# Existence, Uniqueness, and Almost Sure Exponential Stability of Solutions to Nonlinear Stochastic System with Markovian Switching and Lévy Noises

Authors: Chao Wei
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with existence, uniqueness, and almost sure exponential stability of solutions to nonlinear stochastic system with Markovian switching and Lévy noises. Firstly, the existence and uniqueness of solutions to the system is studied. Then, the almost sure exponential stability of the system is derived. Finally, an example is presented to illustrate the results.


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## Body

## 1. Introduction

In practical engineering applications, some linear systems may be affected by external factors or internal causes to appear nonlinear phenomena, such as deformation of the spring due to excessive external force, sensor measurement due to environmental influences, or their own causes leading to the input and output show a nonlinear relationship. This makes nonlinearity an important factor that cannot be ignored. On the contrary, almost all dynamic systems are nonlinear. If the input and output of the system are not proportional, the system can be regarded as a nonlinear system. Moreover, most systems are affected by some kind of noise, and such noise usually shows a certain random characteristic. Therefore, it is reasonable and necessary to model the actual system with a nonlinear stochastic system. With the development of science and technology, nonlinear stochastic systems have attracted the attention of many scholars ([1, 2]). However, in engineering practice, most random disturbances exhibit non-Gauss characteristics. In recent years, with the development of the Lévy process theory, Lévy noise, as an important type of non-Gauss noise, has been widely used in engineering systems, economic systems, and management systems and has attracted great attention from scholars. Bertoin [3] introduced the basic theory of the Lévy process. Applebaum [4] introduced stochastic integral and stochastic differential equation driven by Lévy process. Applebaum [5] proposed that Lévy noise can be decomposed into a continuous part and a jump part which, respectively, corresponds to the diffusion and jump term in systems by Lévy-Itô decomposition. Wong and Guan [6] used fast Fourier transform to solve option pricing in Lévy process. Liao et al. [7] studied pricing power options with generalized jump diffusion.During actual project operation, the system may appear between systems described by homogeneous models with different coefficients due to factors such as component failures and repairs, changes in the connection methods of subsystems, and changes in operating points of nonlinear systems after linearization [8]. Obviously, the deterministic model cannot accurately represent this switching relationship between different subsystems nor can it achieve the effect expected in engineering. By studying the nature of the Markov process, the scholars found that it can well reflect this switching relationship between subsystems, and it has been introduced to simulate such systems. In general, the state of a continuous Markov transition system consists of two parts. One part is the modal state of the system, and the other part is the system state. Therefore, the Markov transition system is sometimes called the hybrid system. Over the past few decades, many authors studied the systems with Markovian switching. For example, Xi [9] studied the stability of system with Markovian switching and jump diffusions. Huang and Mao [10] applied Holder’s inequality, Chebyshev’s inequality, and Borel–Cantelli’s lemma to analyze the almost sure stability of nonlinear time delayed stochastic systems with Markovian switching and jump diffusion. Zhou et al. [11] studied slide mode and delay-dependent adaptive exponential synchronization in pth moment for stochastic delayed neural networks with Markovian switching. Wu et al. [12] discussed the stabilization of stochastic coupled systems with Markovian switching via feedback control based on discrete-time observations.For any practical system, it will always be disturbed by various random factors during its normal operation. In this case, whether the system can maintain the predetermined working state safely is a basic and important research content. In the past few decades, the problem of stability of systems has been investigated by many authors [13–16]. For example, Liu et al. [17] constructed an equivalent set of delay differential algebraic equations to study the stability of systems modeled as neutral delay differential equations. Wu et al. [18] analyzed the almost sure globally asymptotical stability and almost sure exponential stability for switched systems with semi-Markovian switching, Markovian switching, and renewal process switching signals, respectively. Li et al. [19] used piecewise constant Lyapunov matrices and the generalized Kalman–Yakubovich–Popov lemma to discuss strong delay-independent stability of linear time-invariant systems. Song and Xu [20] investigated the stability of the metastable low and high concentration states under non-Gaussian noise based on the stochastic basin of attraction. However, there are few literature studies about the existence, uniqueness, and almost sure exponential stability of solutions to nonlinear stochastic system with Markovian switching and Lévy noises. This paper uses probability analysis methods to study the existence and uniqueness of global solution to nonlinear stochastic system with Markovian switching and Lévy noises under the condition of local Lipschitz condition and local linear growth condition. Then, the almost sure exponential stability of the solution is proved by the generalized Itô formula, the law of large numbers for local martingale, and the ergodic of Markovian chain.This paper is organized as follows. In Section2, we introduce the nonlinear stochastic system with Markovian switching and Lévy noises. Some important assumptions, definitions, and lemmas are given as well. In Section 3, the existence and the almost sure exponential stability of the global unique solution to the system are studied. In Section 4, a numerical example is provided. The conclusion is given in Section 5.


