# Stochastic Functional Differential Equation under Regime Switching

Authors: Ling Bai; Zhang Kai
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We discuss stochastic functional differential equation under regime switchingdx(t)=f(xt,r(t),t)dt+q(r(t))x(t)dW1(t)+σ(r(t))|x(t)|βx(t)dW2(t). We obtain unique global solution of this system without the linear growth condition; furthermore, we prove its asymptotic ultimate boundedness. Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially stable of this system.


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## Body

## 1. Introduction

Recently, many papers devoted their attention to the hybrid system, they concerned that how to change if the system undergoes the environmental noise and the regime switching. For the detailed understanding of this subject, [1] is good reference.In this paper we will consider the following stochastic functional equation:(1.1)dx(t)=f(xt,r(t),t)dt+q(r(t))x(t)dW1(t)+σ(r(t))|x(t)|βx(t)dW2(t).
The switching between these N regimes is governed by a Markovian chain r(t) on the state space 𝕊={1,2,…,N}.  xt∈C([-τ,0];Rn) is defined by xt(θ)=x(t+θ); θ∈[-τ,0]. C([-τ,0];Rn) denote the family of continuous functions from [-τ,0] to Rn, which is a Banach space with the norm ∥ϕ∥=sup-τ≤θ≤0|ϕ(θ)|.  f:C([-τ,0];Rn)×𝕊×R+→Rn satisfies local Lipschitz condition as follows.Assumption A.
For each integerk≥1,…, there is a positive number Hk such that
(1.2)|f(φ1,k)-f(φ2,k)|≤Hk∥φ1-φ2∥
for all t≥0 and those φ1,φ2∈C([-τ,0];Rn) with ∥φ1∥∨∥φ2∥≤k.
Throughout this paper, unless otherwise specified, we let(Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). Let Wi(t) (i=1,2), t≥0, be the standard Brownian motion defined on this probability space. We also denote by R+n={x∈Rn:xi>0forall1≤i≤n}. Let r(t) be a right-continuous Markov chain on the probability space taking values in a finite state space 𝕊={1,2,…,N} with the generator Γ=(γuv)N×N given by
(1.3)P{r(t+δ)=v∣r(t)=u}={γuvδ+o(δ),ifu≠v,1+γuvδ+o(δ),ifu=v,
where δ>0. Here γuv is the transition rate from u to v and γuv≥0 if u≠v while
(1.4)γuu=-∑v≠u‍γuv.
We assume that the Markov chain r(·) is independent on the Brownian motion Wi(·),i=1,2; furthermore, W1 and W2 are independent.
In addition, throughout this paper, letC2,1(Rn×[-τ,∞)×𝕊;R¯+) denote the family of all positive real-valued functions V(x,t,k) on Rn×[-τ,∞)×𝕊 which are continuously twice differentiable in x and once in t. If for the following equation
(1.5)dx(t)=f(xt,r(t),t)dt+g(x,r(t),t)dB(t),
there exists V∈C2,1(Rn×[-τ,∞)×𝕊;R+), define an operator ℓV from C([-τ,0];Rn)×R+×𝕊 to R by
(1.6)ℓV(φ,t,k)=Vt(φ(0),t,k)+Vx(φ(0),t,k)f(φ,k,t)+12trace[gT(φ(0),k,t)Vxx(φ(0),k,t)g(φ(0),k,t)]+∑l=1N‍γklV(φ(0),t,l),
where
(1.7)Vt(x,t,k)=∂V(x,t,k)∂t,Vx(x,t,k)=(∂V(x,t,k)∂x1,…,∂V(x,t,k)∂xn),Vxx(x,t,k)=(∂V2(x,t,k)∂xi∂xj)n×n.
Here we should emphasize that [1, Page 305] the operator ℓV (thought as a single notation rather than ℓ acting on V) is defined on C([-τ,0];Rn)×R+×𝕊 although V is defined on Rn×[-τ,∞)×𝕊.


## 2. Global Solution

Firstly, in this paper, we are concerned about that the existence of global solution of stochastic functional differential equation (1.1).In order to have a global solution for any given initial data for a stochastic functional equation, it is usually required to satisfy the local Lipschitz condition and the linear growth condition [1, 2]. In addition, as a generation of linear condition, it is also mentioned in [3, 4] with one-sided linear growth condition. The authors improve the results using polynomial growth condition in [5, 6]. After that, these conditions were mentioned under regime systems [7–9].Replacing the linear growth condition or the one-sided linear growth condition, we impose the so-called polynomial growth condition on the functionf for (1.1).Assumption B.
