −g(x,y,i)|2≤ α|x−x|2+ |y+y|2

.

(H2) There are positive numbers c1, β, λ1 > λ2 ≥0, functions V ∈

C2(Rn×S;R+)andw1∈C(Rn;R+)such that

for any(x,i) ∈Rn×S and

LV(x,y,i) ≤ −λ1w1(x) + λ2w1(y) + β (1.3)

for any(x,y,i) ∈Rn×Rn×S.

(H3) There exist numbers c2 >0, λ > κ ≥0 and functions U ∈C2

(Rn×S;R+), w2∈C(Rn;R+)such that

c2|x|2≤ w2(x) ∧U(x,i) ∀ (x,i) ∈Rn×S, (1.4)

and

LU(x,y,z1,z2,i) ≤ −λw2(x−y) + κw2(z1−z2)

Then, Eq.(1.1)is stable in distribution.

Although this theorem can be applied to many stochastic differ-ential delay equations with Markovian switching as demonstrated

in [3], some of the conditions seem to be restrictive First of all, it

is seen that the linear growth excludes some important functions

as the coefficients Thus, it should be replaced with a condition of Khas’minskii type (see [5,1]) Moreover, f(x,z1,i) −f(y,z2,i)and

g(x,z1,i)−g(y,z2,i)may depend not only on(x−y)and(z1−z2)

but also on x,y,z1,z2themselves Therefore, the hypothesis (H3),

in whichLU is required to be uniformly upper bounded by a

func-tion of(x−y)and(z1−z2), may not be satisfied for many equa-tions Besides, the hypothesis (H1), which is imposed only for tech-nical reason, should be removed In addition, the statement in [3,

p 198] that the uniform boundedness of the second moment to-gether with Chebyshev’s inequality implies the tightness is not ob-vious since the relative compactness in a function space does not follow from the boundedness The tightness therefore needs to be proved carefully Motivated by these comments, the main goal of this paper is to weaken the aforesaid hypotheses, particularly, we will use the following assumptions

Assumption 1.1 (The Local Lipschitz Condition) For each integer

k≥1 there is a positive constant h ksuch that

|f(x,y,i) −f(x,y,i)|2+ |g(x,y,i) −g(x,y,i)|2

≤h k(|x−x|2+ |y−y|2) (1.6) for all|x| ∨ |y| ∨ |x| ∨ |y| ≤k, i∈S.

Assumption 1.2 There are positive numbers c1, β, λ1 > λ2≥0,

functions V ∈C2(Rn×S;R+)andw1∈C(Rn;R+)such that lim

|x|→∞w1(x) = ∞, c1w1(x) ≤V(x,i) ≤ w1(x) (1.7) for any(x,i) ∈Rn×S and

LV(x,y,i) ≤ −λ1w1(x) + λ2w1(y) + β (1.8) for any(x,y,i) ∈Rn×Rn×S.

Assumption 1.3 There exists a nonnegative numberκand

contin-uous functions U∈C2(Rn×S;R+), w2∈C(Rn;R+), λ ∈C(R4n;

R+)such that U(·,i), w2(·)vanish only at 0 for all i ∈ S, λ(x,y,

z1,z2) > κprovided x̸=y and that

LU(x,y,z1,z2,i) ≤ −λ(x,y,z1,z2)w2(x−y)

+ κw2(z1−z2). (1.9) The idea of our proofs is as follows First, underAssumptions 1.1

and1.2, we will prove the existence of a unique global solution for any initial value and the tightness of the family of transition probabilities, which is necessary for the convergence of transition probabilities Then, in lieu of considering the second moment as

in [3], utilizing the tightness, we can estimate the amount of time when the solution stays in a suitable compact set or lies outside the compact set When the solution is in a compact set, global conditions can be replaced by local ones For this reason, the hypothesis (H1) is not necessary Furthermore, inAssumption 1.3,

we do not need that c2|x|2≤ w2(x) ∧U(x,i) ∀ (x,i) ∈Rn×S but

only the condition thatw2(x)and U(x,i)vanish at only x=0 (they might even tend to 0 at infinity) It means that the class of functions which can be candidates for testing the assumption is broadened considerably Analogously, in many cases, which will later been demonstrated by some examples, we cannot findλ > κsuch that