## 2. Problem Formulation and Preliminaries

LetΩ,ℱ,ℙ be a complete probability space equipped with a right continuous and increasing family of σ-algebras ℱtt≥0. Denoted by C1,2ℝn×ℝ+×S;ℝ+, the family of positive real-valued functions Vx,t,i defined on ℝn×ℝ+×S are continuously twice differentiable in x∈ℝn and once differentiable in t∈ℝ+. Let rt,t≥0 be a right-continuous Markov chain on the probability space taking values in a finite state space S=1,2,…,N with generator Γ=γijN×N given by(1)ℙrt+Δ=jrt=i=γijΔ+oΔ,i≠j,1+γiiΔ+oΔ,i=j,where Δ>0 and γij≥0 is the transition rate from i to j if i≠j while γii=−∑i≠jγij.We consider the following nonlinear stochastic system:(2)dxt=fxt,t,rtdt+gxt,t,rtdBt+∫YGxt−,t,rt−,yNdt,dy,where x0=x0∈ℱ0, r0=r0∈S, xt−=lims↓txs, Bt is an m-dimensional standard Brownian motion, Nt,y is a Poisson random measure on 0,+∞×ℝn with compensator N˜t,y which satisfies N˜t,y=Ndt,dy−νdydt, νdy is a Lévy measure, f:ℝn×ℝ+×S⟶ℝn, g:ℝn×ℝ+×S⟶ℝn×m, and G:ℝn×ℝ+×S×ℝn⟶ℝn. We assume that Bt, Nt,y, and rt in system (1) are independent, and Markov chain is ergodic. Then, Markov chain has a unique stable distribution π=π1,π2,…,πN∈ℝ1×N. π can be solved by the following equation:(3)πΓ=0s.t.∑j=1Nπj=1,∀j∈S.We put the following assumptions, definitions, and lemmas.Assumption 1.
For eachK>0, there exists LK>0 such that, for any t≥0, x1∨x2≤K and i∈S, fx1,t,i−fx2,t,i2+gx1,t,i−gx2,t,i2+∫YGx1,t,i,y−Gx2,t,i,y2νdy≤Lk.Assumption 2.
There existsVx,t,i∈C1,2ℝn×ℝ+×S;ℝ+ and ms∈L1ℝ+;ℝ+ such that limx⟶∞inft≥0,i∈SVx,t,i=∞ and ℒVx,t,i≤ms.Assumption 3.
f0,t,i≡0, g0,t,i≡0, G0,t,i,y≡0, ∀i∈S, and ∀y∈Y.Definition 1.
The solution of system (1) is said to be almost sure exponential stability if there exists λ>0 satisfying(4)limt⟶∞sup1tlogxt;x0,r0<0,for any x0∈ℱ0 and r0∈S.
GivenV∈C1,2ℝn×ℝ+×S;ℝ+, we define the operator ℒV by(5)ℒVx,t,i=Vtx,t,i+Vxx,t,ifx,t,i+12tracegTx,t,iVxxx,t,igx,t,i+∫Y∑k=1lVx+Gkx,t,i,yk,t,i−Vx,t,iνkdyk+∑j=1NγijVx,t,j.Lemma 1 (see [21]).
LetH=Htt≥0 be a real-valued local martingale and H0=0, then(6)limt⟶∞∫0tdH,Hs1+s2<∞,a.s.,⇒limt⟶∞Htt=0,a.s.Remark 1.
According to Assumption1, it is easy to check that system (1) satisfies local linear growth condition. Since, the local solution of system (1) exists and is unique. In the next section, we prove that the global unique solution exists.


