DSpace at VNU: A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching

Contents lists available atScienceDirectNonlinear Analysis journal homepage:www.elsevier.com/locate/na A note on sufficient conditions for asymptotic stability in distribution of stochas

Contents lists available atScienceDirect

Nonlinear Analysis

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

A note on sufficient conditions for asymptotic stability in

distribution of stochastic differential equations with

Markovian switching

Nguyen Hai Dang∗

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

Article history:

Received 7 February 2013

Accepted 30 September 2013

Communicated by S Carl

MSC:

34K50

34K20

65C30

60J10

Keywords:

Stochastic differential equations

Stability in distribution

Itô’s formula

Markovian switching

a b s t r a c t

The aim of this paper is to improve the results on stability in distribution of nonlinear stochastic differential equations in Yuan and Mao (2003) Both conditions for the stability

in distribution given in that paper are weakened

© 2013 Elsevier Ltd All rights reserved

1 SDEs with Markovian switching

For the past decade, stochastic differential equations with Markovian switching and their stability have drawn much attention Most of the papers are concerned with stochastic stability (stability in probability), moment stability and almost sure stability However, the stability in distribution does not seem to be paid as much attention as its importance Among very few papers studying this type of stability, [1] is a notable contribution Its results are summarized as follows

Denote by(Ω,F, (Ft)t≥ 0,P)the probability space satisfying the usual conditions and B(t) = (B1(t),B2(t), ,B m(t))T

the m-dimensional Brownian motion adapted to the filtration(Ft)t≥ 0 Let r(t),t ≥0 be a Markov chain on the probability

space taking values in a finite state space S= {1,2, ,N}with generatorΓ = (γij)N×Ngiven by

P{r(t+∆) =j|r(t) =i} =



γij∆+o(∆) if i̸=j

1+ γij∆+o(∆) if i=j, where∆>0 Hereγij>0 is the transition rate from i to j if i̸=j while

γii= − 

i̸=j

γij.

It is well known that almost every sample path of r(t)is right continuous step functions and r(t)is an ergodic Markov chain

We assume further that Markov chain r(·)is independent of Brownian motion B(·)

∗Tel.: +84 986120076.

E-mail address:dangnh.maths@gmail.com.

0362-546X/$ – see front matter © 2013 Elsevier Ltd All rights reserved.

http://dx.doi.org/10.1016/j.na.2013.09.030

Consider a nonlinear SDE with Markovian switching

where f,g:Rn×S→Rn.For the existence and uniqueness of a global solution with any initial value, the authors imposed the following assumption

Assumption 1.1 The coefficients of Eq.(1.1)satisfy the local Lipschitz condition and the linear growth condition, that is,

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

|f(x,i)| + |g(x,i)| ≤h(1+ |x| ) ∀x, ∈Rn,i∈S,

and for all∀ |x| ∨ |y| ≤k,i∈S,

|f(x,i) −f(y,i)|2+ |g(x,i) −g(y,i)|2≤h k(|x−y|2).

It is known that the process y(t) = (X(t),r(t))is a homogeneous Markov process with the state space Rn×S The

stability in distribution is defined as follows

Definition 1 (See [ 1 ]) The process y(t)is said to be asymptotically stable in distribution if there exists a probability measure π(·, ·)on Rn×S such that the transition probability p(t,x,i,dy× {j} )of y(t)converges weakly toπ(dy× {j} )as t→ ∞for every(x,i) ∈Rn×S Eq.(1.1)is said to be asymptotically stable in distribution if y(t)is asymptotically stable in distribution

Let C2(R×S,R+)denote the family of non-negative functions U(x,i)on Rn×S which is continuously twice differentiable

in x We denote byKthe set of continuous functionsµ :R+→R+vanishing only at 0 Moreover,µis said to belong toK∞

ifµ ∈Kandµ(x) → ∞as x→ ∞ C Yuan and X Mao in [1] proved that Eq.(1.1)is asymptotically stable in distribution

ifAssumption 1.1and the two following ones are satisfied

Assumption 1.2 There exists a function V(x,i) ∈ C2(R×S,R+)and positive constantsα, βsuch that V(x,i) → ∞as

|x| → ∞and that LV(x,i) ≤ −αV(x,i) + βfor any(x,i) ∈Rn×S.

