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The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects

The stability with general decay rate of hybrid stochastic fractional differential equations... 1IntroductionWith the wide application of stochastic differential equations driven by Lévy noise in biology, engineering, finance and economy, more and more experts and scholars pay attention to stochastic differential equations [1,2,3]. The stability has become one of the important topics, such as stochastic stability, stochastic asymptotic stability, moment exponential stability, almost everywhere stability and mean square polynomial stability (see [4,5,6, 7,8,9]). Li and Deng [10] studied the almost sure stability with general decay rate of neutral delay stochastic differential equations with Lévy noise, while Shen et al. [11] studied the stability of solutions of neutral stochastic functional hybrid differential equations with Lévy noise. Shen’s conclusion is more specific and universal than Deng’s.As is known to all, the integer order differential equations determine the local characteristics of the function, while the fractional order differential equations describe the overall information of the function in the form of weighting, so it is more flexible and widely used in the model. Abouagwa et al. [12] studied the existence and uniqueness by using Carathéodory approximation under non-Lipschitz conditions. Shen et al. [13] obtained an averaging principle and stability of hybrid stochastic fractional differential equations driven by Lévy noise. Recently, the classical mathematical modelling approach coupled with the stochastic methods were used to develop stochastic dynamic models for financial data (stock price). In order to extend this approach to more complex dynamic processes in science and engineering operating under internal structural and external environmental perturbations, Pedjeu and Ladde [14] modified the existing mathematical models by incorporating certain significant attributable parameters or variables with state variables, explicitly. Meanwhile, they obtained the existence and uniqueness of the solution by using the Picard-Lindel successive approximations. This motivates us to initiate to partially characterize intrastructural and external environmental perturbations by a set of linearly independent time-scales. For example, t,B(t),tα,N˜t,B\left(t),{t}^{\alpha },\widetilde{N}, where B(t)B\left(t)is the standard Wiener process, and N˜\widetilde{N}is the Lévy process, α∈(0,1]\alpha \in (0,1].In addition, impulsive stochastic differential systems [15,16,17] with Markovian switching have been investigated. Zhu [18] has determined the pth moment exponential stability of impulsive stochastic functional differential equations with Markovian switching. Then Kao et al. [19] proved the pth moment exponential stability, almost exponential stability and instability on the basis of predecessors. Tan et al. [20] discussed the stability of hybrid impulsive and switching stochastic systems with time delay.However, it is worth mentioning that to the best of our knowledge, the stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects has not been investigated yet and this arouses our interest in research. In order to fill this gap, in this paper, combined with previous work, we consider the following stochastic differential equations driven by Lévy noise with impulsive effects (1)dx(t)=u(x(t−),t,r(t))dt+b(x(t−),t,r(t))dB(t)+σ(x(t−),t,r(t))(dt)α+∫∣y∣<ch(x(t−),y,t,r(t))N˜(dt,dy),t≠tk,t≥0,Δx(tk)=Ik(x(tk−),tk),k∈N,\left\{\begin{array}{l}{\rm{d}}x\left(t)=u\left(x\left(t-),t,r\left(t)){\rm{d}}t+b\left(x\left(t-),t,r\left(t)){\rm{d}}B\left(t)+\sigma \left(x\left(t-),t,r\left(t)){\left({\rm{d}}t)}^{\alpha }\\ \hspace{2.0em}\hspace{2.0em}+\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x\left(t-),y,t,r\left(t))\widetilde{N}\left({\rm{d}}t,{\rm{d}}y),\hspace{1em}t\ne {t}_{k},t\ge 0,\\ \Delta x\left({t}_{k})={I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),\hspace{1em}k\in {\mathbb{N}},\end{array}\right.where 0<α≤10\lt \alpha \le 1, x(0)=x0∈Rnx\left(0)={x}_{0}\in {{\mathbb{R}}}^{n}is the initial value satisfying E∣x0∣2<∞{\mathbb{E}}| {x}_{0}\hspace{-0.25em}{| }^{2}\lt \infty , the constant ccis the maximum allowable jump size and the mappings u,σ:Rn×R+×S→Rn,b:Rn×R+×S→Rn×mu,\sigma :{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{n},b:{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{n\times m}, h:Rn×R×R+×S→Rnh:{{\mathbb{R}}}^{n}\times {\mathbb{R}}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{n}are continuous functions. The fixed impulse time sequence {tk}k∈N{\left\{{t}_{k}\right\}}_{k\in {\mathbb{N}}}satisfies 0≤t0<t1<⋯<tk<⋯,tk→∞0\le {t}_{0}\lt {t}_{1}\hspace{0.33em}\lt \cdots \lt {t}_{k}\hspace{0.33em}\lt \cdots \hspace{0.33em},\hspace{0.25em}{t}_{k}\to \infty (as k→∞k\to \infty ), Δx(tk)=Ik(x(tk−),tk)\Delta x\left({t}_{k})={I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k})denotes the state jumping at impulsive time instant tk{t}_{k}. Denote by x(t;0,x0)x\left(t;\hspace{0.33em}0,{x}_{0})the solution to the system which is assumed to be right continuous, i.e., x(tk+)=x(tk)x\left({t}_{k}^{+})=x\left({t}_{k}).In this paper, we utilize the local Lipschitz condition and a weaker condition to replace the linear growth condition to obtain a unique global solution for system (1). According to the method of Lyapunov function, we can prove there is a unique global solution. Then we present a kind of λ\lambda -type function which will be introduced in Section 2. By means of nonnegative semi-martingale convergence theorem and Lyapunov function, we derive a kind of almost sure λ\lambda -type stability, including almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability. The main features of this paper are as follows: (i)The presented hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects have not been considered before.(ii)A more general almost sure stability (including almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability) problem has been investigated under much weaker conditions.