LU(x,y,z1,z2,i) ≤ −λw2(x−y) + κw2(z1−z2),

∀ (x,y,z1,z2,i) ∈R4n×S

but we have for all(x,y,z1,z2,i) ∈R4n×S that

LU(x,y,z ,z,i) ≤ −λ(x,y,z ,z)w (x−y) + κw (z −z)

whereλ(x,y,z1,z2) > κif x ̸= y while this inequality may fail

either at x = y or when one of the variables x,y,z1,z2tends to

infinity

2 A sufficient condition for stability in distribution of SDDEs

with the Markovian switching

Denote by Xξ,i(t)the solution to Eq.(1.1)with initial data X0=

ξ ∈ C([−τ,0];Rn)and r(0) = i We also denote by r i(t) the

Markov chain starting in i To establish new sufficient conditions

for stability in distribution of Eq.(1.1), we will give three lemmas

Lemma 2.1 Let Assumptions 1.1and1.2hold For anyξ ∈C([−τ,

0];Rn) and i ∈ S, there exists a unique global solution Xξ,i(t)to

Eq.(1.1)on[0, ∞) Moreover, for any compact set K ⊂C[− τ,0];

Rn



,

(a) there is M=M(K)such that

c1Ew1(Xξ,i(t)) ≤EV(Xξ,i(t),r i(t)) ≤M

∀t ≥0, (ξ,i) ∈K×S;

(b) for any T > 0, ε > 0, there exists a positive integer H = H

(K,T, ε)such that

P∥Xξ,i

s ∥ ≤H∀s∈ [t;t+T] ≥1− ε,

∀t ≥0, (ξ,i) ∈K×S.

Proof Letλ3=min

λ 1 − λ 2

2 ,1

τ lnλ1 + λ 2

2 λ 2  >0 and for each k∈N,

define the stopping time

σk=inf{t≥0: |Xξ,i(t)| >k}

Note thatAssumption 1.1guarantees the existence and uniqueness

of a maximal local solution to Eq.(1.1) To prove the existence of a

unique global solution, it is sufficient to show that limk→∞σk= ∞

w.p.1

Applying the generalized Itô formula to eλ 3t V(Xξ,i(t),r i(t))and

then usingAssumption 1.2yields

Eeλ 3 (t∧ σk)V(Xξ,i(t∧ σk),r i(t∧ σk)) =EV(Xξ,i(0),r i(0))

+ λ3E

 t∧ σk

0

eλ 3s VXξ,i(s),r i(s)ds

+E

 t∧ σk

0

eλ 3s LV(Xξ,i(s),Xξ,i(s− τ),r i(s))ds

≤V(ξ(0),i) + λ3E

 t∧ σk

0

eλ 3sw1(Xξ,i(s))ds

+E

 t∧ σk

0

eλ 3s− λ1w1(Xξ,i(s))

+ λ2w1(Xξ,i(s− τ)) + βds

≤V(ξ(0),i) + β

 t

0

eλ 3s ds+E

 t∧ σk

0

eλ 3s− λ1+ λ3



× w1(Xξ,i(s)) + λ2w1



Xξ,i(s− τ)ds

≤V(ξ(0),i) + β

 t

0

eλ 3s ds+E

 t∧ σk

0

eλ 3s− λ1+ λ3



× w1(Xξ,i(s))ds+E

 t∧ σk− τ

− τ λ2eλ 3s+ λ 3 τw1(Xξ,i(s))ds.

It is easy to verify thatλ1− λ3> λ2eλ 3 τ, so we have

Eeλ 3 (t∧ σk)V(Xξ,i(t∧ σk),r i(t∧ σk)) ≤EV(Xξ,i(0),r i(0))

λ3

eλ 3t+ λ2eλ 3 τ 0

− τEe

λ 3sw1(Xξ,i(s))ds

λ3

eλ 3t+V(ξ(0),i) + λ2eλ 3 τ 0

τw1(ξ(s))ds. (2.1)

If the claim limk→∞σk= ∞a.s is false, we can findΩ1⊂Ω,t0>

0 such that P(Ω1) >0 and for allω ∈Ω1, σk ≤t0 ∀k ∈N As a

result,

Eeλ 3 (t0 ∧ σk)V(Xξ,i(t0∧ σk),r i(t0∧ σk))

≥E1Ω 1V(Xξ,i(t0∧ σk),r i(t0∧ σk))

≥inf{V(y,i) : |y| >k} ·P(Ω1) → ∞, as k→ ∞ , (2.2) which contradicts(2.1) The existence and uniqueness of a global solution is therefore proved

To prove the item (a), letting k→ ∞, we obtain from(2.1)that

eλ 3t

EV(Xξ,i(t),r i(t)) ≤ λ β

3

eλ 3t+V(ξ(0),i)

+ λ2eλ 3 τ 0

− τw1(ξ(s))ds.