Assumption 1.3 There exist functions U∈C2(R×S;R+), ν, µ ∈K, such that

The two operators LV(x,i)and LU(x,y,i), which will still be used throughout this paper, are defined by

LV(x,i) =

n



j= 1

γij V(x,j) +V x(x,i)f(x,i) +1

2trace



g(x,i)T V xx(x,i)g(x,i)

and

LU(x,y,i) =

n



j= 1

γij U(x−y,j) +U x(x−y,i) (f(x,i) −f(y,i))

+1

2trace

 (g(x,i) −g(y,i))T U xx(x−y,i) (g(x,i) −g(y,i)) This result is very useful to investigate the stability in distribution of Eq.(1.1)as illustrated with some examples in [1

Nevertheless, these conditions can be weakened to cover a larger collection of stochastic differential equations A slight and usual improvement can be made by using the condition of Khasminskii type instead of the linear growth one Moreover,

Assumptions 1.2and1.3should be replaced with weaker ones also In particular,Assumption 1.2guarantees the tightness

of the family of transition probabilities However, as we will point out later that this assumption can be weakened Condition

(1.4)seems be too restrictive, also It demands LU(x,y,i)to be uniformly upper bounded by a negative-definite function

of the difference(x−y) Therefore, it is nearly satisfied only for f and g being Lipschitz functions For this reason, we also

attempt to localize this condition The improvements will be shown in the next section and justified by some examples

2 New sufficient conditions

Instead of usingAssumptions 1.1–1.3, we will prove that Eq.(1.1)is still asymptotically stable in distribution under the three following assumptions

Assumption 2.1 The coefficients of Eq.(1.1)satisfy the local Lipschitz condition, that is, for any k ∈ N, there exists an

h k>0 such that

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

Assumption 2.2 There exist functions V∈C2(Rn×S;R+), ν ∈K∞and a positive numberβsuch that

lim

and

Assumption 2.3 There exist functions U∈C2(R×S;R+), µ1∈KandµM∈Kfor each M>0 satisfying

Theorem 2.1 Let Assumptions 2.1and2.2holds Then, for any initial value(x,i) ∈Rn×S, there exists a unique global solution, denoted by X x,i(t), to(1.1) Moreover, for R>0, we can find K =K(R) >0 such that 1t t

0Eµ(X x,i(s))ds≤K, ∀|x| ≤R,t≥1.

Proof UnderAssumptions 2.1and2.2, the existence and uniqueness of global strong solutions is obvious due to the Khasminskii-type theorem (see [2, Theorem 3.19]) Define stopping times t k=inf{t≥0:X x,i(t) ∨X y,i(t) >k} ,k∈N By virtue of the generalized Itô formula,

EV(X x,i(t∧t k)) ≤V(x,i) +E

 t∧t k

0

(β − µ(X x(s)))ds

which implies that

E

 t∧t k

0

µ(X x(s))ds≤V(x,i) + βt.

Letting k→ ∞and using the Lebesgue monotone convergence theorem, we conclude that

1

tE

 t

0 µ(X x(s))ds≤ β + 1

t V(x,i) ≤K(R) ∀|x| ≤R,t≥1. 

Theorem 2.2 If Assumptions 2.1–2.3 are satisfied, Eq.(1.1)has the property that for any positive number R, ε, δ, there exists a

T =T(R, ε, δ)such that

P{|X x,i(t) −X y,i(t)| ≤ δ ∀t ≥T} ≥1− ε ∀|x| ∨ |y| ≤R.

Proof Suppose that|x| ∨ |y| ≤ R Let arbitrarilyδ > 0 andε >0 We firstly show that for any bounded stopping times

τ1≤ τ2,

EUX x,i(τ2),X y,i(τ2),r(τ2) ≤EUX x,i(τ1),X y,i(τ1),r(τ1) ≤U(x−y,i) (2.6) and

0≤E

 τ 1

0

Indeed, let t kbe defined above, then, applying the generalized Itô formula, we yield

0<EUX x,i(τ1∧t k),X y,i(τ1∧t k),r(τ1∧t k)

=U(x−y,i) +E

 τ 1 ∧t k

0

LU(X x,i(s),Y y,i(s),r(s))ds≤U(x−y,i).

Sinceτ1∧t k→ τ1as k→ ∞, by the Lebesgue monotone convergence theorem, EUX x,i(τ1),X y,i(τ1),r(τ1) ≤U(x−y,i) Similarly we have

EUX x,i(τ2),X y,i(τ2),r(τ2) ≤EUX x,i(τ1),X y,i(τ1),r(τ1)

It also follows from the generalized Itô formula that

−E

 τ 1 ∧t k

LU(X x,i

s ,Y y,i

s ,r(s))ds≤U(x−y,i).