(iii)The upper bound of each coefficient at any mode is obtained.The article is organized as follows. In Section 2, we present several definitions and preliminaries. In Section 3, the conditions for the existence and uniqueness of the global solution and the sufficient conditions for λ\lambda -type stability are established, respectively. In Section 4, we prove the λ\lambda -type stability about the upper bound of each coefficient at any mode by using the theory of the MM-matrix. Finally, an example is given to illustrate the obtained theory.2PreliminariesThroughout this paper, unless otherwise specified, we use the following notations. Rn{{\mathbb{R}}}^{n}denotes the nn-dimensional Euclidean space, and ∣x∣| x| denotes the Euclidean norm of a vector x. R=(−∞,+∞){\mathbb{R}}=\left(-\infty ,+\infty )and R+=[0,+∞){{\mathbb{R}}}^{+}={[}0,+\infty ). Diag(ζ1,…,ζN{\zeta }_{1},\ldots ,{\zeta }_{N}) denotes a diagonal matrix with diagonal entries ζ1,…,ζN{\zeta }_{1},\ldots ,{\zeta }_{N}. Let (Ω,ℱ,{ℱt}t≥0,P)\left(\Omega ,{\mathcal{ {\mathcal F} }},{\left\{{{\mathcal{ {\mathcal F} }}}_{t}\right\}}_{t\ge 0},{\mathbb{P}})be a complete probability space with a filtration {ℱt}t≥0{\left\{{{\mathcal{ {\mathcal F} }}}_{t}\right\}}_{t\ge 0}satisfying the usual conditions (i.e., it is right continuous and ℱ0{{\mathcal{ {\mathcal F} }}}_{0}contains all PP-null sets). B(t)=(B1(t),B2(t),…,Bm(t))TB\left(t)={\left({B}_{1}\left(t),{B}_{2}\left(t),\ldots ,{B}_{m}\left(t))}^{T}be an mm-dimensional ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted Brownian motion defined on the complete probability space (Ω,ℱ,ℱt≥0,P)\left(\Omega ,{\mathcal{ {\mathcal F} }},{{\mathcal{ {\mathcal F} }}}_{t\ge 0},{\mathbb{P}}), and N(t,z)N\left(t,z)be a ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted Poisson random measure on [0,+∞)×R{[}0,+\infty )\times {\mathbb{R}}with a σ\sigma -finite intensity measure ν\nu (dz), the compensator martingale measure N˜(t,z)\widetilde{N}\left(t,z)satisfies N˜(dt,dz)=N(dt,dz)−ν(dz)dt\widetilde{N}\left({\rm{d}}t,{\rm{d}}z)=N\left({\rm{d}}t,{\rm{d}}z)-\nu \left({\rm{d}}z){\rm{d}}t. Let r(t)r\left(t), t≥0t\ge 0be a right-continuous Markov chain defined on the probability space taking values in a finite state S={1,2,…N}{\mathbb{S}}=\left\{1,2,\ldots N\right\}with generator Γ=(γij)N×N\Gamma ={\left({\gamma }_{ij})}_{N\times N}given by P{r(t+Δ)=j∣r(t)=i}=γijΔ+o(Δ),ifi≠j,1+γijΔ+o(Δ),ifi=j,P\left\{r\left(t+\Delta )=j| r\left(t)=i\right\}=\left\{\begin{array}{ll}{\gamma }_{ij}\Delta +o\left(\Delta ),& {\rm{if}}\hspace{0.33em}i\ne j,\\ 1+{\gamma }_{ij}\Delta +o\left(\Delta ),& {\rm{if}}\hspace{0.33em}i=j,\end{array}\right.where Δ>0\Delta \gt 0, and γij≥0{\gamma }_{ij}\ge 0is the transition rate from iito jjif i≠ji\ne jwhile γii=−∑i≠jγij{\gamma }_{ii}=-{\sum }_{i\ne j}{\gamma }_{ij}. And we assume that the Markov chain r(t)r\left(t)is ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted but independent of the Brownian motion B(t)B\left(t).Next, we give some definitions about fractional calculus and λ\lambda -type function, which will be used in this paper.Definition 2.1[21] (Riemann-Liouville fractional integrals, Samko et al., 1993): For any α∈(0,1)\alpha \in \left(0,1)and function f∈L1[[a,b];Rn]f\in {L}^{1}\left[\left[a,b];\hspace{0.33em}{{\mathbb{R}}}^{n}], the left-sided and right-sided Riemann-Liouville fractional integrals of order α\alpha are defined for almost all a<t<ba\lt t\lt bby (Ia+αf)(t)=1Γ(α)∫at(t−s)α−1f(s)ds,t>a\left({I}_{a+}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(\alpha )}\underset{a}{\overset{t}{\int }}{\left(t-s)}^{\alpha -1}f\left(s){\rm{d}}s,\hspace{1.0em}t\gt aand (Ib−αf)(t)=1Γ(α)∫tb(t−s)α−1f(s)ds,t<b,\left({I}_{b-}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(\alpha )}\underset{t}{\overset{b}{\int }}{\left(t-s)}^{\alpha -1}f\left(s){\rm{d}}s,\hspace{1.0em}t\lt b,where Γ(α)=∫0∞sα−1e−sds\Gamma \left(\alpha )={\int }_{0}^{\infty }{s}^{\alpha -1}{e}^{-s}{\rm{d}}sis the Gamma function and L1[a,b]{L}^{1}\left[a,b]is the space of integrable functions in a finite interval [a,b]\left[a,b]of R{\mathbb{R}}.Definition 2.2[21] (Riemann-Liouville fractional derivatives, Samko et al., 1993): For any α∈(0,1)\alpha \in \left(0,1)and well-defined absolutely continuous function ffon an interval [a,b]\left[a,b], the left-sided and right-sided Riemann-Liouville fractional derivatives are defined, respectively, by (Da+αf)(t)=1Γ(1−α)f(a)(t−a)α+∫at(t−s)−αf′(s)ds\left({D}_{a+}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(1-\alpha )}\left[\frac{f\left(a)}{{\left(t-a)}^{\alpha }}+\underset{a}{\overset{t}{\int }}{\left(t-s)}^{-\alpha }f^{\prime} \left(s){\rm{d}}s\right]and (Db−αf)(t)=1Γ(1−α)f(b)(b−t)α−∫tb(s−t)−αf′(s)ds.\left({D}_{b-}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(1-\alpha )}\left[\frac{f\left(b)}{{\left(b-t)}^{\alpha }}-\underset{t}{\overset{b}{\int }}{\left(s-t)}^{-\alpha }f^{\prime} \left(s){\rm{d}}s\right].Definition 2.3[22] (Jumarie, 2005): Let σ(t)\sigma \left(t)be a continuous function, then its integration with respect to (dt)α{\left({\rm{d}}t)}^{\alpha }, 0<α≤10\lt \alpha \le 1, is defined by ∫0tσ(s)(ds)α=α∫0t(t−s)α−1σ(s)ds.\underset{0}{\overset{t}{\int }}\sigma \left(s){\left({\rm{d}}s)}^{\alpha }=\alpha \underset{0}{\overset{t}{\int }}{\left(t-s)}^{\alpha -1}\sigma \left(s){\rm{d}}s.Definition 2.4The function λ:R→(0,∞)\lambda :{\mathbb{R}}\to \left(0,\infty )is said to be λ\lambda -type function if the function satisfies the following three conditions: (1)It is continuous and nondecreasing in R{\mathbb{R}}and differentiable in R+{{\mathbb{R}}}^{+},(2)λ(0)=1,λ(∞)=∞\lambda \left(0)=1,\lambda \left(\infty )=\infty and r=supt≥0λ′(t)λ(t)<∞r={\sup }_{t\ge 0}\left[\frac{\lambda ^{\prime} \left(t)}{\lambda \left(t)}\right]\lt \infty ,(3)For any s,t≥0,λ(t)≤λ(s)λ(t−s)s,t\ge 0,\lambda \left(t)\le \lambda \left(s)\lambda \left(t-s).Remark 2.5It is obvious that the functions λ(t)=et,λ(t)=(1+t+)\lambda \left(t)={e}^{t},\lambda \left(t)=\left(1+{t}^{+})and log(1+t+)\log \left(1+{t}^{+})are λ\lambda -type functions since they satisfy the aforementioned three conditions. Next, we give the definition of the almost sure stability with general decay rate based on Definition 2.4.Definition 2.6Let the function λ(t)∈C(R+;R+)\lambda \left(t)\in C\left({{\mathbb{R}}}^{+};\hspace{0.33em}{{\mathbb{R}}}^{+})be a λ\lambda -type function. Then for any initial x0∈Rn{x}_{0}\in {{\mathbb{R}}}^{n}, the trivial solution is said to be almost surely stable with decay λ(t)\lambda \left(t)of order γ\gamma if limsupt→∞log∣x(t,x0)∣logλ(t)≤−γ,a.s.\mathop{\mathrm{limsup}}\limits_{t\to \infty }\frac{\log | x\left(t,{x}_{0})| }{\log \lambda \left(t)}\le -\gamma ,\hspace{1.0em}\hspace{0.1em}\text{a.s.}\hspace{0.1em}Remark 2.7It is obvious that this almost sure λ\lambda -type stability implies the almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability when λ(t)\lambda \left(t)is replaced by et,1+t+,log(1+t+){e}^{t},1+{t}^{+},\log \left(1+{t}^{+}), respectively. Because we have a wide choice for λ\lambda -type functions, thus our results will be more general than some classical results.Let C2,1(Rn×R+×S→R+){C}^{2,1}\left({{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{+})denote the family of all functions V(x,t,i)V\left(x,t,i)on Rn×R+×S{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}, which are continuously twice differentiable in xxand once in tt. Define three functions L1V,L2V,L3V:Rn×R+×S→R{L}_{1}V,{L}_{2}V,{L}_{3}V:{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\hspace{0.25em}\times {\mathbb{S}}\to {\mathbb{R}}by L1V(x,t,i)=Vt(x,t,i)+Vx(x,t,i)u(x,t,i)+∑j=1NγijV(x,t,j)+12trace[bT(x,t,i)Vxx(x,t,i)b(x,t,i)],L2V(x,t,i)=∫∣y∣<c[V(x+h(x,y,t,i),t,i)−V(x,t,i)−Vx(x,t,i)h(x,y,t,i)]ν(dy),L3V(x,t,i)=Vx(x,t,i)σ(x,t,i),\begin{array}{rcl}{L}_{1}V\left(x,t,i)& =& {V}_{t}\left(x,t,i)+{V}_{x}\left(x,t,i)u\left(x,t,i)+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}V\left(x,t,j)+\frac{1}{2}\hspace{0.1em}\text{trace}\hspace{0.1em}\left[{b}^{T}\left(x,t,i){V}_{xx}\left(x,t,i)b\left(x,t,i)],\\ {L}_{2}V\left(x,t,i)& =& \mathop{\displaystyle \int }\limits_{| y| \lt c}\left[V\left(x+h\left(x,y,t,i),t,i)-V\left(x,t,i)-{V}_{x}\left(x,t,i)h\left(x,y,t,i)]\nu \left({\rm{d}}y),\\ {L}_{3}V\left(x,t,i)& =& {V}_{x}\left(x,t,i)\sigma \left(x,t,i),\end{array}where Vt(x,t,i)=∂V(x,t,i)∂t{V}_{t}\left(x,t,i)=\frac{\partial V\left(x,t,i)}{\partial t}, Vx(x,t,i)=∂V(x,t,i)∂x1,…,∂V(x,t,i)∂xn{V}_{x}\left(x,t,i)=\left(\frac{\partial V\left(x,t,i)}{\partial {x}_{1}},\ldots ,\frac{\partial V\left(x,t,i)}{\partial {x}_{n}}\right), Vxx(x,t,i)=∂V2(x,t,i)∂xk∂xln×n{V}_{xx}\left(x,t,i)={\left(\frac{\partial {V}^{2}\left(x,t,i)}{\partial {x}_{k}\partial {x}_{l}}\right)}_{n\times n}. By the generalized Itô formula, for t∈[tj−1,tj)t\in {[}{t}_{j-1},{t}_{j})V(x(t),t,r(t))=V(x(tj−1),tj−1,r(tj−1))+∫tj−1tL1V(x(s),s,r(s))ds+∫tj−1tL2V(x(s),s,r(s))ds+∫tj−1tα(t−s)α−1L3V(x(s),s,r(s))ds+G(t),\begin{array}{rcl}V\left(x\left(t),t,r\left(t))& =& V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s+G\left(t),\end{array}where G(t)=∫tj−1tVx(x(s),s,r(s))b(x(s−),s,r(s))dB(s)+∫tj−1t∫∣y∣<c[V(x(s)+h(x(s−),y,s,r(s)),s,r(s))−V(x(s),s,r(s))]N˜(ds,dy).G\left(t)=\underset{{t}_{j-1}}{\overset{t}{\int }}{V}_{x}\left(x\left(s),s,r\left(s))b\left(x\left(s-),s,r\left(s)){\rm{d}}B\left(s)+\underset{{t}_{j-1}}{\overset{t}{\int }}\mathop{\int }\limits_{| y| \lt c}\left[V\left(x\left(s)+h\left(x\left(s-),y,s,r\left(s)),s,r\left(s))-V\left(x\left(s),s,r\left(s))]\tilde{N}\left({\rm{d}}s,{\rm{d}}y).In fact, if a stochastic process is a martingale, then it is a local martingale. Hence, we can easily know that {G(t)}t≥0{\left\{G\left(t)\right\}}_{t\ge 0}is a local martingale.3Main resultBefore we state our main results in this section, the following hypotheses are imposed.Assumption 3.1(Local Lipschitz condition). For arbitrary x1,x2∈Rn{x}_{1},{x}_{2}\in {{\mathbb{R}}}^{n}, and ∣x1∣∨∣x2∣≤n| {x}_{1}| \vee | {x}_{2}| \le n, there is a positive constant Ln{L}_{n}such that ∣u(x1,t,i)−u(x2,t,i)∣∨∣b(x1,t,i)−b(x2,t,i)∣∨∣σ(x1,t,i)−σ(x2,t,i)∣∨∫∣y∣<c∣h(x1,y,t,i)−h(x2,y,t,i)∣ν(dy)≤Ln(∣x1−x2∣2).| u\left({x}_{1},t,i)-u\left({x}_{2},t,i)| \vee | b\left({x}_{1},t,i)-b\left({x}_{2},t,i)| \vee | \sigma \left({x}_{1},t,i)-\sigma \left({x}_{2},t,i)| \vee \mathop{\int }\limits_{| y| \lt c}| h\left({x}_{1},y,t,i)-h\left({x}_{2},y,t,i)| \nu \left({\rm{d}}y)\le {L}_{n}\left(| {x}_{1}-{x}_{2}\hspace{-0.25em}{| }^{2}).Assumption 3.2For any (t,i)∈R×S,u(0,t,i)=b(0,t,i)=σ(0,t,i)=h(0,y,t,i)=0\left(t,i)\in {\mathbb{R}}\times {\mathbb{S}},u\left(0,t,i)=b\left(0,t,i)=\sigma \left(0,t,i)=h\left(0,y,t,i)=0.For the stability analysis, Assumption 3.2 implies that x(t)=0x\left(t)=0is the trivial solution.Assumption 3.3There are several nonnegative number CCand pu{p}_{u}, real number Ku{K}_{u}and βu{\beta }_{u}(1≤u≤U1\le u\le U, for positive integer UU), a Lyapunov function V∈C2,1(Rn×R+×S;R+)V\in {C}^{2,1}\left({{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}};\hspace{0.33em}{{\mathbb{R}}}^{+}), such that (i)lim∣x∣→∞inf0≤t<∞V(x(t),t,i)=∞{\mathrm{lim}}_{| x| \to \infty }{\inf }_{0\le t\lt \infty }V\left(x\left(t),t,i)=\infty .(ii)L1V(x,s,i)+L2V(x,s,i)+α(t−s)α−1L3V(x,s,i)≤C+∑u=1UKu∣x∣pu{L}_{1}V\left(x,s,i)+{L}_{2}V\left(x,s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\le C+{\sum }_{u=1}^{U}{K}_{u}| x\hspace{-0.25em}{| }^{{p}_{u}}.(iii)V(x(tk−)+Ik(x(tk−),tk),tk,r(tk))≤d(x(tk−),tk)+V(x(tk−),tk,r(tk)).V\left(x\left({t}_{k}^{-})+{I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),{t}_{k},r\left({t}_{k}))\le d\left(x\left({t}_{k}^{-}),{t}_{k})+V\left(x\left({t}_{k}^{-}),{t}_{k},r\left({t}_{k})).(iv)∑j=nkd(x(tj−1−),tj−1)≤∑u=1Uβu∫tj−1tk∣x∣puds{\sum }_{j=n}^{k}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\le {\sum }_{u=1}^{U}{\beta }_{u}{\int }_{{t}_{j-1}}^{{t}_{k}}| x\hspace{-0.25em}{| }^{{p}_{u}}{\rm{d}}s.For any x∈Rn,y∈R,0≤s<t,i∈Sx\in {{\mathbb{R}}}^{n},y\in {\mathbb{R}},0\le s\lt t,i\in {\mathbb{S}}.According to the above assumptions, let us state the following existence and uniqueness theorem:Theorem 3.4Assume that Assumptions 3.1–3.3 hold, then for any initial data x0∈Rn{x}_{0}\in {{\mathbb{R}}}^{n}, there is a unique global solution x(t;0,x(0))x\left(t;\hspace{0.33em}0,x\left(0))on t>0t\gt 0to system (1).ProofApplying the standing truncation technique, Assumptions 3.1 and 3.2 admit a unique maximal local solution to system (1)). Let x(t)(t∈[0,ϱ∞))x\left(t)\left(t\in {[}0,{\varrho }_{\infty }))be the maximal local solution to system (1) and ϱ∞{\varrho }_{\infty }be the explosion time. And let a0∈R+{a}_{0}\in {{\mathbb{R}}}^{+}be sufficiently large for ∣x0∣≤a0| {x}_{0}| \le {a}_{0}. For any integer a≥a0a\ge {a}_{0}, define the stopping time τa=inf{t∈[0,ϱ∞);∣x(t)∣≥a},{\tau }_{a}=\inf \left\{t\in {[}0,{\varrho }_{\infty });\hspace{0.33em}| x\left(t)| \ge a\right\},where infϕ=∞\inf \phi =\infty . Obviously, the sequence τa{\tau }_{a}is increasing. So we have a limit τ∞=lima→∞τa{\tau }_{\infty }={\mathrm{lim}}_{a\to \infty }{\tau }_{a}, whence τ∞≤ϱ∞{\tau }_{\infty }\le {\varrho }_{\infty }. If we can show that τ∞=∞{\tau }_{\infty }=\infty a.s., then we have ϱ∞=∞{\varrho }_{\infty }=\infty a.s. Therefore, we only need to devote to prove τ∞=∞{\tau }_{\infty }=\infty a.s., which is equivalent to proving that P(τa≤t)→0P\left({\tau }_{a}\le t)\to 0as a→∞a\to \infty for any t>0t\gt 0. In fact, by the generalized Itô formula, for t∈[tj−1,tj)t\in {[}{t}_{j-1},{t}_{j}), we have (2)E(Iτa≤tV(x(τa),τa,r(τa)))≤EV(x(t∧τa),t∧τa,r(t∧τa))=EV(x(tj−1),tj−1,r(tj−1))+E∫tj−1t∧τaL1V(x(s),s,r(s))ds+E∫tj−1t∧τaL2V(x(s),s,r(s))ds+E∫tj−1t∧τaα(t−s)α−1L3V(x(s),s,r(s))ds.\begin{array}{l}{\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a})))\\ \hspace{1.0em}\le {\mathbb{E}}V\left(x\left(t\wedge {\tau }_{a}),t\wedge {\tau }_{a},r\left(t\wedge {\tau }_{a}))\\ \hspace{1.0em}={\mathbb{E}}V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))+{\mathbb{E}}\underset{{t}_{j-1}}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{\mathbb{E}}\underset{{t}_{j-1}}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{\mathbb{E}}\underset{{t}_{j-1}}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s.\end{array}By condition (iii), we have V(x(tj−1),tj−1,r(tj−1))=V(x(tj−1−)+Ij−1(x(tj−1−),tk),tj−1−,r(tj−1))≤d(x(tj−1−),tj−1)+V(x(tj−1−),tj−1−,r(tj−1)).\hspace{-35.5em}\begin{array}{rcl}V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))& =& V\left(x\left({t}_{j-1}^{-})+{I}_{j-1}\left(x\left({t}_{j-1}^{-}),{t}_{k}),{t}_{j-1}^{-},r\left({t}_{j-1}))\\ & \le & d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})+V\left(x\left({t}_{j-1}^{-}),{t}_{j-1}^{-},r\left({t}_{j-1})).\end{array}Hence, for all t≥0t\ge 0, (3)E(Iτa≤tV(x(τa),τa,r(τa)))≤EV(x(0),0,r(0))+E∫0t∧τaL1V(x(s),s,r(s))ds+E∫0t∧τaL2V(x(s),s,r(s))ds+E∫0t∧τaα(t−s)α−1L3V(x(s),s,r(s))ds+∑j:0<tj≤tEd(x(tj−1−),tj−1)≤EV(x(0),0,r(0))+Ct+∑u=1UKuE∫0t∧τa∣x(s)∣puds+∑j:0<tj≤tEd(x(tj−1−),tj−1).\begin{array}{rcl}{\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a})))& \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s+\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\mathbb{E}}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\\ & \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+\mathop{\displaystyle \sum }\limits_{u=1}^{U}{K}_{u}{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s+\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\mathbb{E}}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1}).