## 3. Existence, Uniqueness, and Almost Sure Exponential Stability of Solutions to Nonlinear Stochastic System with Markovian Switching and Lévy Noises

In the following theorem, the existence and uniqueness of solutions to nonlinear stochastic system with Markovian switching and Lévy noises are derived.Theorem 1.
Suppose that Assumptions1 and 2 hold. Then, the system has a global unique solution xt,t≥0.Proof.
For the given initial valuesx0 and r0, we assume that x0≤η. For k≥η, k∈ℕ, let(7)fkx,t,i=fx∧kxx,t,i,gkx,t,i=gx∧kxx,t,i,where x∧k/xx=0 when x=0.
It is easy to check thatfk and gk satisfy the local Lipschitz condition and linear growth condition. Then, we obtain that(8)dxkt=fkxkt,t,rtdt+gkxkt,t,rtdBt+∫YGxkt−,t,rt−Ndt,dyhas the global unique solution.
Fork∈ℕ, we define the stopping time:(9)αk=inft≥0:xkt≥k,where infϕ=∞.
It can be checked that when0≤t≤αk and xkt=xk+1, namely, αk is increasing. Therefore, there exists a stopping time α such that(10)α=limn⟶∞αk.
Define(11)xt=limk⟶∞xkt,0≤t<α.
Obviously, when0≤t<α, xt is the unique solution of system (1).
We now show thatxt.
Fort≥0, according to Itô formula, we have(12)EVxkt∧αk,t∧αk,rt∧αk=EVxk0,0,r0+E∫0t∧αkℒkVxks,s,rsds,where ℒkVxks,s,rs=ℒVxks,s,rs when 0≤s≤t∧αk.
Then, from Assumption1, we obtain(13)EVxkt∧αk,t∧αk,rt∧αk≤EVx0,0,r0+∫0tmsds<∞.
Moreover, since(14)Pαk≤tinfx≥n,t≥0,i∈SVx,t,i≤∫αk≤tVxkt∧αk,t∧αk,rt∧αkdP≤EVxkt∧αk,t∧αk,rt∧αk,we have(15)Pαk≤t≤EVx0,0,r0+∫0tmsdsinfx≥n,t≥0,i∈SVx,t,i.
Whent⟶∞, it follows that(16)Pα≤t=0.
Therefore,(17)Pα=∞=1.
The proof is complete.In the following theorem, the almost sure exponential stability for the nonlinear stochastic system with Markovian switching and Lévy noises is derived. The conditions of stability only depend on the state of Markovian chain and some constants.Theorem 2.
Suppose that Assumptions1 and 3 hold. For any xt∈ℝn, if there exists a symmetric positive definite matrix F and some constants γi∈ℝ,δi,ξi,σi≥0,i∈S satisfy(18)2xTFfx,t,i+trgTx,t,iFgx,t,i≤γixTFx,(19)xTFgx,t,i2≥δixTFx2,(20)ξix≤Gx,t,i,y+x≤σix,(21)∑i=1Nπiγi−2δi+logλmaxFσi2λminF<0,system (2) is almost sure exponential stability.Proof.
According to the generalized Itô formula, we have(22)logxTtFxt=logx0TFx0+∫0t2xTsFfxs,s,rs+trgTxs,s,rsFgxs,s,rsxTsFxs−2xTsFgxs,s,rs2xTsFxs2+∑j=1NγijlogxTtFxtds+∫0t2xTsFgxs,s,rsxTsFxsdBs+∫0t∫Ylogxs+GHxs−,s,rs−,yTFxs+Gxs−,s,rs−,y−logxTsFxsNds,dy=logx0TFx0+∫0t2xTsFfxs,s,rs+trgTxs,s,rsFgxs,s,rsxTsFxs−2xTsFgs2xTsFxs2ds+∫0t2xTsFgxs,s,rsxTsFxsdBs+∫0t∫Ylogxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,yxTsFxsN˜ds,dy+∫0t∫Ylogxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,yxTsFxsνdyds.
Let(23)H1=∫0t2xTsFgxs,s,rsxTsFxsdBs,H2=∫0t∫Ylogxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,yxTsFxsN˜ds,dy.
It is obvious thatH1 and H2 are local martingale.
Then, the quadratic variation ofH1 satisfies(24)∫0tdH1,H1s1+s2=∫0t4xTsFgxs,s,rs2xTsFxs21+s2ds≤∫0t4Lkxs4F2λmin2Fxs41+s2ds≤4LkF2λmin2F∫0∞11+s2ds<∞.
From equation (12), we obtain(25)logλminFξrs−2λmaxF≤logλminFxs+Gxs−,s,rs−,y|2λmaxFxs2≤logxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,yxTsFxs≤logλmaxFxs+Gxs−,s,rs−,y2λminFxs2≤logλmaxFσrs−2λminF.
Hence,(26)logxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,yxTsFxs2≤H,where H=logλminFξrs−2/λmaxF∨logλmaxFσrs−2/λminF.
Thus,(27)∫0tdH2,H2s1+s2=∫0tlogxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,y/xTsFxs21+s2ds≤L∫0∞11+s2ds<∞.
Therefore, according to Lemma1, it follows that(28)limt⟶∞H1tt=0,(29)limt⟶∞H2tt=0.
From equation (12), we have(30)∫0t∫Ylogxs+Gxs−,s,rs−,yTFxs+Gxs−,s,rs−,yxTsFxsνdyds≤∫0t∫YlogλmaxFxs+Gxs−,s,rs−,y2λminFxs2νdyds≤∫0tlogλmaxFσrs−2λminFds.
Together with equations (10) and (11), it can be checked that(31)logxTtFxt≤logxT0Fx0+H1t+H2t+∫0tγrs−2δrs+logλmaxFσrs−2λminFds.
According to the ergodic of the Markovian chain, we have(32)limt⟶∞1t∫0tγrs−2δrs+logλmaxFσrs−2λminFds=∑i=1Nπiγi−2δi+logλmaxFσi2λminF.
From equations (15)–(18), we obtain(33)limt⟶∞sup1tlogxt=12limt⟶∞sup1tlogxTtFxt≤12∑i=1Nπiγi−2δi+logλmaxFσi2λminF<0.
The proof is complete.