−

 τ 1 ∧t k

0

LU(X x,i

s ,Y y,i

s ,r(s))ds↗ −

 τ 1 0

LU(X x,i

s ,Y y,i

s ,r(s))ds as k→ ∞ ,

we obtain(2.7)

Since lim| ξ|→∞µ(ξ) = ∞, for anyε >0, there exists an M such that inf| ξ|>Mµ(ξ) ≥ 2K

ε 2, where K is defined in Theo-rem 2.1 Thus,

P{|X x,i

t | ∨ |Y y,i

t | >M} ≤P{|X x,i

t | >M} +P{|X y,i

t | >M} ≤ ε2

2K



Eµ X x,i +Eµ X y,i

.

On the other hand, since U is continuous and U(0,i) = 0∀i∈ S, there exists anαsuch that 0 < α < δand sup|x|≤α,i∈S

U(x,i)

µ 1 (δ) ≤ ε Define the stopping time

τα=inf{t ≥0: |X x,i(t) −X y,i(t)| ≤ αand|X x,i(t)| ∨ |X y,i(t)| ≤M} ,

Note that

E

 τ α ∧t

0

LU(X x,i(s),X y,i(s),r(s))ds= E

 t

0

1{ 0 ≤s≤ τ α }LU(X x,i(s),X y,i(s),r(s))ds

 t

0

1{|X x,i(s)|∨|X y,i(s)|≤M}1{ 0 ≤s≤ τ α }LU



X x,i(s),X y,i(s),r(s)ds

≤ − µM(α)E

 t

0

1{|X x,i(s)|∨|X y,i(s)|≤M}1{ 0 ≤s≤ τ α }ds.

On the other hand,

E

 t

0

1{ 0 ≤s≤ τ α }ds−E

 t

0

1{|X x,i(s)|∨|X y,i(s)|}≤M1{ 0 ≤s≤ τ α }ds=E

 t

0

1{|X x,i(s)|∨|X y,i(s)|>M}1{ 0 ≤s≤ τ α }ds.

By virtue of Holder’s inequality,

 t

0

E1{|X x,i(s)|∨|X y,i(s)|>M}1{ 0 ≤s≤ τ α }ds

2

≤

 t

0

E1{|X x,i(s)|∨|X y,i(s)|>M}

 t

0

E1{ 0 ≤s≤ τ α }ds

≤ tε2

2K

 t

0



Eµ X x,i(s) +Eµ X y,i(s)ds≤ (tε)2 ∀t≥1. Consequently, Et

01{|X x,i(s)|∨|X y,i(s)|≤M}1{ 0 ≤s≤ τ α }ds≥E(τα∧t) − εt, which implies

E

 t

0

LU(X x,i(s),X y,i(s)),r(s)ds≤ − µM(α)E

 t

0

1|X x,i(s)|∨|X y,i(s)|≤M1{ 0 ≤s≤ τ α }ds

≤ − µM(α) (E(τα∧t) − εt)

In view of(2.7), 0≤U(x−y,i) +E

τ α ∧t

0 LU(X x,i(s),X y,i(s),r(s))ds Hence U(x−y,i) + µM(α)εt− µM(α)E(τα∧t) ≥0 Consequently

tP(τα≥t) ≤E(τα∧t) ≤ U( µx M− (α)y,i) + εt ∀t≥1.

This implies the existence of T >1 satisfying

Define the stopping time

σ =inf{t≥ τα: |X x,i(t) −Y y,i(t)| ≥ δ}

and it is easy to verify that

σα=



σ ifτα≤T

τα otherwise.

is also a stopping time Sinceσα≥ τα, it follows from(2.6)

EUX x,i(σα∧t) −X y,i(σα∧t),r(σα∧t) ≤EUX x,i(τα∧t) −X y,i(τα∧t),r(σα∧t)

E1{ τ α >T}UX x,i(σα∧t) −X y,i(σα∧t),r(σα∧t) =E1{ τ α >T}UX x,i(τα∧t) −X y,i(τα∧t),r(σα∧t)

Consequently,

E1{ τ α ≤T}UX x,i(σα∧t) −X y,i(σα∧t),r(σα∧t) ≤E1{ τ α ≤T}UX x,i(τα∧t) −X y,i(τα∧t),r(σα∧t)