\end{array}For almost all ω∈Ω\omega \in \Omega , there is an integer m0=m0(ω){m}_{0}={m}_{0}\left(\omega ), for any m≥m0m\ge {m}_{0}and 0≤t∧τa<m0\le t\wedge {\tau }_{a}\lt m, define tkm=max{tk:tk≤t∧τa}.\hspace{-17em}{t}_{{k}_{m}}={\rm{\max }}\left\{{t}_{k}:{t}_{k}\le t\wedge {\tau }_{a}\right\}.Combining with condition (iv), (4)∑j:0<tj≤td(x(tj−1−),tj−1)=∑j=1tkmd(x(tj−1−),tj−1)=∑u=1Uβu∫0tkm∣x(s)∣puds.\sum _{j:0\lt {t}_{j}\le t}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})=\mathop{\sum }\limits_{j=1}^{{t}_{{k}_{m}}}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})=\mathop{\sum }\limits_{u=1}^{U}{\beta }_{u}\underset{0}{\overset{{t}_{{k}_{m}}}{\int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s.\hspace{8.5em}Substituting (4) to (3), for 0<t∧τa<m0\lt t\wedge {\tau }_{a}\lt m, (5)E(Iτa≤tV(x(τa),τa,r(τa)))≤EV(x(0),0,r(0))+Ct+∑u=1UKuE∫0t∧τa∣x(s)∣puds+∑u=1UβuE∫0tkm∣x(s)∣puds≤EV(x(0),0,r(0))+Ct+∑u=1U(Ku+βu)E∫0t∣x(s)∣puds.\begin{array}{rcl}{\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a})))& \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+\mathop{\displaystyle \sum }\limits_{u=1}^{U}{K}_{u}{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s+\mathop{\displaystyle \sum }\limits_{u=1}^{U}{\beta }_{u}{\mathbb{E}}\underset{0}{\overset{{t}_{{k}_{m}}}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s\\ & \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+\mathop{\displaystyle \sum }\limits_{u=1}^{U}\left({K}_{u}+{\beta }_{u}){\mathbb{E}}\underset{0}{\overset{t}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s.\end{array}By lim∣x∣→∞inf0≤t<∞V(x(t),t,r(t))=∞{\mathrm{lim}}_{| x| \to \infty }{\inf }_{0\le t\lt \infty }V\left(x\left(t),t,r\left(t))=\infty , let μa=inf∣x∣≥a,0≤t<∞V(x(t),t,r(t)){\mu }_{a}={\inf }_{| x| \ge a,0\le t\lt \infty }V\left(x\left(t),t,r\left(t)), for a≥a0a\ge {a}_{0}. Therefore, we have P(τa≤t)μa≤E(Iτa≤tV(x(τa),τa,r(τa))).P\left({\tau }_{a}\le t){\mu }_{a}\le {\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a}))).\hspace{0.16em}Using the idea of Theorem 3.1 in [8], letting a→∞a\to \infty , by Fatou’s lemma, we can derive (6)0≤P(τ∞≤t)≤lima→∞P(τa≤t)=lima→∞EV(x(0),0,r(0))+Ct+∑u=1U(Ku+βu)E∫0t∣x(s)∣pudsμa=0.\begin{array}{rcl}0\le P\left({\tau }_{\infty }\le t)& \le & \mathop{\mathrm{lim}}\limits_{a\to \infty }P\left({\tau }_{a}\le t)\\ & =& \mathop{\mathrm{lim}}\limits_{a\to \infty }\frac{{\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+{\displaystyle \sum }_{u=1}^{U}\left({K}_{u}+{\beta }_{u}){\mathbb{E}}{\displaystyle \int }_{0}^{t}| x\left(s){| }^{{p}_{u}}{\rm{d}}s}{{\mu }_{a}}\\ & =& 0.\end{array}This implies that there exists a unique global solution x(t;0,x(0))x\left(t;\hspace{0.33em}0,x\left(0))for system (1).□Next, in order to obtain sufficient conditions of the almost sure stability with general decay rate, we need the following lemmas.Lemma 3.5[23] Let {Mt}t≥0{\left\{{M}_{t}\right\}}_{t\ge 0}be a local martingale and {Nt}t≥0{\left\{{N}_{t}\right\}}_{t\ge 0}be a locally bounded predictable process, then the stochastic integral ∫0tNsdMs{\int }_{0}^{t}{N}_{s}{\rm{d}}{M}_{s}is also a local martingale.Lemma 3.6(Nonnegative semi-martingale convergence theorem) [10] Assume {At}\left\{{A}_{t}\right\}and {Ut}\left\{{U}_{t}\right\}are two continuous predictable increasing processes vanishing at t=0t=0a.s. and {Mt}\left\{{M}_{t}\right\}is a real-valued continuous local martingale with M0=0{M}_{0}=0a.s. Let X(t)X\left(t)be a nonnegative adapted process and ξ\xi be a nonnegative ℱ0{{\mathcal{ {\mathcal F} }}}_{0}-measurable random variable satisfyingXt≤ξ+At−Ut+Mt,t≥0.{X}_{t}\le \xi +{A}_{t}-{U}_{t}+{M}_{t},\hspace{1.0em}t\ge 0.If A∞≔limt→∞At<∞{A}_{\infty }:= {\mathrm{lim}}_{t\to \infty }{A}_{t}\lt \infty , a.s., then, we havelimt→∞supXt<∞,a.s.\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup {X}_{t}\lt \infty ,\hspace{1em}{a.s.}With these above assumptions and lemmas, we can now state our results about the almost sure stability with general decay rate.Theorem 3.7Let Assumptions 3.1 and 3.2 hold. If there are positive numbers c1,c2,α1,α2,q>2{c}_{1},{c}_{2},{\alpha }_{1},{\alpha }_{2},q\gt 2, such that the functions V(x,t,i)V\left(x,t,i)and LjV(x,t,i)(j=1,2,3){L}_{j}V\left(x,t,i)\left(j=1,2,3)satisfy(i)c1∣x∣2≤V(x,t,i)≤c2∣x∣2.{c}_{1}| x\hspace{-0.25em}{| }^{2}\le V\left(x,t,i)\le {c}_{2}| x\hspace{-0.25em}{| }^{2}.(ii)L1V(x,s,i)+L2V(x,s,i)+α(t−s)α−1L3V(x,s,i)≤−α1∣x∣2−α2∣x∣q.{L}_{1}V\left(x,s,i)+{L}_{2}V\left(x,s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\le -{\alpha }_{1}| x\hspace{-0.25em}{| }^{2}-{\alpha }_{2}| x\hspace{-0.25em}{| }^{q}.(iii)V(x(tk−)+Ik(x(tk−),tk),tk,r(tk))≤d(x(tk−),tk)+V(x(tk−),tk,r(tk)).V\left(x\left({t}_{k}^{-})+{I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),{t}_{k},r\left({t}_{k}))\le d\left(x\left({t}_{k}^{-}),{t}_{k})+V\left(x\left({t}_{k}^{-}),{t}_{k},r\left({t}_{k})).(iv)Σj=nkλε(tj−1)d(x(tj−1−),tj−1)≤α3∫tj−1tkλε(s)∣x∣2ds+α4∫tj−1tkλε(s)∣x∣qds{\Sigma }_{j=n}^{k}{\lambda }^{\varepsilon }\left({t}_{j-1})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\le {\alpha }_{3}{\int }_{{t}_{j-1}}^{{t}_{k}}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{2}{\rm{d}}s+{\alpha }_{4}{\int }_{{t}_{j-1}}^{{t}_{k}}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{q}{\rm{d}}s.If there exists a small enough ε>0\varepsilon \gt 0such that α1−c2εα−α3>0,α2−α4>0{\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3}\gt 0,{\alpha }_{2}-{\alpha }_{4}\gt 0.Therefore, for any initial data x0{x}_{0}, the inequalitylimt→∞suplog∣x(t,x0)∣logλ(t)<−ε2\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup \frac{\log | x\left(t,{x}_{0})| }{\log \lambda \left(t)}\lt -\frac{\varepsilon }{2}holds, that is, the trivial solution of system (1) is almost surely stable with decay λ(t)\lambda \left(t)of order ε2\frac{\varepsilon }{2}.ProofNote that conditions (i)–(iv) are stronger than Assumption 3.3, so there is a unique global solution for system (1). Let λ(t)\lambda \left(t)be a λ\lambda -type function, and applying the generalized Itô formula to λε(t)V(x(t),t,r(t)){\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t)), for t∈[tj−1,tj)t\in {[}{t}_{j-1},{t}_{j})(7)λε(t)V(x(t),t,r(t))=λε(tj−1)V(x(tj−1),tj−1,r(tj−1))+∫tj−1tελ′(s)λ(s)λε(s)V(x(s),s,r(s))ds+∫tj−1tλε(s)L1V(x(s),s,r(s))ds+∫tj−1tλε(s)L2V(x(s),s,r(s))ds+∫tj−1tλε(s)α(t−s)α−1L3V(x(s),s,r(s))ds+Mtj−1t,\hspace{0.8em}\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& =& {\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}\varepsilon \frac{\lambda ^{\prime} \left(s)}{\lambda \left(s)}{\lambda }^{\varepsilon }\left(s)V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{M}_{{t}_{j-1}}^{t},\end{array}where Mtj−1t=∫tj−1tλε(s)dG(s){M}_{{t}_{j-1}}^{t}={\int }_{{t}_{j-1}}^{t}{\lambda }^{\varepsilon }\left(s){\rm{d}}G\left(s)is a real-valued continuous local martingale with M0=0{M}_{0}=0by Lemma 3.5. By condition (iii), we have (8)λε(tj−1)V(x(tj−1),tj−1,r(tj−1))=λε(tj−1)V(x(tj−1−)+Ij−1(x(tj−1−),tk),tj−1−,r(tj−1))≤λε(tj−1)[d(x(tj−1−),tj−1)+V(x(tj−1−),tj−1−,r(tj−1))]=λε(tj−1)V(x(tj−1−),tj−1−,r(tj−1))+λε(tj−1)d(x(tj−1−),tj−1).