## 4. Example

LetBt be a one-dimensional Brownian motion. The character measure υ of Poisson jump satisfies υdy=ζϕdy, where ζ=1.5 is the intensity of Poisson distribution and ϕ is the probability intensity of the standard normal distributed variable y, F=I2, rt∈S=1,2, and(34)Γ=γij2×2=−553−3.Consider the following nonlinear stochastic system with Markovian switching and Lévy noises:(35)dxt=fxt,t,rtdt+gxt,t,rtdBt+∫YHxt−,t,rt−,yNdt,dy,where(36)fxt,t,1=−2xt+13sinxt,gxt,t,1=12xt,fxt,t,2=−72xt+15sinxt,gxt,t,2=14xt,Hxt,t,1,y=xt2,Hxt,t,2,y=−xt.Then, we obtain(37)π1=0.35,π2=0.65,γ1=−9.26,γ2=−5.43,δ1=0.25,δ2=0.15,ξ1=1.46,ξ2=2.10,σ1=2,σ2=3.Thus, it is easy to check that(38)12∑i=12πiγi−2δi+logλmaxFσi2λminF=−0.1565<0.Therefore, the system is almost sure exponential stability. Figure1 shows the simulation results.Figure 1
State trajectory.


## 5. Conclusion

The aim of this paper is to study the existence, uniqueness, and almost sure exponential stability of solution to nonlinear stochastic system with Markovian switching and Lévy noises. The existence of the global unique solution to the system has been derived. Almost sure exponential stability of the solution has been analyzed by the generalized Itô formula, the law of large numbers for local martingale, and the ergodic of Markovian chain. Further research topics will include the stability for the nonlinear delayed stochastic system with Markovian switching andα stable noises [22].



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