As a result

EV(Xξ,i(t),r i(t))

λ3

eλ 3t



V(ξ(0),i) + λ2eλ 3 τ 0

− τw1(ξ(s))ds



Putting M = sup(ξ,i)∈K×S



β

λ 3 +V(ξ(0),i) + λ2eλ 3 τ0

− τw1(ξ(s))

ds < ∞, we obtain the desired result of item (a)

Now, we move on to the item (b) For any(ξ,i) ∈K×S, define

σt

k=inf{s≥t : ∥Xξ,i

s ∥ >k}

Applying Itô’s formula for eλ 3 (s−t)V(Xξ,i(s),r i(s))on the interval

[t, (t+T) ∧ σk], similar to(2.1), we have

Eeλ 3 ((t+T)∧σt

k−t)V

Xξ,i((t+T) ∧ σt

k),r i((t+T) ∧ σt

k)

λ3

eλ 3T+EV(Xξ,i(t),r i(t))

+ λ2eλ 3 τ t

t− τEw1(Xξ,i(s))ds. (2.3)

In view of the claim of item (a),

β

λ3

eλ 3T+EV(Xξ,i(t),r i(t)) + λ2eλ 3 τ t

t− τEw1(Xξ,i(s))ds

λ3

eλ 3T+M+ λ2eλ 3 τMτ

Let H =H(K,T, ε) ∈N satisfy

inf

|y|≥H,j∈S V(y,j) ≥ 1ε  β

λ3

eλ 3T+M+ λ2eλ 3 τMτ

c1



Employing(2.3)and(2.4)yields



inf

|y|≥H,j∈S V(y,j)  ·P{ σt

H<t+T} (2.5)

≤Eeλ 3 ((t+T)∧σt

H−t)V

Xξ,i((t+T) ∧ σt

H),

r i((t+T) ∧ σt

H)

λ3

eλ 3T+M+ λ2eλ 3 τMτ

This implies that P(σt

H<t+T) ≤ ε The proof is complete 

In the next stage, we investigate the tightness of the family of transition probability distributions{p(t, ξ,i,dζ × {j} ) : t ≥ 0} Note that, in [3], under the linear growth and local Lipschitz con-ditions as well as the hypothesis(H2), it is proved that sup0≤t<∞

E∥Xξ,i

t ∥2< ∞, ∀ (ξ,i) ∈C([−τ,0];Rn) ×S Moreover, from the

Kolmogorov–Centsov theorem, we can deduce that the family{p(t,

ξ,i,dζ × {j} ) : t ≥ 0}is tight However, the situation here

be-comes more difficult since we useAssumptions 1.1and1.2instead

In fact, we shall employ the following theorem (see [6] for details)

Theorem 2.2 A family{Pa : a ∈ A}of probability measures on

C([−τ,0];Rn)is tight if and only if the two following conditions hold:

(1) For each positiveε, there exists an M such that

sup

a∈ A(Pa[x: |x(−τ)| ≥M] ) ≤ ε.

(2) For each pair of positive number(ε1, ε2), there existsδ0, 0 <

δ0≤ τsuch that

sup

a∈ A



Pa[x: sup

|s1 −s2 |≤ δ 0

|x(s1) −x(s2)| ≥ ε1]



≤ ε2.

It should be noted thatTheorem 2.2is a slight generalization

of [6, Theorem 7.3] which is stated for a sequence of probability

measures only Indeed, for necessity, if the family{Pa:a∈A}is

tight, the two conditions (1) and (2) ofTheorem 2.2are deduced

directly from tightness{Pa : a ∈ A}in combination with the

Arzelà–Ascoli theorem Conversely, for sufficiency, we can prove

in the same manner as in the proof of [6, Theorem 7.3] in which

the countability of the index set is not necessary

Lemma 2.3 Let Assumptions 1.1and1.2hold Then for any compact

set K ⊂C[− τ,0];Rn, the family

{p(t, ξ,i,dζ × {j} ) : (t,i, ξ) ∈R+×S×K}

is tight.

Proof First, we show that for anyε1, ε2 > 0, there existsδ0 =

δ0(ε1, ε2,K) >0 such that

P



sup

t≤s1≤s2≤t+ τ

s2−s1<δ 0

|Xξ,i(s2) −Xξ,i(s1)| ≥ ε1



≤ ε2,

∀ (ξ,i) ∈K×S,t≥0. (2.7)

It follows from item (b) of Lemma 2.1 that there exists H =

H



K,ε2

2, τ such that

P



sup

t≤s≤t+ τ∥X

ξ,i

s ∥ ≤H



≥1− ε2

2, ∀ (ξ,i) ∈K×S,t ≥0.