Thus,

P({τα≤T} ∩ { σ <t} ) µ1(δ) ≤E1{ τ α ≤T}∩{ σ <t}U

X x,i(σ ∧t) −X y,i(σ ∧t),r(σ ∧t)

≤E1{τ α ≤T}UX x,i(σ ∧t) −X y,i(σ ∧t),r(σ ∧t)

=E1{τ α ≤T}U

X x,i(σα∧t) −X y,i(σα∧t),r(σα∧t)

≤E1{τ α ≤T}UX x,i(τα∧t) −X y,i(τα∧t),r(τα∧t)

≤P{ τα≤T} × sup

|x|≤ α,i∈S

U(x,i).

Since sup|x|≤α,i∈S U(x,i) ≤ εµ1(δ), it follows that P({τα≤T} ∩ { σ <t} ) < ε ∀t ≥T Letting t → ∞yields P({τα≤T}

∩{ σ < ∞}) ≤ ε By this estimate and(2.8), P({τα≤T} ∩ { σ = ∞}) ≥1−3ε Note that ifω ∈ {τα ≤T} ∩ { σ < ∞},

|X x,i

t −X y,i

t | < δ ∀t ≥T We conclude that for any positive numbersε, δand R, there exists a T =T(R, ε, δ)such that

P



sup

t≥T

|X x,i(t) −X y,i(t)| ≤ δ



>1−3ε ∀|x| ∨ |y| ≤R,i∈S

as desired 

To take further steps, we need to introduce more notations LetP(Rn×S)denote all probability measures on Rn×S.

For P1,P2∈P(Rn×S), we define metric dHas follows

dH[P1,P2] =sup

f∈ H







n



i= 1



Rn

f(x,i)P1(dx,i) −

n



i= 1



Rn

f(x,i)P2(dx,i)





 , where

H= {f :Rn×S→R: |f(x,i) −f(y,j)| ≤ |x−y| + |i−j| , |f(·, ·)| ≤1}

It is well known that the weak convergence of probability measures is equivalent to the convergence in this metric (see [3])

Lemma 2.1 Let Assumptions 2.1–2.3 hold Then, for any compact subset K of R n ,

lim

uniformly in x,y∈K and i,j∈S.

Proof UnderAssumptions 2.1and2.2, it is easy to point out that for any positive numbersε,R and T , we can find aR =

R(R,T, ε) >0 such that

P



sup

0 ≤t≤T

X x,i(t) ≤ R



>1− ε ∀|x| ≤R,i∈S. Having this property as well as the conclusion ofTheorem 2.2, we can obtain the desired assertion by employing the proof

of [1, Lemma 3.2] 

Note that, the twoAssumptions 2.1and2.2guarantee the Feller property of the solution Furthermore, they also imply the existence of an invariant probability measureπon Rn×S owing to the conclusion ofTheorem 2.1(see [4, Theorem 4.5]

or [5] for details) We now prove our main result with simplified notations Particularly, we denote by p(t,u,dv)the

transition probability instead of p(t,x,i,dy× {j} )and S=Rn×S.

Theorem 2.3 Let Assumptions 2.1–2.3 hold Then, the transition probabilities p(t,z,du) converge weakly to the invariant probability measureπ(du)for all z∈S.

Proof Sinceπ(·)is invariant, thenπ(du) = Sp(t, v,du)π(dv) ∀t ≥0 which implies



f(u)π(du) =



f(u)



p(t, v,du)π(dv)



=

 

f(u)p(t, v,du)



For any z∈S, we estimate







S

f(u)p(t,z,du) − 

S

f(u)π(du)



 =







S



S

f(u)p(t,z,du) − 

S

f(u)p(t, v,du)

 π(dv)





≤



Bk







S

f(u)p(t,z,du) −



S

f(u)p(t, v,du)



 π(dv)

+



Bc k







S

f(u)p(t,z,du) −



S

f(u)p(t, v,du)



 π(dv)

≤ sup v∈ Bk

{dH[p(t,z, ·),p(t, v, ·)]} π(Bk) +2π(Bc k) (2.11) where Bk= {x∈Rn: |x| ≤k} ×S and B c k= (Rn\Bk) ×S and k∈N is chosen to be sufficiently large such thatπ(Bk) ≥