For eachi∈𝕊, there exist nonnegative constants α,κi,κ¯i,γ,K and probability measures μ, on [-τ,0] such that
(2.1)〈ϕ(0),f(φ,i,t)〉≤|φ(0)|2[κi|φ(0)|α+κ¯i∫-τ0‍|φ(θ)|αdμ(θ)+γ]+K|ϕ(0)|
for any φ∈C([-τ,0];Rn).Theorem 2.1.
Under the conditions of AssumptionsA and B, if 2β>α, and σ(i)≠0 for i=1,2,…,n, there almost surely exists a unique globally solution x(t) to (1.1) on t≥-τ for any given initial data φ∈C([-τ,0];Rn).Proof.
Since the coefficients of (1.1) are locally Lipschitz, there is a unique maximal local solution x(t) on t∈[-τ,τe), where τe is the explosion time. In order to prove this solution is global, we need to show that τe=∞ a.s. Let m0>0 be sufficiently large such that 1/m0<min-τ≤θ≤0|ξ(θ)|<max-τ≤θ≤0|ξ(θ)|<m0. For each m≥m0, we define the stopping time
(2.2)τm=inf{t∈[-τ,τe):xi(t)∉(1m,m)forsomei=1,2,…,n}.
Clearly τm is increasing as m→∞. Set τ∞=limm→∞τm, if we can obtain that τ∞=∞ a.s., then τe=∞ a.s. for all t≥0. That is, to complete the proof, also equivalent to prove that, for any t>0,P(τm≤t)→0 as m→∞. If this conclusion is false, there is a pair of constants T>0 and ɛ∈(0,1) such that
(2.3)P{τ∞≤T}>ɛ.
So there exists an integer m1≥m0 such that
(2.4)P{τm≤T}>ɛ∀m≥m1.
To prove the conclusion that we desired, for anyp∈(0,1/2), define a C2-function: Rn×𝕊→R¯+ by
(2.5)V(x,k)=c(k)[1+|x|2]p,
where {c(k),1≤k≤N} is positive constant sequence. Applying the generalized Itô formula,
(2.6)dV(x,k)=d[c(k)(1+|x|2)p]=ℓV(φ,k)dt+2c(k)p(1+|x|2)p-1[q(k)|x|2dW1(t)+σ(k)|x|β+2dW2(t)],
where ℓV is computed as
(2.7)ℓV(φ,k)=pc(k)(1+|x|2)p-2[(k)|x|2β+42(1+|x|2)〈x,f(φ,k,t)〉+q2(k)|x|2+σ2(k)|x|2β+2+(2p-1)(q2(k)|x|4+σ2(k)|x|2β+4)]+∑l=1N‍γklV(φ(0),l).
Letq~=maxk,l∈𝕊{c(l)/c(k)}. For any k,l∈𝕊, we get
(2.8)V(x,l)=c(l)(1+|x|2)p≤q~c(k)(1+|x|2)p=q~V(x,k).
Therefore,(2.9)∑l=1N‍γklV(x,l)≤q~∑l=1N‍|γkl|V(x,k).
According to AssumptionB, the first term in (2.7)
(2.10)pc(k)(1+|x|2)p-2[2(1+|x|2)〈x,f(φ,k,t)〉]≤2c(k)p(1+|x|2)p-1[κk|x|α+2+γ|x|2+K|x|]+2c(k)κ¯kp(1+|x|2)p-1|x|2∫-τ0‍|x(t+θ)|αdμ(θ).
By the Young inequality and noting thatp∈(0,1/2), it is obvious that
(2.11)2c(k)κ¯kp(1+|x|2)p-1|x|2∫-τ0‍|x(t+θ)|αdμ(θ)≤2c(k)κ¯kp(1+|x|2p)∫-τ0‍|x(t+θ)|αdμ(θ)≤2c(k)κ¯kp∫-τ0‍|x(t+θ)|αdμ(θ)+4c(k)κ¯kp2α+2p|x|α+2p+2c(k)ακ¯kpα+2p∫-τ0‍|x(t+θ)|α+2pdμ(θ)≤2c(k)κ¯kp(|x|α+|x|α+2p)+2c(k)κ¯kp{∫-τ0‍|x(t+θ)|αdμ(θ)-|x|α}+2c(k)ακ¯kpα+2p{∫-τ0‍|x(t+θ)|α+2pdμ(θ)-|x|α+2p},
where the first inequality we have used the elementary inequality: for any a,b≥0 and r∈(0,1), (a+b)r≤ar+br.  Therefore we have
(2.12)d[c(k)(1+|x|2)p]≤{Hk(x)+2c(k)κ¯kp[∫-τ0‍|x(t+θ)|αdμ(θ)-|x|α]+2c(k)ακ¯kpα+2p[∫-τ0‍|x(t+θ)|α+2pdμ(θ)-|x|α+2p]}dt+2c(k)p(1+|x|2)p-1[q(k)|x|2dW1(t)+σ(k)|x|β+2dW2(t)],
where
(2.13)Hk(x)=2c(k)κ¯kp(|x|α+|x|α+2p)+pc(k)(1+|x|2)p-2[q2(k)|x|2+σ2(k)|x|2β+2+(2p-1)(q2(k)|x|4+σ2(k)|x|2β+4)]+2c(k)p(1+|x|2)p-1[κk|x|α+2+γ|x|2+K|x|]+∑l=1N‍γklV(φ(0),l).