We also suppose that∥ ξ∥ ≤H for allξ ∈K Let

|x|∨|y|≤H,i∈S

{|f(x,y,i)|, |g(x,y,i)|}

For each s≥0, define the stopping time

τs=inf

u≥s: ∥Xξ,i

u ∥ >H

.

Let 0≤ δ ≤ τ By the Burkholder–Davis–Gundy inequality and the

basic inequality(|x|+|y| )4≤8(|x|4+|y|4)for any s1∈ [t,t+ τ −δ],

we have

E



sup

s2 ∈[s1 ,s1 + δ]

|Xξ,i(s2∧ τs1) −Xξ,i(s1)|4

≤8E



sup

s2 ∈[s1 ,s1 + δ]







 s2

s1

1{s≤ τs1}f(Xξ,i(s),Xξ,i(s− τ),r i(s))ds







4

+8E



sup

s2 ∈[s1 ,s1 + δ]







 s2

s1

1{s≤τs1}g(Xξ,i(s),

Xξ,i(s− τ),r i(s))dB(s)





4

≤8E



 s1 + δ

s1

1{s≤ τs1}



f(Xξ,i(s),Xξ,i(s− τ),r i(s))  ds

4

+32E







 s1 + δ

s1

1{s≤ τs1}|g(Xξ,i(s),Xξ,i(s− τ),r i(s)) |2ds





 2

≤C H′δ2.

Consequently, for any 0≤ δ ≤ τ, 1

δP



1{ τt≥t+ τ} sup

s2 ∈[s1 ,s1 + δ]

|Xξ,i(s2) −Xξ,i(s1)| ≥ ε1

3



δP



1{ τt≥t+ τ} sup

s2 ∈[s1 ,s1 + δ]

|Xξ,i(s2∧ τs1) −Xξ,i(s1)| ≥ ε1

3



≤ 81

ε4

1δE



1{ τt≥t+ τ} sup

s2 ∈[s1 ,s1 + δ]

|Xξ,i(s2∧ τs1) −Xξ,i(s1)|4



≤ 81

ε4 1

Lettingδ = ˜ ε4ε2

162C H′, it follows from the corollary on p 83 of [6] that

for each t≥0

P



sup

t≤s1≤s2≤t+τ

s2−s1≤˜ δ

|Xξ,i(s2) −Xξ,i(s1)| ≥ ε1



≤P{ τt<t+ τ}

+P



1{τt≥t+ τ} sup

t≤s1≤s2≤t+τ

s2−s1≤˜ δ

|Xξ,i(s2) −Xξ,i(s1)| ≥ ε1



≤ ε2

2 +

ε2

Since K is a compact set, by the Arzelà–Ascoli theorem there exists

δ >0 such that for allξ ∈K

sup

− τ≤s1≤s2≤ 0

s2−s1≤ δ

|Xξ,i(s2) −Xξ,i(s1)| = sup

− τ≤s1≤s2≤ 0

s2−s1≤ δ

| ξ(s2) − ξ(s1)|

≤ ε2. (2.10) Puttingδ0= ˜ δ ∧ δ, and combining(2.9)with(2.10), we have

p(t, ξ,i,M×S) =P



sup

t− τ≤s1≤s2≤t s2−s1≤δ 0

|Xξ,i(s2) −Xξ,i(s1)| ≥ ε1



≤ ε2

where



x∈C([−τ,0];Rn) : sup

t− τ≤s1 ≤s2 ≤s1 + δ 0 ≤t

|x(s2) −x(s1)|



which implies that the condition (2) ofTheorem 2.2holds for the family{p(t, ξ,i,dζ ×S) : (t,i, ξ) ∈R+×S×K} We also note that the condition (1) is derived from item (b) ofLemma 2.1

Con-sequently, it is tight Since the transition probabilities of r i(t)are

by themselves tight, we can claim that the family of{p(t, ξ,i,dζ × {j} ) : (t,i, ξ) ∈R+×S×K}is tight The proof is complete 

Lemma 2.4 Let Assumptions 1.1–1.3 be satisfied Then for anyε >0

and any compact set K ⊂C[− τ,0];Rn

, there exists T =T(ε,K)

such that for all t>T ,

P{∥Xξ,i

t −Xη,i

t ∥ < ε} ≥1− ε, ∀ ξ, η ∈K,i∈S.