1−ε

4.On the other hand, byLemma 2.1, there is a T=T(Bk, ε) >0 such that

sup

v∈ Bk

{dH[p(t,z, ·),p(t, v, ·)]} ≤ ε

Since f is taken arbitrarily, it follows from(2.11)and(2.12)that

which implies the desired conclusion 

3 Examples

We now illustrate our results by the two following examples

Example 3.1 We consider stability in distribution of the stochastic logistic model

where a(i),b(i),c(i)is positive constants

It is known that (see [6]) if X0>0, a.s then 0<X t < ∞ ∀t>0 with probability 1 Put Z t =ln X t, the equation becomes

dZ t=



a(r(t)) −b(r(t))e Z t− c2(r(t))

2Z t



For this equation, the local Lipschitz condition is obviously satisfied Moreover, consider the function V(z,i) =e z+e−z>

0∀ (z,i) ∈R×S,we have

LV(z,i) =a(i)(e z−e−z) −b(i)(e 2z−1) +c2(i)

z.

Thus,β =sup(z,i)∈ R ×S{LV(z,i) + εV} < +∞, wheneverε <min{a(i)}.It follows that LV(z,i) ≤ β − εV(z) ∀(z,i) ∈R×S

thenAssumption 2.2holds

Put U(z,i) =z2 We consider two solutions of(3.2)with the initial values are(u,i), (v,i) ∈R×S.We have

LU(u, v,i) = −2(u− v)



b(i)(e u−ev) +c2

2(e 2u−e2v)

 +c2(i)(e u−ev)2

= −2b(i)(u− v)(e u−ev) −c2(i)(u− v)(e u−ev)



e u+ev−e u−ev

u− v



It is easy to see that e u+ev−e u−ev

u− v >0∀u, v ∈R, hence

LU(u, v,i) ≤ −2b(i)(u− v)(e u−ev) ∀(u, v,i) ∈R×R×S.

On the other hand, LU(u, v,i) →0 whenever u→ −∞andv → −∞ Thus, there is noµ ∈ K such that LU(u, v,i) ≤

− µ(|u− v|) It means that the function U(z,i) =z2does not satisfyAssumption 1.3 However, for any M>0,LU(u, v,i) ≤

−2b(i)(u− v)(e u−ev) ≤ −k M(u− v)2where k M = 2 min{b(i)}e−M Consequently,Assumption 2.3holds for this func-tion and it follows that Eq.(3.2)is asymptotically stable in distribution As a results, Eq.(3.1)is asymptotically stable in distribution on the state space(0, +∞) ×S also.

Example 3.2 Consider another equation

dX(t) = a(r(t)) −b(r(t))arctan(X(t))ln

X2(t) +1

where a(i),b(i),c(i)are constant for each i∈S and b(i)are positive Using V(x,i) =x2, we see that

LV(x,i) =2a(i)x+c2(i) −2b(i)x arctan x ln(x2+1)

= 

2a(i)x+c2(i) − (2b(i) − α)x arctan x ln(x2+1) − αx arctan x ln(x2+1) ≤ β − µ(x), (3.4) whereα =min{b(i) :i∈S} , µ(x) = αx arctan x ln(x2+1)and

(x,i)∈ R ×S

{2a(i)x+c2(i) − (2b(i) − α)x arctan x ln(x2+1)} < ∞.

The estimate(3.4)means thatAssumption 2.2holds On the other hand, it is easy to see that,Assumption 1.2cannot be

satisfied for V(x,i) =x2 In order to checkAssumption 2.3, we compute

LV(x,y,i) = −2b(i)(x−y) arctan x ln(x2+1) −arctan y ln(y2+1) (3.5) Using arguments analogous to those in Eq.(3.1), we can show that for any M>0, there is aσM>0 such that LV(x,y,i) ≤

− σM(x−y)2∀|x| ∨ |y| ≤M although there is noµ ∈Ksuch that LV(x,y,i) ≤ −µ(x−y) We therefore still conclude that

Eq.(3.3)is asymptotically stable in distribution

References

[1] C Yuan, X Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process Appl 79 (2003) 45–67.

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

[3] N Ikeda, S Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.

[4] S.P Meyn, R.L Tweedie, Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes, Adv Appl Probab 25 (1993) 518–548.

[5] L Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, LCDS Report No 86-16, April 1986, Brown University, Providence, 1986.

[6] Q Luo, X Mao, Stochastic population dynamics under regime switching, J Math Anal Appl 334 (2007) 69–84.