Using the elementary inequality: for any a,b≥0 and r≥1, (a+b)r≤2r-1(ar+br), we obtain that (1+|x|2)2-p≤21-p[1+|x|4-2p], also we have
(2.14)(1+|x|2)2-p∑l=1N‍γklV(x,l)≤q~∑l=1N‍|γkl|21-p(1+|x|4-2p)(1+|x|2p).
Therefore,
(2.15)Hk(x)≤(1+|x|2)2-pHk(x)≤22-pc(k)κ¯kp[1+|x|4-2p](|x|α+|x|α+2p)+pc(k)[q2(k)|x|2+σ2(k)|x|2β+2+(2p-1)(q2(k)|x|4+σ2(k)|x|2β+4)]+2c(k)p(1+|x|2)[κk|x|α+2+γ|x|2+K|x|]+q~∑l=1N‍|γkl|21-p(1+|x|4-2p)(1+|x|2p).
Noting that p∈(0,1/2) and 2β>α≥0, by the boundedness property of polynomial functions, there exists a positive constant Mk such that Hk(x)≤(1+|x|2)2-pHk(x)≤Mk.  Taking expectation from two sides of (2.6) leads to
(2.16)𝔼V(x(τm∧T),r(τm∧T))=𝔼V(x(0),r(0))+𝔼∫0τm∧T‍ℓV(xs,r(s))ds,
and from (2.12) and (2.15), we have
(2.17)𝔼V(x(τm∧T),r(τm∧T))≤𝔼V(x(0),r(0))+MkT+2cˇκ¯kp𝔼{∫0τm∧T‍[∫-τ0‍|x(t+θ)|αdμ(θ)-|x|α]ds}+2cˇακ¯kpα+2p𝔼{∫0τm∧T‍[∫-τ0‍|x(t+θ)|α+2pdμ(θ)-|x|α+2p]ds},
where we denote cˇ=maxk∈𝕊,1≤i≤nci(k).
By the Fubini theorem and a substitution technique, we may compute that(2.18)∫0τm∧T‍∫-τ0‍|x(s+θ)|α+2pdμ(θ)ds-∫0τm∧T‍|x(s)|α+2pds=∫-τ0‍dμ(θ)∫θτm∧T+θ‍|x(s)|α+2pds-∫0τm∧T‍|x(s)|α+2pds≤∫-τ0‍dμ(θ)∫-ττm∧T‍|x(s)|α+2pds-∫0τm∧T‍|x(s)|α+2pds≤∫-τ0‍|ξ(s)|α+2pds.
Similarly,
(2.19)∫0τm∧T‍∫-τ0‍|x(s+θ)|αdμ(θ)ds-∫0τm∧T‍|x(s)|αds≤∫-τ0‍|ξ(s)|αds.
Therefore we rewrite (2.17) into
(2.20)𝔼V(x(τm∧T),r(τm∧T))≤V(ξ(0),r(0))+MT+2cˇκ¯kp∫-τ0‍𝔼|ξ(s)|αds+2cˇακ¯kpα+2p∫-τ0‍𝔼|ξ(s)|α+2pds≔KT,
where KT is bounded and KT is independent of m.
By the definition ofτm,x(τm)=m or 1/m, so let Ωm={τm≤T} for m≥m1 and by (2.6), noting that for every ω∈Ωm, there is some m such that xm(τm,ω) equals either m or 1/m hence
(2.21)P(τm≤T)mini∈𝕊{c(i)(1+m2)p∧c(i)(1+1m2)p}≤P(τm≤T)V(x(τm),i)≤𝔼[I{τm≤T}V(x(τm∧T),r(τm∧T))]≤𝔼V(x(τm∧T),r(τm∧T))≤KT.
Letting m→∞ implies that
(2.22)limsupm→∞P(τm≤T)=0.
So we must obtain τ∞=∞ a.s., as required. The proof is complete.