Proof Step 1 SinceAssumption 1.3is weaker than the hypothesis (H3), in lieu of proving

limE|Xξ,i(t) −Xη,i(t)|2→0

as in [3], we will show that for anyσ, ¯h>0,

lim

t→∞PAσ ,¯h

t  →0,

where Aσ ,¯h

t = ω : ∥Xξ,i

t ∥ ∨ ∥Xη,i

t ∥ ≤ ¯h, |Xξ,i(t) −Xη,i(t)| ≥ σ 

For each n∈N, define the stopping time

T n=inf

s>0: ∥Xξ,i

s ∥ ∨ ∥Xη,i

s ∥ >n ∧t.

To simplify the notation, denote

λ(Xξ,i

s ,Xη,i

s ) = λXξ,i(s),Xη,i(s),Xξ,i(s− τ),Xη,i(s− τ),

cσ,¯h

2 =min{ w2(x−y) : |x| ∨ |y| ≤ ¯h, |x−y| ≥ σ },

and

λσ,¯h

=min{ λ(x,y,z1,z2) : |x| ∨ |y| ∨ |z1| ∨ |z2| ≤ ¯h,

|x−y| ≥ σ } − κ.

Note that cσ ,¯h

2 andλσ,¯h

are positive sinceλ(·), w2(·)are the con-tinuous functions andw2(x−y) >0, λ(x,y,z1,z2) >0, ∀x̸=y.

We have the estimate

EU(Xξ,i(T n) −Xη,i(T n),r i(T n))

≤EU(ξ(0) − η(0),i) −E

 T n

0

λ(Xξ,i

s ,Xη,i

s )

× w2(Xξ,i(s) −Xη,i(s))ds

+ κE

 T n

0

w2(Xξ,i(s− τ) −Xη,i(s− τ))ds

≤EU(ξ(0) − η(0),i) + κE

 0

− τw2(ξ(s) − η(s))ds

−E

 T n

0



λ(Xξ,i

s ,Xη,i

s ) − κw2(Xξ,i(s) −Xη,i(s))ds

≤U(ξ(0) − η(0),i) + κ

 0

− τw2(ξ(s) − η(s))ds

− λσ ,¯h

E

 T n

0

1{Aσ , h¯

s }w2(Xξ,i(s) −Xη,i(s))ds. (2.11)

Letting n → ∞, which implies that T n → t, and then letting

t→ ∞, we derive that for anyh¯ , σ >0,

0

P{Aσ ,¯h

s }ds≤ 1

cσ ,¯h

2

0

E



1{Aσ , h¯

s }w2(Xξ,i(s) −Xη,i(s)) ds

cσ ,¯h

2 λσ ,¯h



U(ξ(0) − η(0),i)

+ κ

 0

− τw2(ξ(s) − η(s))ds



Suppose that

lim sup

t→∞ P{Aσ ,¯h

t } >0.

Thus, there exists a constantℓ > 0 and a sequence t n, n= 1,2,

,t n↑ ∞such that

P{Aσ,¯h

t n } =P∥Xξ,i

t n ∥ ∨ ∥Xη,i

t n ∥ ≤ ¯h, |Xξ,i(t n)

−Xη,i(t n)| ≥ σ  > ℓ, ∀n∈N. (2.13)

Applying(2.7)forε1= σ

3, ε2= ℓ

8, there is 0< δ0< τsuch that P



sup

t n≤s<t n+ δ 0

|Xζ,i(s) −Xζ ,i(t n)| ≥ σ

3



8,

∀ (ζ ,i) ∈K×S, n∈N. (2.14)

It follows from(2.13)and(2.14)that

P



sup

t n≤s<t n+ δ 0

|Xξ,i(s) −Xη,i(s)| ≥ σ

3



> ℓ −2ℓ

8 =

3

4ℓ,

In view of the tightness of the family{p(t, ξ,i,dζ ×{j} ) : (t, ξ,i) ∈

R+×K×S}, we can find H1=H1(K, ℓ)satisfying

P{∥Xζ,i

t ∥ ≤H1} ≥1− ℓ

4, ∀t≥0, (ζ ,i) ∈K×S. (2.16) Combining(2.15)and(2.16), we deduce that for t n≤s<t n+ δ0

P



∥Xξ,i

s ∥ ∨ ∥Xη,i

s ∥ ≤H1; |Xξ,i(s) −Xη,i(s)| ≥ σ

3



≥3

4ℓ −2ℓ

4 =

ℓ

It means thatt n+δ 0

t n P{A

σ

3 ,H1

s }ds≥ δ 0 ℓ

4 , ∀n∈N Consequently,

0

P{A

σ

3 ,H1

s }ds= ∞

which is a contradiction since the inequality(2.12)holds for any

σ , ¯h > 0, that is, it must hold for the pair(σ

3,H1) We therefore conclude that

lim

t→∞P{Aσ,¯h

Next, we will prove the uniformity of the limit above forξ, η ∈K ,

that is, for anyσ , ¯h, ε >0, there is Tσ,¯h

ε =Tεσ,¯h(K) >0 such that

for all t>Tσ,¯h

ε we have

P



∥Xξ,i

t ∥ ∨ ∥Xη,i

t ∥ ≤ ¯h, |Xξ,i(t) −Xη,i(t)| ≥ σ  < ε,

∀ ξ, η ∈K,i∈S.