\begin{array}{rcl}{\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))& =& {\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}^{-})+{I}_{j-1}\left(x\left({t}_{j-1}^{-}),{t}_{k}),{t}_{j-1}^{-},r\left({t}_{j-1}))\\ & \le & {\lambda }^{\varepsilon }\left({t}_{j-1})\left[d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})+V\left(x\left({t}_{j-1}^{-}),{t}_{j-1}^{-},r\left({t}_{j-1}))]\\ & =& {\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}^{-}),{t}_{j-1}^{-},r\left({t}_{j-1}))+{\lambda }^{\varepsilon }\left({t}_{j-1})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1}).\end{array}\hspace{0.25em}Hence, for all t≥0t\ge 0, we have (9)λε(t)V(x(t),t,r(t))=V(x(0),0,r(0))+∫0tελ′(s)λ(s)λε(s)V(x(s),s,r(s))ds+∫0tλε(s)L1V(x(s),s,r(s))ds+∫0tλε(s)L2V(x(s),s,r(s))ds∫0tλε(s)α(t−s)α−1L3V(x(s),s,r(s))ds+∑j:0<tj≤tλε(tj−1)d(x(tj−1−),tj−1)+M0t.\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& =& V\left(x\left(0),0,r\left(0))+\underset{0}{\overset{t}{\displaystyle \int }}\varepsilon \frac{\lambda ^{\prime} \left(s)}{\lambda \left(s)}{\lambda }^{\varepsilon }\left(s)V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\lambda }^{\varepsilon }\left({t}_{j-1})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})+{M}_{0}^{t}.\end{array}For almost all ω∈Ω\omega \in \Omega , there is an integer m0=m0(ω){m}_{0}={m}_{0}\left(\omega ), for any m≥m0m\ge {m}_{0}and 0≤t<m0\le t\lt m, define tkm=max{tk:tk≤t},\hspace{1.22em}{t}_{{k}_{m}}={\rm{\max }}\left\{{t}_{k}:{t}_{k}\le t\right\},\hspace{9.1em}while 0<t<m0\lt t\lt m, (10)∑j:0<tj≤tλε(tj)d(x(tj−1−),tj−1)=∑j=1tkmλε(tj)d(x(tj−1−),tj−1)≤α3∫0tkmλε(s)∣x∣2ds+α4∫0tkmλε(s)∣x∣qds.≤α3∫0tλε(s)∣x∣2ds+α4∫0tλε(s)∣x∣qds.\begin{array}{rcl}\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\lambda }^{\varepsilon }\left({t}_{j})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})& =& \mathop{\displaystyle \sum }\limits_{j=1}^{{t}_{{k}_{m}}}{\lambda }^{\varepsilon }\left({t}_{j})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\\ & \le & {\alpha }_{3}\underset{0}{\overset{{t}_{{k}_{m}}}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{{t}_{{k}_{m}}}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{q}{\rm{d}}s.\\ & \le & {\alpha }_{3}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{q}{\rm{d}}s.\end{array}\hspace{3.25em}By condition (ii) and the definition of λ\lambda -type function, we have (11)λε(t)V(x(t),t,r(t))≤V(x(0),0,r(0))+εα∫0tλε(s)V(x(s),s,r(s))ds−α1∫0tλε(s)∣x(s)∣2ds−α2∫0tλε(s)∣x(s)∣qds+α3∫0tλε(s)∣x(s)∣2ds+α4∫0tλε(s)∣x(s)∣qds+M0t.\hspace{-43em}\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& \le & V\left(x\left(0),0,r\left(0))+\varepsilon \alpha \underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)V\left(x\left(s),s,r\left(s)){\rm{d}}s-{\alpha }_{1}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s\\ & & -{\alpha }_{2}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{\alpha }_{3}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{M}_{0}^{t}.\end{array}Moreover, (12)λε(t)V(x(t),t,r(t))≤V(x(0),0,r(0))+c2εα∫0tλε(s)∣x(s)∣2ds−α1∫0tλε(s)∣x(s)∣2ds−α2∫0tλε(s)∣x(s)∣qds+α3∫0tλε(s)∣x(s)∣2ds+α4∫0tλε(s)∣x(s)∣qds+M0t≤V(x(0),0,r(0))−(α1−c2εα−α3)∫0tλε(s)∣x(s)∣2ds−(α2−α4)∫0tλε(s)∣x(s)∣qds+M0t.\hspace{0.2em}\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& \le & V\left(x\left(0),0,r\left(0))+{c}_{2}\varepsilon \alpha \underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s-{\alpha }_{1}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s\\ & & -{\alpha }_{2}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{\alpha }_{3}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{M}_{0}^{t}\\ & \le & V\left(x\left(0),0,r\left(0))-\left({\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3})\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s-\left({\alpha }_{2}-{\alpha }_{4})\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{M}_{0}^{t}.\end{array}Consequently, by inequality α1−c2εα−α3>0,α2−α4>0{\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3}\gt 0,{\alpha }_{2}-{\alpha }_{4}\gt 0and (12) imply λε(t)V(x(t),t,r(t))≤V(x(0),0,r(0))+M0t.{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))\le V\left(x\left(0),0,r\left(0))+{M}_{0}^{t}.Applying Lemma 3.6, we have limt→∞supλε(t)V(x(t),t,r(t))<∞,a.s.\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup {\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))\lt \infty ,\hspace{1em}\hspace{0.1em}\text{a.s.}\hspace{0.1em}\hspace{0.2em}Thus, there exists a positive constant HHsuch that for any t>0t\gt 0, c1λε(t)∣x(t)∣2≤λε(t)V(x(t),t,r(t))≤H.{c}_{1}{\lambda }^{\varepsilon }\left(t)| x\left(t){| }^{2}\le {\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))\le H.\hspace{0.25em}Moreover, λε(t)∣x(t)∣2≤Hc1<∞.{\lambda }^{\varepsilon }\left(t)| x\left(t){| }^{2}\le \frac{H}{{c}_{1}}\lt \infty .\hspace{7.62em}Hence, limt→∞suplog∣x(t,x0)∣logλ(t)<−ε2.□\hspace{14.5em}\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup \frac{\log | x\left(t,{x}_{0})| }{\log \lambda \left(t)}\lt -\frac{\varepsilon }{2}.\hspace{19em}\square 4Almost surely stable with decay λ(t)\lambda \left(t)We find that conditions (i) and (ii) of Theorem 3.7 are somewhat inconvenient in applications since they are not related to the coefficient u,b,σ,hu,b,\sigma ,hexplicitly. In this section, we will give the visualized conditions for the coefficient u,b,σ,hu,b,\sigma ,hto study the λ\lambda -type stability.We first recall some definitions and preliminaries.Definition 4.1A square matrix A={aij}N×NA={\left\{{a}_{ij}\right\}}_{N\times N}is called a ZZ-matrix if its off-diagonal entries are less than or equal to zero, namely aij≤0{a}_{ij}\le 0for i≠ji\ne j.Definition 4.2((M-matrix) [24]) Let AAbe a N×NN\times Nreal ZZ-matrix. And the matrix AAis also a nonsingular MM-matrix if it can be expressed in the form A=sI−BA=sI-Bwhile all the elements of B=(bij)B=\left({b}_{ij})are nonnegative and s≥ρ(B)s\ge \rho \left(B), where IIis an identity matrix and ρ(B)\rho \left(B)the spectral radius of BB.Noting that if A={aij}N×NA={\left\{{a}_{ij}\right\}}_{N\times N}is an MM-matrix, then it has positive diagonal entries and nonpositive off-diagonal entries, that is, aii≥0{a}_{ii}\ge 0while aij≥0,i≠j.{a}_{ij}\ge 0,i\ne j.Lemma 4.3([24]) If A={aij}N×NA={\left\{{a}_{ij}\right\}}_{N\times N}is a ZZ-matrix, then the following statements are equivalent: (1)AAis a nonsingular MM-matrix.(2)Every real eigenvalue of AAis positive.(3)All of the principle minors of AAare positive.(4)A−1{A}^{-1}exist and its elements are all nonnegative.Assumption 4.4For each i∈S,x∈Rni\in {\mathbb{S}},x\in {{\mathbb{R}}}^{n}and 0≤s<t0\le s\lt t, there exists a positive integer q>2q\gt 2, suppose the following conditions hold. (1)There exist constants αi1,αi2{\alpha }_{i1},{\alpha }_{i2}such that xT(u(x,s,i)+∫∣y∣<ch(x,y,s,i)ν(dy))≤−αi1∣x∣2+αi2∣x∣q.{x}^{T}\left(u\left(x,s,i)+\mathop{\int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y))\le -{\alpha }_{i1}| x\hspace{-0.25em}{| }^{2}+{\alpha }_{i2}| x\hspace{-0.25em}{| }^{q}.(2)There exist constants βi1,βi2{\beta }_{i1},{\beta }_{i2}such that xT(σ(x,s,i)α(t−s)α−1−∫∣y∣<ch(x,y,s,i)ν(dy))≤βi1∣x∣2−βi2∣x∣q.