## 3. Asymptotic Boundedness

Theorem2.1 shows that the solution of SDE (1.1) exists globally and will not explode under some reasonable conditions. In the study of stochastic system, stochastically ultimate boundedness is more important topic comparing with nonexplosion of the solution, which means that the solution of this system will survive under finite boundedness in the future. Here we examine the 2pth moment boundedness.Lemma 3.1.
Under the conditions of Theorem2.1, for any p∈(0,1/2), there exists a constant Kp independent on the initial data such that the global solution x(t) of SDE (1.1) has the property that
(3.1)limsupt→∞𝔼(|x(t)|2p)≤Kp.Proof.
First, Theorem2.1 indicates that the solution x(t) of (1.1) almost surely remain in Rn for all t≥-τ with probability 1.
Applying the Itô  formula toeɛtV(x,k) and taking expectation yields
(3.2)𝔼V(x,k)=e-ɛtV(ξ(0),r(0))+e-ɛt𝔼∫0t‍eɛs[ℓV(xs,k)+ɛV(x(s),k)]ds.
Here ℓV(φ,k) is defined as before (2.7).
Now we consider the function(3.3)Φk(x)=2c(k)κ¯kp(eɛτ|x|α+αeɛτ+2pα+2p|x|α+2p)+pc(k)(1+|x|2)p-2[q2(k)|x|2+σ2(k)|x|2β+2+(2p-1)(q2(k)|x|4+σ2(k)|x|2β+4)]+2c(k)p(1+|x|2)p-1[κk|x|α+2+γ|x|2+K|x|]+[cˇq~∑l=1N‍|γkl|+ɛcˇ](1+|x|2p).
Similar to the proof of Theorem2.1, then we know (3.3) is upper bounded; there exists constant ψk such that Φk(x)≤(1+|x|2)2-pΦk(x)≤ψk≤Ψ:=max1≤k≤Nψk; therefore, (3.2) implies that
(3.4)𝔼V(x,k)≤e-ɛtV(ξ(0),r(0))+e-ɛt𝔼∫0t‍eɛs[Ψ+2c(k)κ¯kp(∫-τ0‍|x(s+θ)|αdμ(θ)-eɛτ|x(s)|α)+2c(k)ακ¯kpα+2p(∫-τ0‍|x(s+θ)|α+2pdμ(θ)-eɛτ|x(s)|α+2p)]ds.
We have the following calculus transformation:(3.5)∫0t‍eɛs∫-τ0‍|x(s+θ)|αdμ(θ)ds-∫0t‍eɛ(s+τ)|x(s)|αds=∫-τ0‍dμ(θ)∫θt+θ‍eɛ(s-θ)|x(s+θ)|αds-∫0t‍eɛ(s+τ)|x(s)|αds≤∫-τ0‍dμ(θ)∫-τt‍eɛ(s+τ)|x(s)|αds-∫0t‍eɛ(s+τ)|x(s)|αds≤eɛτ∫-τ0‍eɛθ|ξ(θ)|αdθ<∞.
Similarly,
(3.6)∫0t‍∫-τ0‍eɛs|x|α+2p(s+θ)dμ(θ)ds-∫0t‍eɛ(s+τ)|x|α+2p(s)ds≤eɛτ∫-τ0‍eɛθξiα+2p(θ)dθ<∞.
We therefore have from (3.4)
(3.7)𝔼V(x,k)≤e-ɛtV(ξ(0),r(0))+Ψɛ-1(1-e-ɛt)+2cˇκkpe-ɛ(t-τ)𝔼∫-τ0‍eɛθ|ξ(θ)|αdθ+e-ɛ(t-τ)2cˇακkpα+2p𝔼∫-τ0‍eɛθ|ξ(θ)|α+2pdθ.
Clearly,(3.8)limsupt→∞𝔼V(x,k)≤Ψɛ-1;
denote c^=min{ci(k):1≤i≤n,k∈𝕊}; therefore,
(3.9)c^limsupt→∞𝔼|x|2p≤limsupt→∞𝔼[c(k)(1+|x|2)p]≤limsupt→∞𝔼V(x,k),
which yields
(3.10)limsupt→∞𝔼|x|2p≤Ψc^ɛ=:Kp.
This means that the solution is bounded in the 2pth moment; the stochastically ultimate boundedness will follow directly.Definition 3.2.
The solutionsx(t) of SDE (1.1) are called stochastically ultimately bounded, if for any ϵ∈(0,1), there is a positive constant χ(=χ(ϵ)), such that the solution of SDE (1.1) with any positive initial value has the property that
(3.11)limsupt→+∞P{|x(t)|>χ}<ϵ.Theorem 3.3.