In view of tightness, let H2=H2(ε,K) > ¯h such that

P{∥Xζ,i

t ∥ >H2} < ε

6, ∀ (ζ ,i) ∈K×S, t >0. (2.19)

Put mσ,H2 =min{U(x−y,i) : |x−y| ≥ σ , |x| ∨ |y| ≤H2} Since

U(0,i) = w2(0) =0, for anyε >0, we can findδ1>0 such that

U(ξ(0) − η(0),i) + κ

 0

− τw2(ξ(s) − η(s))ds< εmσ,H2

6 provided that∥ ξ − η∥ ≤ δ1 It follows from(2.11)that

P{∥Xξ,i

t ∥ ∨ ∥Xη,i

t ∥ ≤H2, |Xξ,i(t) −Xη,i(t)| ≥ σ }

≤P{|Xξ,i(t)| ∨ |Xη,i(t)| ≤H2, |Xξ,i(t) −Xη,i(t)| ≥ σ }

mσ,H2EU



Xξ,i(t) −Xη,i(t) ≤ ε

6,

By the compactness of K , there existζ1, , ζnsuch that for any

ζ ∈K , we can findζk,k∈ {1, ,n}such that∥ ζ − ζk∥ ≤ δ1 By

(2.18), there exists Tσ,H2

ε >0 such that for all 1≤u, v ≤n,

P



∥Xζu,i

t ∥ ∨ ∥Xζ v ,i

t ∥ ≤H2, |Xζu,i(t) −Xζ v ,i(t)| ≥ σ

3



≤ ε

6,

∀t ≥Tσ,H2

ε .

For anyξ, η ∈K , we can findζl, ζmsuch that∥ ξ − ζl∥ ≤ δ1, ∥η −

ζm∥ ≤ δ1 As a result, for any t ≥Tσ ,H2

ε ,

P



∥Xξ,i

t ∥ ∨ ∥Xη,i

t ∥ ≤ ¯h, |Xξ,i(t) −Xη,i(t)| ≥ σ 

≤P



∥Xξ,i

t ∥ ∨ ∥Xη,i

t ∥ ≤H2, |Xξ,i(t) −Xη,i(t)| ≥ σ 

≤P



∥Xξ,i

t ∥ ∨ ∥Xζl,i

t ∥ ≤H2, |Xξ,i(t) −Xζl,i(t)| ≥ σ

3



+P



∥Xζl,i

t ∥ ∨ ∥Xζm,i

t ∥ ≤H2, |Xζl,i(t) −Xζm,i(t)| ≥ σ

3



+P



∥Xζm,i

t ∥ ∨ ∥Xη,i

t ∥ ≤H2, |Xζm,i(t) −Xη,i(t)| ≥ σ

3



+P ∥Xζm,i

t ∥ >H2} +P{∥Xζl,i

t ∥ >H2 < ε, (2.21)

as desired

Step 2 Let arbitrarilyε > 0 By virtue of tightness, there is H3 =

H3(K, ε)such that

P{∥Xζ,i

t ∥ ≤H3} ≥1− ε

32, ∀ (ζ ,i,t) ∈K×S×R+.

Note that if∥Xζ,i

t ∥ ≤H3and∥Xζ,i

t+ τ∥ ≤H3, then∥Xζ ,i

s ∥ ≤H3∀s,

t ≤s≤t+ τ Consequently, for anyξ, η ∈K ,

P{ τt <t+ τ} ≤ P{∥Xξ,i

t ∥ ∨ ∥Xη,i

t ∥ ∨ ∥Xξ,i

t+ τ∥ ∨ ∥X tη,+iτ∥ >H3}

8,

where

τt = (t+ τ) ∧inf{s≥t: ∥Xξ,i

s ∥ ∨ ∥Xη,i

s ∥ >H3}

Using the same arguments as in the proof ofLemma 2.3, for any

0< δ < τand t≤s1<s1+ δ ≤t+ τwe have (see(2.8))

1

δP



1{ τt≥t+ τ} sup

s2 ∈[s1 ,s1 + δ]

|Xζ,i(s2) −Xζ ,i(s1)| ≥ ε

3



≤ 81

where C3, a constant depending on K,H3, ε, can be constructed like

C H′inLemma 2.3

Let m0∈N such that81

ε 4C3δ ≤ ε

8 τwithδ = τ

m0 In view of(2.22), forρ =0, ,m0−1, we have

P



{ τt ≥t+ τ} ∩



sup

s2 ∈[t+ ρδ,t+ (ρ+ 1 )δ]

|Xξ,i(s2)

−Xξ,i(t+ ρδ)| ≥ ε

3



≤ δε

8τ .