{x}^{T}\left(\sigma \left(x,s,i)\alpha {\left(t-s)}^{\alpha -1}-\mathop{\int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y))\le {\beta }_{i1}| x\hspace{-0.25em}{| }^{2}-{\beta }_{i2}| x\hspace{-0.25em}{| }^{q}.(3)There exist constants ηi1,ηi2{\eta }_{i1},{\eta }_{i2}such that ∣b(x,s,i)∣2≤ηi1∣x∣2+ηi2∣x∣q.\hspace{-13.8em}| b\left(x,s,i){| }^{2}\le {\eta }_{i1}| x\hspace{-0.25em}{| }^{2}+{\eta }_{i2}| x\hspace{-0.25em}{| }^{q}.(4)There exist constants ρi1,ρi2{\rho }_{i1},{\rho }_{i2}such that ∫∣y∣<c∣h(x,y,s,i)∣2ν(dy)≤ρi1∣x∣2+ρi2∣x∣q.\mathop{\int }\limits_{| y| \lt c}| h\left(x,y,s,i){| }^{2}\nu \left({\rm{d}}y)\le {\rho }_{i1}| x\hspace{-0.25em}{| }^{2}+{\rho }_{i2}| x\hspace{-0.25em}{| }^{q}.Remark 4.5Note that q>2q\gt 2in Assumption 4.4 which means that we can allow the coefficients u,b,σ,hu,b,\sigma ,hof system (1) to be high-order nonlinear.Assumption 4.6According to the definition of MM-matrix, let A≔−diag(−2α11+2β11+η11+ρ11,…,−2αN1+2βN1+ηN1+ρN1)−Γ{\mathcal{A}}:= -{\rm{diag}}\left(-2{\alpha }_{11}+2{\beta }_{11}+{\eta }_{11}+{\rho }_{11},\ldots ,-2{\alpha }_{N1}+2{\beta }_{N1}+{\eta }_{N1}+{\rho }_{N1})-\Gamma be a nonsingular MM-matrix, where Γ=(γij)N×N\Gamma ={\left({\gamma }_{ij})}_{N\times N}is given in Section 2. Moreover, from the fourth equivalent condition in Lemma 4.3, we obtain (θ1,θ2,…,θN)T≔A−11→>0,{\left({\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{N})}^{T}:= {{\mathcal{A}}}^{-1}\overrightarrow{1}\gt 0,where 1→=(1,1,…,1)T.\overrightarrow{1}={\left(1,1,\ldots ,1)}^{T}.That is, θi>0{\theta }_{i}\gt 0for all i∈Si\in {\mathbb{S}}.Theorem 4.7Let Assumptions 3.1–3.2, 4.4 and 4.6 hold. Those constants satisfy the following inequalities0<a̲<a¯,0\lt \underline{a}\lt \overline{a},wherea̲=mini∈S[θi(−2αi2+2βi2−ηi2−ρi2)],a¯=maxi∈S[θi(−2αi2+2βi2−ηi2−ρi2)].\begin{array}{rcl}\underline{a}& =& \mathop{\min }\limits_{i\in {\mathbb{S}}}\left[{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})],\\ \overline{a}& =& \mathop{\max }\limits_{i\in {\mathbb{S}}}\left[{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})].\end{array}Then for any given initial data x0{x}_{0}, there is a unique global solution x(t;x0)x\left(t;\hspace{0.33em}{x}_{0})and the trivial solution is almost surely stable with decay λ(t)\lambda \left(t).ProofLet function V:Rn×R+×S→R+V:{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times S\to {{\mathbb{R}}}^{+}by V(x,s,i)=θi∣x∣2V\left(x,s,i)={\theta }_{i}| x\hspace{-0.25em}{| }^{2}, choose two positive numbers c1=mini∈S{θi}{c}_{1}={\min }_{i\in {\mathbb{S}}}\left\{{\theta }_{i}\right\}, c2=maxi∈S{θi}{c}_{2}={\max }_{i\in {\mathbb{S}}}\left\{{\theta }_{i}\right\}, such that c1∣x∣2≤V(x,s,i)≤c2∣x∣2.{c}_{1}| x\hspace{-0.25em}{| }^{2}\le V\left(x,s,i)\le {c}_{2}| x\hspace{-0.25em}{| }^{2}.By the definition of operator LjV{L}_{j}V, j=1,2,3j=1,2,3in Section 2, we obtain ∑j=12LiV(x(s),s,i)+α(t−s)α−1L3V(x,s,i)=2θixTu(x,s,i)+θi∣b(x,s,i)∣2+∑j=1Nγijθj∣x∣2+2θixTσ(x,s,i)α(t−s)α−1+θi∫∣y∣<c[∣x+h(x,y,s,i)∣2−∣x∣2]ν(dy)−∫∣y∣<c2θixTh(x,y,s,i)ν(dy)≤2θixT[u(x,s,i)+∫∣y∣<ch(x,y,s,i)ν(dy)]+θi∣b(x,s,i)∣2+∑j=1Nγijθj∣x∣2+θi∫∣y∣<c∣h(x,y,s,i)∣2ν(dy)+2θixT[σ(x,s,i)α(t−s)α−1−∫∣y∣<ch(x,y,s,i)ν(dy)].\begin{array}{l}\mathop{\displaystyle \sum }\limits_{j=1}^{2}{L}_{i}V\left(x\left(s),s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\\ \hspace{1.0em}=2{\theta }_{i}{x}^{T}u\left(x,s,i)+{\theta }_{i}| b\left(x,s,i){| }^{2}+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}+2{\theta }_{i}{x}^{T}\sigma \left(x,s,i)\alpha {\left(t-s)}^{\alpha -1}\\ \hspace{1.0em}\hspace{1.0em}+{\theta }_{i}\mathop{\displaystyle \int }\limits_{| y| \lt c}\left[| x+h\left(x,y,s,i){| }^{2}-| x\hspace{-0.25em}{| }^{2}]\nu \left({\rm{d}}y)-\mathop{\displaystyle \int }\limits_{| y| \lt c}2{\theta }_{i}{x}^{T}h\left(x,y,s,i)\nu \left({\rm{d}}y)\\ \hspace{1.0em}\le 2{\theta }_{i}{x}^{T}\left[u\left(x,s,i)+\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y)]+{\theta }_{i}| b\left(x,s,i){| }^{2}+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}\\ \hspace{1.0em}\hspace{1.0em}+{\theta }_{i}\mathop{\displaystyle \int }\limits_{| y| \lt c}| h\left(x,y,s,i){| }^{2}\nu \left({\rm{d}}y)+2{\theta }_{i}{x}^{T}\left[\sigma \left(x,s,i)\alpha {\left(t-s)}^{\alpha -1}-\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y)].\end{array}\hspace{10.5em}By Assumption 4.4, we have (13)∑j=12LjV(x(s),s,i)+α(t−s)α−1L3V(x,s,i)≤2θi(−αi1∣x∣2+αi2∣x∣q)+2θi(βi1∣x∣2−βi2∣x∣q)+θi(ηi1∣x∣2+ηi2∣x∣q)+θi(ρi1∣x∣2+ρi2∣x∣q)+∑j=1Nγijθj∣x∣2=(−2θiαi1+2θiβi1+θiηi1+θiρi1)∣x∣2+∑j=1Nγijθj∣x∣2−(−2θiαi2+2θiβi2−θiηi2−θiρi2)∣x∣q=[θi(−2αi1+2βi1+ηi1+ρi1)+∑j=1Nγijθj]∣x∣2−θi(−2αi2+2βi2−ηi2−ρi2)∣x∣q.\begin{array}{l}\mathop{\displaystyle \sum }\limits_{j=1}^{2}{L}_{j}V\left(x\left(s),s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\\ \hspace{1.0em}\le 2{\theta }_{i}\left(-{\alpha }_{i1}| x\hspace{-0.25em}{| }^{2}+{\alpha }_{i2}| x\hspace{-0.25em}{| }^{q})+2{\theta }_{i}\left({\beta }_{i1}| x\hspace{-0.25em}{| }^{2}-{\beta }_{i2}| x\hspace{-0.25em}{| }^{q})+{\theta }_{i}\left({\eta }_{i1}| x\hspace{-0.25em}{| }^{2}+{\eta }_{i2}| x\hspace{-0.25em}{| }^{q})+{\theta }_{i}\left({\rho }_{i1}| x\hspace{-0.25em}{| }^{2}+{\rho }_{i2}| x\hspace{-0.25em}{| }^{q})+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}\\ \hspace{1.0em}=\left(-2{\theta }_{i}{\alpha }_{i1}+2{\theta }_{i}{\beta }_{i1}+{\theta }_{i}{\eta }_{i1}+{\theta }_{i}{\rho }_{i1})| x\hspace{-0.25em}{| }^{2}+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}-\left(-2{\theta }_{i}{\alpha }_{i2}+2{\theta }_{i}{\beta }_{i2}-{\theta }_{i}{\eta }_{i2}-{\theta }_{i}{\rho }_{i2})| x\hspace{-0.25em}{| }^{q}\\ \hspace{1.0em}=\left[{\theta }_{i}\left(-2{\alpha }_{i1}+2{\beta }_{i1}+{\eta }_{i1}+{\rho }_{i1})+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}]| x\hspace{-0.25em}{| }^{2}-{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})| x\hspace{-0.25em}{| }^{q}.\end{array}By Assumption 4.6, we have (14)∑j=12LjV(x(s),s,i)+α(t−s)α−1L3V(x,s,i)≤−∣x∣2−θi(−2αi2+2βi2−ηi2−ρi2)∣x∣q≤−∣x∣2−a̲∣x∣q.\mathop{\sum }\limits_{j=1}^{2}{L}_{j}V\left(x\left(s),s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\le -| x\hspace{-0.25em}{| }^{2}-{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})| x\hspace{-0.25em}{| }^{q}\le -| x\hspace{-0.25em}{| }^{2}-\underline{a}| x\hspace{-0.25em}{| }^{q}.By Assumptions 3.1–3.2, conditions (iii)–(iv) of Assumption 3.3, inequality (14) and Theorem 3.4, there is a unique global solution for any initial data. By inequality (14) and Theorem 3.7, the trivial solution is almost surely λ\lambda -type stable.□5ExampleWe consider two scalar stochastic fractional hybrid differential equations driven by Lévy noise with impulsive effects, and 2-state Markovian switching such that our results have been illustrated simply.Example 5.1(15)dx(t)=u(x(t−),t,r(t))dt+b(x(t−),t,r(t))dB(t)+σ(x(t−),t,r(t))(dt)α+∫∣y∣<ch(x(t−),y,t,r(t))N˜(dt,dy),t≠tk,t≥0,Δx(tk)=Ik(x(tk−),tk),k∈N,\left\{\begin{array}{l}{\rm{d}}x\left(t)=u\left(x\left(t-),t,r\left(t)){\rm{d}}t+b\left(x\left(t-),t,r\left(t)){\rm{d}}B\left(t)+\sigma \left(x\left(t-),t,r\left(t)){\left({\rm{d}}t)}^{\alpha }+\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x\left(t-),y,t,r\left(t))\widetilde{N}\left({\rm{d}}t,{\rm{d}}y),\hspace{1em}t\ne {t}_{k},t\ge 0,\\ \Delta x\left({t}_{k})={I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),\hspace{1em}k\in {\mathbb{N}},\end{array}\right.where α=0.5\alpha =0.5, c=1c=1, and Lévy measure ν\nu satisfies ν(dy)=∣y∣dy\nu \left({\rm{d}}y)=| y| {\rm{d}}y, r(t)r\left(t)is a Markov chain in the state space S=1,2{\mathbb{S}}=1,2with the generator Γ\Gamma Γ=−112−2.