The solution of (1.1) is stochastically ultimately bounded under the condition Lemma 3.1; that is, for any ϵ∈(0,1), there is a positive constant χ(=χ(ϵ)), such that for any positive initial value the solution of Lemma 3.1 has the property that
(3.12)limsupt→+∞P{|x(t)|>χ}<ϵ.Proof.
This can be easily verified by Chebyshev's inequality and Lemma3.1 by choosing χ=(Kp/ϵ)1/p sufficiently large because of the following
(3.13)limsupt→+∞P(|x(t)|>χ)≤limsupt→+∞𝔼[|x|p]χp.


## 4.  Stabilization of Noise

From Sections2 and 3, we know that under the condition σ(i)≠0 and 2β>α, the Brownian noise σ(i)|x(t)|βx(t)dW1(t) can suppress the potential explosion of the solution and guarantee this global solution to be bounded in the sense of the 2pth moment. Clearly, the boundedness results are also dependent only on the choice of β under the condition σ(i)≠0 and independent of q(i). This implies that the noise W1(t) plays no role to guarantee existence and boundedness of the global solution to (1.1). This section is devoted to consider the effect of noise q(i)x(t)dW1(t), we will show that the system (1.1) is exponential stability if for some sufficiently large q(i).For the purpose of stability study, we impose the following the general polynomial growth condition.Assumption C.
For eachi∈𝕊, there exist nonnegative constants α,κi,κ¯i,γ,andK and probability measures μ, on [-τ,0] such that
(4.1)|f(φ,i,t)|≤|φ(0)|[κi|φ(0)|α+κ¯i∫-τ0‍|φ(θ)|αdμ(θ)+γ]
for any φ∈C([-τ,0];Rn).
Clearly, AssumptionC is stronger than the one-sided polynomial growth condition Assumption B. Therefore, Theorems 2.1 and Lemma 3.1 still hold under Assumption C.
In [10, Page 165], for a given nonlinear SDE with Markovian switching
(4.2)dx(t)=f(x(t),r(t),t)dt+g(x(t),r(t),t)dB(t),
any solution starting form a nonzero state will remain to be non-zero. But for the system (1.1)
(4.3)dx(t)=f(xt,r(t),t)dt+q(r(t))x(t)dW1(t)+σ(r(t))|x(t)|βx(t)dW2(t),
the drift coefficient is a functional, so we will prove this non-zero property under Assumption C.Lemma 4.1.
Letx(t) be the global solution of (1.1). Under Assumption C, if 2β>α and σ(i)≠0, for any non-zero initial data x(0)≠0(4.4)P(x(t)≠0)=1,t≥0;
that is, almost all the sample path of any solution starting from a non-zero state will never reach the origin.Proof.
For any initial dataξ∈C([-τ,0];Rn) satisfying x(0)≠0, for sufficiently large positive number i0, such that |x(0)|>1/i0. For each integer i≥i0, define the stopping time
(4.5)ρi=inf{t≥0:|x(t)|≤1i}.
Clearly, ρi is increasing as i→∞ and ρi→ρ∞ a.s. If we can show that ρ∞=∞ a.s., the desired result P(x(t)≠0)=1 on t≥0 follows. This is equivalent to proving that, for any t>0,P(ρi≤t)→0 as i→∞.
To prove this statement, define aC2-function
(4.6)V1(x,k)=c(k)[x-12logx],
where V1(·)>0 and V1(0+)=∞. Applying the Itô formula and taking the expectation yield
(4.7)𝔼V1(|x(t∧ρi),r(t∧ρi)|)=𝔼V1(|x(0,r(0))|)+𝔼∫0t∧ρi‍ℓV1(xs,r(s))ds,
where ℓV1 is defined as
(4.8)ℓV1(φ,k)=12c(k)(|φ(0)|-3/2-|φ(0)|-2)φT(0)f(φ,t,k)-σ2(k)c(k)8(|φ(0)|2β+(1/2)-2|φ(0)|2β)-q2(k)c(k)8(|φ(0)|1/2-2)+∑l=1N‍γklV(φ(0),l),
for any φ∈C([-τ,0];Rn). By Assumption C and the Young inequality, the first term of (4.8) will be written as
(4.9)(|φ(0)|-3/2-|φ(0)|-2)φT(0)f(φ,t,k)≤(|φ(0)|-1/2+|φ(0)|-1)|f(φ,t,k)|≤(|φ(0)|1/2+1)[κk|φ(0)|α+κ¯k∫-τ0‍|φ(θ)|αdμ(θ)+γ]≤(κk+κ¯k)(|φ(0)|α+1/2+|φ(0)|α)+γ|φ(0)|1/2+γ+ακ¯kα+(1/2)[∫-τ0‍|φ(0)|α+(1/2)dμ(θ)-|φ(0)|α+(1/2)]+κ¯k[∫-τ0‍|φ(0)|αdμ(θ)-|φ(0)|α].