As a result, P { τt≥t+ τ} ∩Cξ,i

t



≤ε

8, where

Cξ,i



∃ ρ ∈ {0, ,m0−1} :

sup

s2 ∈[t+ ρδ,t+ (ρ+ 1 )δ]

|Xξ,i(s2) −Xξ,i(t+ ρδ)| ≥ ε

3



Hence

P



{ τt ≥t+ τ} \Cξ,i

t



≥1− ε

8−

ε

8 =1−

ε

4. (2.23)

Similarly,

P

 { τt≥t+ τ} \Cη,i

t



≥1− ε

Owing to the uniform convergence shown in Step 1, we can find

T =T(K, ε)such that for any t>T ,

m0 − 1



ρ= 0

P



∥Xξ,i

t+ ρδ∥ ∨ ∥X tη,+iρδ∥ ≤H3,

|Xξ,i(t+ ρδ) −Xη,i(t+ ρδ)| ≥ ε

3



≤ ε

4,

which implies that P{D t} ≤ ε

4where



∃ ρ ∈ {0, ,m0−1} : ∥Xξ,i

t+ ρδ∥ ∨ ∥X tη,+iρδ∥

≤H3and|Xξ,i(t+ ρδ) −Xη,i(t+ ρδ)| ≥ ε

3



Thus, for t>T ,

P

 { τt≥t+ τ} \D t ≥1− ε

8−

ε

4 =1−

3ε

8.

Since{∥Xξ,i

t+ ρδ∥∨∥X tη,+iρδ∥ ≤H3} ⊃ { τt ≥t+ τ}, it is easy to see that P



{ τt≥t+ τ} ∩



|Xξ,i(t+ ρδ) −Xη,i(t+ ρδ)| < ε

3,

∀0≤ ρ ≤m0−1



=P



{ τt≥t+ τ} \



∃0≤ ρ ≤m0−1:

|Xξ,i(t+ ρδ) −Xη,i(t+ ρδ)| ≥ ε

3



=P

 { τt ≥t+ τ} \D t



≥1−3ε

Note that if the three events|Xξ,i(t + ρδ) −Xη,i(t + ρδ)| <

ε

3, ∀ ρ =0, ,m0−1 , {τt ≥t+ τ}\Cξ,i

t and{ τt ≥t+ τ}\Cη,i

t

occur simultaneously, we have∥Xξ,i

t+ τ−X tη,+iτ∥ < ε This statement, combining with(2.23)–(2.25)implies that

P



∥Xξ,i

t+ τ−X tη,+iτ∥ < ε >1− ε, ∀t >T,

as required 

In view ofLemma 2.4and a slight modification of [3, Lemma 3.4], we can employ [3, Theorem 3.1] to obtain the main result

Theorem 2.5 Eq. (1.1) is stable in distribution under Assump-tions 1.1–1.3

3 Examples

In this section, some examples will be presented in order to illustrate our improvements

Example 3.1 We now consider equation

dX(t) =



a(r(t)) −1

2X(t)



dt

+ (b(r(t)) +sin X(t− τ))dB(t). (3.1)

Our main purpose is to prove that the hypothesis(H3)of

Theo-rem 1.1is possibly not satisfied, while it is easier to verify

Assump-tion 1.3 Let V(x,i) =x2, we have

LV(x,y,i) =2a(i)x−x2+ (b(i) +sin y)2≤ −x2

2 + β,

whereβ =sup(x,y,i)∈ R × R ×S



2a(i)x−x2

2+ (b(i)+sin y)2 < ∞ This means thatAssumption 1.2is satisfied Now, letting U(x,i) = x2

again, it is computed that

LU(x,y,z1,z2,i) = −(x−y)2+ (sin z1−sin z2)2

= − (x−y)2+4 sin2z1−z2

2 cos

2z1+z2

Suppose that there areλ > κ >0, w2(·)such that

LU(x,y,z1,z2,i) ≤ −λw2(x−y) + κw2(z1−z2).