Substituting (4.9) into (4.8) gives
(4.10)ℓV1(φ,k)≤H^k(φ(0))+c(k)ακ¯kα+(1/2)[∫-τ0‍|φ(0)|α+(1/2)dμ(θ)-|φ(0)|α+(1/2)]+c(k)κ¯k[∫-τ0‍|φ(0)|αdμ(θ)-|φ(0)|α],
where
(4.11)H^k(x)≤-σ2(k)c(k)8(|x|2β+(1/2)-2|x|2β)-q2(k)c(k)8(|x|1/2-2)+c(k)(κk+κ¯k)(|x|α+(1/2)+|x|α)+γ|x|1/2+γ+q~∑l=1N‍|γkl|c(k)(|x|1/2-log|x|)≤-σ2(k)c(k)8(|x|2β+(1/2)-2|x|2β)-q2(k)c(k)8(|x|1/2-2)-q~∑l=1N‍|γkl|c(k)log|x|+c(k)(κk+κ¯k)(|x|α+1/2+|x|α)+γ|x|1/2+γ+q~∑l=1N‍|γkl|c(k)|x|1/2.
Noting (2.9) and 2β>α,σ(k)≠0, using the boundedness property of polynomial functions, there exists a constant M^k such that H^k(x)≤M^k≤M^=:max1≤k≤N{M^k}.
Furthermore, we may estimate that(4.12)∫0t∧ρi‍[∫-τ0‍|x(s+θ)|α+(1/2)dμ(θ)-|x(s)|α+(1/2)]ds≤∫-τ0‍|ξ(s)|α+(1/2)ds,∫0t∧ρi‍[∫-τ0‍|x(s+θ)|αdμ(θ)-|x(s)|α]ds≤∫-τ0‍|ξ(s)|αds.
It therefore follows that
(4.13)𝔼V1(|x(t∧ρi),r(t∧ρi)|)≤𝔼V1(|x(0,r(0))|)+M^t+∫-τ0‍|ξ(s)|α+(1/2)ds+∫-τ0‍|ξ(s)|αds=:K^t.
We know that(4.14)12logiℙ(ρi≤t)≤ℙ(ρi≤t)(1i-12log1i)≤𝔼[I{ρi≤t}]V1(|x(ρi≤t)|)≤𝔼V1(|x(ρi≤t)|)≤K^t,
which implies that
(4.15)limsupi→∞ℙ(ρi≤t)≤limi→∞2K^tlogi=0,
as required. The proof is completed.This lemma shows that almost all the sample path of any solution of (1.1) starting from a non-zero state will never reach the origin. Because of this nice property, the Lyapunov functions we can choose need not be imposed globally but only in a deleted neighborhood of the origin.Especially, the hybrid system  always switch from any regime to another regime,   so it is reasonable to assume that the Markov chainr(t) is irreducible.  It means  to  the condition  that irreducible Markov chain has a unique stationary probability distribution π=(π1,π2,…,πN)∈R1×N which can be determined by solving the following linear equation πΓ=0 subject to ∑k=1Nπk=1 and πk>0 for any k∈𝕊, where Γ is generator Γ=(γuv)N×N.Theorem 4.2.
Suppose the Markov chainr(t) is irreducible, under Assumption A and C, if for δ∈(0,1),k∈𝕊,σ(k)≠0 and 2β>α, the solution x(t) of SDE (1.1) with any initial data ξ∈C([-τ,0];Rn) satisfying x(0)≠0 has the property
(4.16)limsupt→∞1tlog|x(t)|≤∑j=1N‍πj[ϕj-qj22]a.s.,
where
(4.17)ϕ(k)=maxx>0{-σ2(k)2|x|2β+(κk+κ¯k)|x|α+γ}.
In particular, the nonlinear hybrid system (1.1) is almost surely exponentially stable if
(4.18)∑j=1N‍πj[ϕj-qj22]<0.Proof.
By Theorem2.1 and Lemma 4.1, (1.1) almost surely admits a global solution x(t) for all t≥0 and x(t)≠0 almost surely. Applying the Itô formula to the function log|x(t)| leads to
(4.19)log|x(t)|=log|x(0)|+∫0t‍{|x(t)|-2〈x(s),f(xs,r(s),s)〉-12[σ(r(s))2|x(s)|2β+q2(r(s))]}ds+∫0t‍q(r(s))dW1(s)+σ(r(s))|x(s)|βdW2(s).
Define M(t)=∫0t‍σ(r(s))|x(s)|βdW2(s); clearly M(t) is a continuous local martingale with the quadratic variation
(4.20)〈M(t),M(t)〉=∫0t‍σ2(r(s))|x(s)|2βds.
For any δ∈(0,1), choose ϑ>0 such that δϑ>1 and each positive integer n>0; the exponential martingale inequality yields
(4.21)ℙ{sup0≤t≤n[M(t)-δ2∫0t‍σ2(r(s))|x(s)|2βds]≥lognϑ}≤1nδϑ.
Since∑n=1∞n-δϑ<∞, by the Borel-Cantelli lemma, there exists an Ω0⊂Ω with ℙ(Ω0)=1 such that for any ω∈Ω0, there exists an integer n(ω), when n>n(ω) and n-1≤t≤n,
(4.22)M(t)≤δ2∫0t‍σ2(r(s))|x(s)|2βds+ϑδlog(t+1).
This, together with Assumption C, denote q~=:maxk∈𝕊{q(k)} and κ¯~=:maxk∈𝕊{κ¯(k)}; noting the definition of (4.17), we therefore have
(4.23)log|x(t)|≤log|x(0)|+∫0t‍[-σ2(r(s))(1-δ)2|x(s)|2β+κ(r(s))|x(s)|α+κ¯(r(s))∫-τ0‍|x(s+θ)|αdμ(θ)+γ-q2(r(s))2]ds+q~W1(t)+ϑδlog(t+1)≤log|x(0)|+∫0t‍[ϕδ(r(s))-q2(r(s))2]ds+κ¯~∫-τ0‍|ξ(s)|αds+q~W1(t)+ϑδlog(t+1),
where
(4.24)ϕδ(k)=-σ2(k)(1-δ)2|x|2β+(κ(k)+κ¯(k))|x|α+γ∫0t‍∫-τ0‍|x(s+θ)|αdμ(θ)ds-∫0t‍|x(s)|αds≤∫-τ0‍|ξ(s)|αds.
Applying the strong law of large number [2, Page 12] to the Brownian motion, we therefore have
(4.25)limt→∞W1(t)t=0a.s..
Moreover, letting δ→0, by the ergodic property of the Markov chain, we have
(4.26)limt→∞1t∫0t‍[ϕ(r(s))-q2(r(s))2]ds=∑j=1N‍πj[ϕj-qj22]a.s..
Combined (4.25) and (4.26), it follows from (4.23)
(4.27)limsupt→∞1tlog|x(t)|≤∑j=1N‍πj[ϕj-qj22]a.s..
Thus the assertion (4.16) follows.
Clearly, if(4.28)∑j=1N‍πj[ϕj-qj22]<0,
system (1.1) is almost surely exponentially stable; the proof is completed.Remark 4.
This results is generation of Theorem 4.2 in [6]. The author consider functional differential equation:
(4.29)dx(t)=f(xt,t)dt+qx(t)dW1(t)+σ|x(t)|βx(t)dW2(t);
for example,
(1.1)1dx(t)=x(t)[h+ax(t)+b∫-τ0‍x(t+θ)dμ(θ)]dt+qx(t)dW1(t)+σx2(t)dW2(t),
with β=1,α=1,κ=|a|,κ¯=|b|,γ=|h|,σ=1,
(4.30)ϕ=maxx>0{-12x2+(|a|+|b|)x+|h|}=|h|+12(|a|+|b|)2;
hence, we choose q2/2>|h|+1/2(|a|+|b|)2 satisfying ϕ-(q2/2)<0; the stochastic functional system (1.1)1 is almost surely exponentially stable. We can observe the numerical simulation in Figure 1.Figure 1
We chooseh=a=b=1/4, q=2, σ=1 in (1.1)1Example 4.3.
Let us assume that the Markov chainr(t) is on the state space S={1,2} with the generator
(4.31)Γ=(-γ12γ12γ21-γ21),
where γ12>0 and γ21>0. It is easy to see that the Markov chain has its stationary probability distribution π=(π1,π2) given by
(4.32)π1=γ21γ12+γ21,π2=γ12γ12+γ21.
As pointed out in Section,we may regard SDE (1.1) as the result of the following two equations:
(1.1)21dx(t)=f(xt,1,t)dt+q(1)x(t)dW1(t)+σ(1)|x(t)|βx(t)dW2(t),(1.1)22dx(t)=f(xt,2,t)dt+q(2)x(t)dW1(t)+σ(2)|x(t)|βx(t)dW2(t).