Substituting z1 = z2and x = y (separately) into the inequality

above we haveλw2(x−y) ≤ (x−y)2, (sin z1−sin z2)2< κw2(z1−

z2) Thus, for u ̸= v,(sin u− sin v) 2

(u− v) 2 ≤ κw 2 (u− v)

(u− v) 2 ≤ κ

λ < 1 However,

this inequality does not hold if we take u→ v It means that the

hypothesis (H3) ofTheorem 1.1is not satisfied for U(x,i) =x2

Nevertheless, definew2(u) =4 sin2 u

2if|u| ≤ πandw2(u) =4 otherwise,λ(x,y,z1,z2) = (x−y) 2

w 2 (x−y)if x̸=y andλ(x,y,z1,z2) =1

if x=y It is easy to verify thatλ(x,y,z1,z2) > κ :=1 for all x̸=y.

In view of(3.2), we can estimate

LU(x,y,z1,z2,i) ≤ −λ(x,y,z1,z2)w2(x−y) + w2(z1−z2),

that is,Assumption 1.3is satisfied The equation is therefore stable

in distribution

Example 3.2 We consider another equation

dX(t) = a(i) −X3(t) −bX(t) +cX(t− τ)dt

+ d(i)X2(t)dB(t), (3.3)

where b>c>0,a(i) >0, 2

17 <d(i) <3

2, ∀ i∈S.

Obviously, the coefficients of this equation do not satisfy the

lin-ear growth condition, but the existence and uniqueness of a global

solution can easily be verified using [5, Theorem 2.4]

Unfortu-nately, as shown later, the hypothesis(H1)do not hold Therefore,

we cannot apply the result in [3] The purpose of this example is

to demonstrate that Eq.(3.3)is stable in distribution although the

condition(H1)is not satisfied Indeed, by calculating we have

2(x−x)(f(x,y,i) −f(x,y,i)) +17|g(x,y,i) −g(x,y,i)|2

=2(x−x) −(x3−x3) −b(x−x) +c(y−y)

+17d(i)(x2−x2)2

= (x−x)2

(17d(i) −2)(x2+x2) +2(17d(i) −1)xx

−2b(x−x)2+2c(x−x)(y−y).

Let x=x+l,l̸=0 Since 17d(i) >2∀ i∈S,

lim

x→+∞

 (17d(i) −2)(x2+x2) +2(17d(i) −1)xx = +∞ ,

which easily implies that there is noα > 0 such that condition (H1) holds However, we will show that the coefficients of Eq.(3.3)

satisfyAssumptions 1.2and1.3 Let V(x,i) =x2, we have

LV(x,y,i) =2a(i)x−2x4−2bx2+2cxy+d(i)x4

≤ − (2−d(i))x4+2a(i)x−b



b

2

−bx2+c2

b y

2≤ −bx2+c2

b y

2+ β,

where β = sup(x,y,i)∈ R × R ×S



− (2 − d(i))x4 + 2a(i)x− bx −

cy b

2

< +∞ Hence,Assumption 1.2is satisfied Next,

employ-ing U(x,i) =x2again, we have the estimate

LU(x,y,z1,z2,i)

=2(x−y) −(x3−y3) −b(x−y) +c(z1−z2)

+d(i)(x2−y2)2

= − (x−y)2

(2−d(i))(x2+y2) +2(1−d(i))xy

−2b(x−y)2+2c(x−y)(z1−z2)

= −b(x−y)2+c2

b(z1−z2)2

−b(x−y) −c

b(z1−z2) 2− (x−y)2

×  (2−d(i))(x2+y2) +2(1−d(i))xy

Since d(i) ≤ 3

2 ∀i∈S,(2−d(i))(x2+y2)+2(1−d(i))xy≥0∀x,y.

Therefore,LU(x,y,z1,z2,i) ≤ −b(x−y)2+ c2

b(z1−z2)2, which means thatAssumption 1.3holds Consequently, Eq.(3.3)is stable

in distribution

Acknowledgment

This work was done under the support of the Grand NAFOSTED,

No 101.02-2011.21

References

[1] X Mao, C Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.

[2] R.Z Has’minskii, Stochastic Stability of Differential Equations, Sijthoff Noord-hoff, 1980.

[3] C Yuan, J Zou, X Mao, Stability in distribution of stochastic differential delay equations with Markovian switching, Systems Control Lett 50 (2003) 195–207 [4] W.J Anderson, Continuous—Times Markov Chain, Springer, Berlin, 1991 [5] X Mao, M.J Rassias, Khasminskii-type theorems for stochastic differential delay equations, J Stoch Anal Appl 23 (2005) 1045–1069.

[6] P Billingsley, Convergence of Probability Measures, John Wiley and Sons, Inc., New York, NY, 1999.