Noting that∑i=1nπi[ϕ(i)-(q2(i)/2)] has the form
(4.33)∑i=12‍πi[ϕ(i)-q2(i)2]=γ21γ12+γ21[ϕ(1)-q2(1)2]+γ12γ12+γ21[ϕ(2)-q2(2)2],
that is, for a given state 1, (1.1)21 may be written as
(4.34)dx(t)=x(t)[1+x(t)+∫-τ0‍x(t+θ)dμ(θ)]dt+x(t)dW1(t)+x2(t)dW2(t),
with x(0)=1 when t≥0. Applied the condition (4.17) with q(1)=1,σ(1)=κ(1)=κ¯(1)=γ=α=β=1 to the system (4.34) yields
(4.35)ϕ(1)=maxx>0{-12x2+2x+1}=3,
satisfying ϕ(1)-(q2(1)/2)>0, which shows that the trajectory of (4.34) will not satisfied the conditions of Theorem 4.2 in [6] although it has global solution.
However, as the result of Markovian switching, the overall behavior, that is SDE (1.1) will be almost surely exponentially stable as long as
(4.36)ϕ(2)-q2(2)2<0,∑i=12‍πi[ϕ(i)-q2(i)2]<0,
namely, the transition rate γ21 from (1.1)22 to (1.1)21 is less than the transition rate γ12 from (1.1)21 to (1.1)22, which ensure that
(4.37)γ21γ12+γ21[ϕ(1)-q2(1)2]+γ12γ12+γ21[ϕ(2)-q2(2)2]<0.Example 4.4.
Consider another stochastic differential equation with Markovian switching, wherer(t) is a Markov chain taking values in S={1,2,3}. Here subsystem of (1.1) is writtern as three different equations:
(1.1)31dx(t)=x(t)[14+14x(t)+14∫-τ0‍x(t+θ)dμ(θ)]dt+2x(t)dW1(t)+x2(t)dW2(t),
where q(1)=2, σ(1)=1, κ(1)=κ¯(1)=1/4, γ=1/4, α=β=1,
(1.1)32ϕ(1)=maxx>0{-12x2+(14+14)x+14}=38,ϕ(1)-12q2(1)=-138<0;dx(t)=x(t)[1+x2(t)+∫-τ0‍x2(t+θ)dμ(θ)]dt+2x(t)dW1(t)+x3(t)dW2(t),
where q(2)=2, σ(2)=κ(2)=κ¯(2)=γ=1, α=β=2,
(1.1)33ϕ(2)=maxx>0{-12x4+2x2+1}=3,ϕ(2)-12q2(2)=1>0;dx(t)=x(t)[1-2x2(t)-6∫-τ0‍x2(t+θ)dμ(θ)]dt+4x(t)dW1(t)+2x3(t)dW2(t),
where q(3)=4, σ(3)=κ(3)=2, κ¯(3)=6, γ=1, α=β=2,
(4.38)ϕ(3)=maxx>0{-2x4+8x2+1}=9,
we compute
(4.39)ϕ(3)-12q2(3)=1>0.Case 1.
Let the generator of the Markov chainr(t) be
(4.40)Γ=(-2113-4111-2).
By solving the linear equation πΓ=0 subject to ∑k=1Nπk=1 and πk>0 for any k∈𝕊, we obtain the unique stationary (probability) distribution
(4.41)π=(π1,π2,π3)=(715,15,13).
Then
(4.42)∑i=13‍πi[ϕ(i)-12q2(i)]=-27120<0.Case 2.
Suppose the generator of the Markov chainr(t) be
(4.43)Γ=(-5231-1030-3).
By solving the linear equationπΓ=0 subject to ∑k=1Nπk=1 and πk>0 for any k∈𝕊, we obtain the unique stationary distribution
(4.44)π=(π1,π2,π3)=(14,12,14).
Then(4.45)∑i=13‍πi[ϕ(i)-12q2(i)]=1132>0.
Therefore, by Theorems 4.2, System (1.1) is almost surely exponentially stable in Case 1.
We can see the impact of the Markov chainr(t). The distribution (π=(π1,π2,…,πn)) of r(t) plays a very important role, which, combined with ∑i=1nπi[ϕ(i)-(1/2)q2(i)]<0, determine that system (1.1) is almost surely exponentially stable. If r(t) spends enough time in the “good” states (the state where ϕ(i)-(q2(i)/2)<0 for some i), even if there exist some “bad” states (the states where ϕ(i)-(q2(i)/2)>0 for some i), the system (1.1) will still be almost surely exponentially stable.



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*Source: 206438-2012-10-14.xml*