\Gamma =\left(\begin{array}{cc}-1& 1\\ 2& -2\\ \end{array}\right).Let u(x,t,1)=−6x−3x2,u(x,t,2)=−5x−1.5x2,b(x,t,1)=22x+22x32,b(x,t,2)=0.5x+0.5x32,σ(x,t,1)=−2x,σ(x,t,2)=−2x3,h(x,y,t,1)=32x32y−2x,h(x,y,t,2)=−x.\begin{array}{ll}u\left(x,t,1)=-6x-3{x}^{2},& u\left(x,t,2)=-5x-1.5{x}^{2},\\ b\left(x,t,1)=\frac{\sqrt{2}}{2}x+\frac{\sqrt{2}}{2}{x}^{\tfrac{3}{2}},& b\left(x,t,2)=0.5x+0.5{x}^{\tfrac{3}{2}},\\ \sigma \left(x,t,1)=-2x,& \sigma \left(x,t,2)=-2{x}^{3},\\ h\left(x,y,t,1)=\frac{\sqrt{3}}{2}{x}^{\tfrac{3}{2}}y-2x,& h\left(x,y,t,2)=-x.\end{array}By computing, we have α11=8,α12=−3,α21=6,α22=−1.5,β11=2,β12=0,β21=1,β22=0,η11=1,η12=1,η21=0.5,η22=0.5,ρ11=4,ρ12=0.375,ρ21=1,ρ22=0\begin{array}{rlll}{\alpha }_{11}=8,& {\alpha }_{12}=-3,& {\alpha }_{21}=6,& {\alpha }_{22}=-1.5,\\ {\beta }_{11}=2,& {\beta }_{12}=0,& {\beta }_{21}=1,& {\beta }_{22}=0,\\ \hspace{0.075em}{\eta }_{11}=1,& {\eta }_{12}=1,& {\eta }_{21}=0.5,& {\eta }_{22}=0.5,\\ {\rho }_{11}=4,& {\rho }_{12}=0.375,& {\rho }_{21}=1,& {\rho }_{22}=0\end{array}and A=8−1−210.5,A−1=18210.5128θ1=0.1402,θ2=0.1220.\begin{array}{rcll}{\mathcal{A}}& =& \left(\begin{array}{cc}8& -1\\ -2& 10.5\\ \end{array}\right),& \hspace{0.8em}{{\mathcal{A}}}^{-1}=\frac{1}{82}\left(\begin{array}{cc}10.5& 1\\ 2& 8\\ \end{array}\right)\\ {\theta }_{1}& =& 0.1402,& \hspace{0.75em}{\theta }_{2}=0.1220.\end{array}Then we take Lyapunov functions V(x,t,1)=0.1402∣x(t)∣2,V(x,t,2)=0.1220∣x(t)∣2,V\left(x,t,1)=0.1402| x\left(t){| }^{2},\hspace{1.0em}V\left(x,t,2)=0.1220| x\left(t){| }^{2},by Theorem 4.7, we derive that a̲=0.305,a¯=0.6484.\underline{a}=0.305,\hspace{1em}\overline{a}=0.6484.Let Ik(x(tk−),tk)=22−1x+22x(tk−tk−1)12,λε(t)=e0.01t.{I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k})=\left(\frac{\sqrt{2}}{2}-1\right)x+\frac{\sqrt{2}}{2}x{\left({t}_{k}-{t}_{k-1})}^{\tfrac{1}{2}},\hspace{1.0em}{\lambda }^{\varepsilon }\left(t)={e}^{0.01t}.By a simple calculation, we have d(x(tk−1−),tk−1)=0.7x2(tk−tk−1),Σj=nke0.01tj−10.7x2(tj−tj−1)≤0.7∫tn−1tke0.01s∣x∣2ds,d\left(x\left({t}_{k-1}^{-}),{t}_{k-1})=0.7{x}^{2}\left({t}_{k}-{t}_{k-1}),\hspace{1.0em}\underset{j=n}{\overset{k}{\Sigma }}{e}^{0.01{t}_{j-1}}0.7{x}^{2}\left({t}_{j}-{t}_{j-1})\le 0.7\underset{{t}_{n-1}}{\overset{{t}_{k}}{\int }}{e}^{0.01s}| x\hspace{-0.25em}{| }^{2}{\rm{d}}s,which verifies the condition of Theorem 3.7 with α1=1,α2=0.305,c2=0.1402,ε=0.01,α=1,α3=0.7,α4=0.{\alpha }_{1}=1,\hspace{1em}{\alpha }_{2}=0.305,\hspace{1em}{c}_{2}=0.1402,\hspace{1em}\varepsilon =0.01,\hspace{1em}\alpha =1,\hspace{1em}{\alpha }_{3}=0.7,\hspace{1em}{\alpha }_{4}=0.Moreover, α1−c2εα−α3=0.2986>0,α2−α4=0.305>0.{\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3}=0.2986\gt 0,\hspace{1em}{\alpha }_{2}-{\alpha }_{4}=0.305\gt 0.Therefore, system (5.1) is almost surely stable with decay λ(t)\lambda \left(t)of order ε2\frac{\varepsilon }{2}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects

Open Mathematics , Volume 20 (1): 13 – Jan 1, 2022

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de Gruyter
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© 2022 Tingting Hou and Hui Zhang, published by De Gruyter
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2391-5455
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2391-5455
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10.1515/math-2022-0004
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Abstract

1IntroductionWith the wide application of stochastic differential equations driven by Lévy noise in biology, engineering, finance and economy, more and more experts and scholars pay attention to stochastic differential equations [1,2,3]. The stability has become one of the important topics, such as stochastic stability, stochastic asymptotic stability, moment exponential stability, almost everywhere stability and mean square polynomial stability (see [4,5,6, 7,8,9]). Li and Deng [10] studied the almost sure stability with general decay rate of neutral delay stochastic differential equations with Lévy noise, while Shen et al. [11] studied the stability of solutions of neutral stochastic functional hybrid differential equations with Lévy noise. Shen’s conclusion is more specific and universal than Deng’s.As is known to all, the integer order differential equations determine the local characteristics of the function, while the fractional order differential equations describe the overall information of the function in the form of weighting, so it is more flexible and widely used in the model. Abouagwa et al. [12] studied the existence and uniqueness by using Carathéodory approximation under non-Lipschitz conditions. Shen et al. [13] obtained an averaging principle and stability of hybrid stochastic fractional differential equations driven by Lévy noise. Recently, the classical mathematical modelling approach coupled with the stochastic methods were used to develop stochastic dynamic models for financial data (stock price). In order to extend this approach to more complex dynamic processes in science and engineering operating under internal structural and external environmental perturbations, Pedjeu and Ladde [14] modified the existing mathematical models by incorporating certain significant attributable parameters or variables with state variables, explicitly. Meanwhile, they obtained the existence and uniqueness of the solution by using the Picard-Lindel successive approximations. This motivates us to initiate to partially characterize intrastructural and external environmental perturbations by a set of linearly independent time-scales. For example, t,B(t),tα,N˜t,B\left(t),{t}^{\alpha },\widetilde{N}, where B(t)B\left(t)is the standard Wiener process, and N˜\widetilde{N}is the Lévy process, α∈(0,1]\alpha \in (0,1].In addition, impulsive stochastic differential systems [15,16,17] with Markovian switching have been investigated. Zhu [18] has determined the pth moment exponential stability of impulsive stochastic functional differential equations with Markovian switching. Then Kao et al. [19] proved the pth moment exponential stability, almost exponential stability and instability on the basis of predecessors. Tan et al. [20] discussed the stability of hybrid impulsive and switching stochastic systems with time delay.However, it is worth mentioning that to the best of our knowledge, the stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects has not been investigated yet and this arouses our interest in research. In order to fill this gap, in this paper, combined with previous work, we consider the following stochastic differential equations driven by Lévy noise with impulsive effects (1)dx(t)=u(x(t−),t,r(t))dt+b(x(t−),t,r(t))dB(t)+σ(x(t−),t,r(t))(dt)α+∫∣y∣<ch(x(t−),y,t,r(t))N˜(dt,dy),t≠tk,t≥0,Δx(tk)=Ik(x(tk−),tk),k∈N,\left\{\begin{array}{l}{\rm{d}}x\left(t)=u\left(x\left(t-),t,r\left(t)){\rm{d}}t+b\left(x\left(t-),t,r\left(t)){\rm{d}}B\left(t)+\sigma \left(x\left(t-),t,r\left(t)){\left({\rm{d}}t)}^{\alpha }\\ \hspace{2.0em}\hspace{2.0em}+\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x\left(t-),y,t,r\left(t))\widetilde{N}\left({\rm{d}}t,{\rm{d}}y),\hspace{1em}t\ne {t}_{k},t\ge 0,\\ \Delta x\left({t}_{k})={I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),\hspace{1em}k\in {\mathbb{N}},\end{array}\right.where 0<α≤10\lt \alpha \le 1, x(0)=x0∈Rnx\left(0)={x}_{0}\in {{\mathbb{R}}}^{n}is the initial value satisfying E∣x0∣2<∞{\mathbb{E}}| {x}_{0}\hspace{-0.25em}{| }^{2}\lt \infty , the constant ccis the maximum allowable jump size and the mappings u,σ:Rn×R+×S→Rn,b:Rn×R+×S→Rn×mu,\sigma :{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{n},b:{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{n\times m}, h:Rn×R×R+×S→Rnh:{{\mathbb{R}}}^{n}\times {\mathbb{R}}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{n}are continuous functions. The fixed impulse time sequence {tk}k∈N{\left\{{t}_{k}\right\}}_{k\in {\mathbb{N}}}satisfies 0≤t0<t1<⋯<tk<⋯,tk→∞0\le {t}_{0}\lt {t}_{1}\hspace{0.33em}\lt \cdots \lt {t}_{k}\hspace{0.33em}\lt \cdots \hspace{0.33em},\hspace{0.25em}{t}_{k}\to \infty (as k→∞k\to \infty ), Δx(tk)=Ik(x(tk−),tk)\Delta x\left({t}_{k})={I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k})denotes the state jumping at impulsive time instant tk{t}_{k}. Denote by x(t;0,x0)x\left(t;\hspace{0.33em}0,{x}_{0})the solution to the system which is assumed to be right continuous, i.e., x(tk+)=x(tk)x\left({t}_{k}^{+})=x\left({t}_{k}).In this paper, we utilize the local Lipschitz condition and a weaker condition to replace the linear growth condition to obtain a unique global solution for system (1). According to the method of Lyapunov function, we can prove there is a unique global solution. Then we present a kind of λ\lambda -type function which will be introduced in Section 2. By means of nonnegative semi-martingale convergence theorem and Lyapunov function, we derive a kind of almost sure λ\lambda -type stability, including almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability. The main features of this paper are as follows: (i)The presented hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects have not been considered before.(ii)A more general almost sure stability (including almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability) problem has been investigated under much weaker conditions.(iii)The upper bound of each coefficient at any mode is obtained.The article is organized as follows. In Section 2, we present several definitions and preliminaries. In Section 3, the conditions for the existence and uniqueness of the global solution and the sufficient conditions for λ\lambda -type stability are established, respectively. In Section 4, we prove the λ\lambda -type stability about the upper bound of each coefficient at any mode by using the theory of the MM-matrix. Finally, an example is given to illustrate the obtained theory.2PreliminariesThroughout this paper, unless otherwise specified, we use the following notations. Rn{{\mathbb{R}}}^{n}denotes the nn-dimensional Euclidean space, and ∣x∣| x| denotes the Euclidean norm of a vector x. R=(−∞,+∞){\mathbb{R}}=\left(-\infty ,+\infty )and R+=[0,+∞){{\mathbb{R}}}^{+}={[}0,+\infty ). Diag(ζ1,…,ζN{\zeta }_{1},\ldots ,{\zeta }_{N}) denotes a diagonal matrix with diagonal entries ζ1,…,ζN{\zeta }_{1},\ldots ,{\zeta }_{N}. Let (Ω,ℱ,{ℱt}t≥0,P)\left(\Omega ,{\mathcal{ {\mathcal F} }},{\left\{{{\mathcal{ {\mathcal F} }}}_{t}\right\}}_{t\ge 0},{\mathbb{P}})be a complete probability space with a filtration {ℱt}t≥0{\left\{{{\mathcal{ {\mathcal F} }}}_{t}\right\}}_{t\ge 0}satisfying the usual conditions (i.e., it is right continuous and ℱ0{{\mathcal{ {\mathcal F} }}}_{0}contains all PP-null sets). B(t)=(B1(t),B2(t),…,Bm(t))TB\left(t)={\left({B}_{1}\left(t),{B}_{2}\left(t),\ldots ,{B}_{m}\left(t))}^{T}be an mm-dimensional ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted Brownian motion defined on the complete probability space (Ω,ℱ,ℱt≥0,P)\left(\Omega ,{\mathcal{ {\mathcal F} }},{{\mathcal{ {\mathcal F} }}}_{t\ge 0},{\mathbb{P}}), and N(t,z)N\left(t,z)be a ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted Poisson random measure on [0,+∞)×R{[}0,+\infty )\times {\mathbb{R}}with a σ\sigma -finite intensity measure ν\nu (dz), the compensator martingale measure N˜(t,z)\widetilde{N}\left(t,z)satisfies N˜(dt,dz)=N(dt,dz)−ν(dz)dt\widetilde{N}\left({\rm{d}}t,{\rm{d}}z)=N\left({\rm{d}}t,{\rm{d}}z)-\nu \left({\rm{d}}z){\rm{d}}t. Let r(t)r\left(t), t≥0t\ge 0be a right-continuous Markov chain defined on the probability space taking values in a finite state S={1,2,…N}{\mathbb{S}}=\left\{1,2,\ldots N\right\}with generator Γ=(γij)N×N\Gamma ={\left({\gamma }_{ij})}_{N\times N}given by P{r(t+Δ)=j∣r(t)=i}=γijΔ+o(Δ),ifi≠j,1+γijΔ+o(Δ),ifi=j,P\left\{r\left(t+\Delta )=j| r\left(t)=i\right\}=\left\{\begin{array}{ll}{\gamma }_{ij}\Delta +o\left(\Delta ),& {\rm{if}}\hspace{0.33em}i\ne j,\\ 1+{\gamma }_{ij}\Delta +o\left(\Delta ),& {\rm{if}}\hspace{0.33em}i=j,\end{array}\right.where Δ>0\Delta \gt 0, and γij≥0{\gamma }_{ij}\ge 0is the transition rate from iito jjif i≠ji\ne jwhile γii=−∑i≠jγij{\gamma }_{ii}=-{\sum }_{i\ne j}{\gamma }_{ij}. And we assume that the Markov chain r(t)r\left(t)is ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted but independent of the Brownian motion B(t)B\left(t).Next, we give some definitions about fractional calculus and λ\lambda -type function, which will be used in this paper.Definition 2.1[21] (Riemann-Liouville fractional integrals, Samko et al., 1993): For any α∈(0,1)\alpha \in \left(0,1)and function f∈L1[[a,b];Rn]f\in {L}^{1}\left[\left[a,b];\hspace{0.33em}{{\mathbb{R}}}^{n}], the left-sided and right-sided Riemann-Liouville fractional integrals of order α\alpha are defined for almost all a<t<ba\lt t\lt bby (Ia+αf)(t)=1Γ(α)∫at(t−s)α−1f(s)ds,t>a\left({I}_{a+}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(\alpha )}\underset{a}{\overset{t}{\int }}{\left(t-s)}^{\alpha -1}f\left(s){\rm{d}}s,\hspace{1.0em}t\gt aand (Ib−αf)(t)=1Γ(α)∫tb(t−s)α−1f(s)ds,t<b,\left({I}_{b-}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(\alpha )}\underset{t}{\overset{b}{\int }}{\left(t-s)}^{\alpha -1}f\left(s){\rm{d}}s,\hspace{1.0em}t\lt b,where Γ(α)=∫0∞sα−1e−sds\Gamma \left(\alpha )={\int }_{0}^{\infty }{s}^{\alpha -1}{e}^{-s}{\rm{d}}sis the Gamma function and L1[a,b]{L}^{1}\left[a,b]is the space of integrable functions in a finite interval [a,b]\left[a,b]of R{\mathbb{R}}.Definition 2.2[21] (Riemann-Liouville fractional derivatives, Samko et al., 1993): For any α∈(0,1)\alpha \in \left(0,1)and well-defined absolutely continuous function ffon an interval [a,b]\left[a,b], the left-sided and right-sided Riemann-Liouville fractional derivatives are defined, respectively, by (Da+αf)(t)=1Γ(1−α)f(a)(t−a)α+∫at(t−s)−αf′(s)ds\left({D}_{a+}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(1-\alpha )}\left[\frac{f\left(a)}{{\left(t-a)}^{\alpha }}+\underset{a}{\overset{t}{\int }}{\left(t-s)}^{-\alpha }f^{\prime} \left(s){\rm{d}}s\right]and (Db−αf)(t)=1Γ(1−α)f(b)(b−t)α−∫tb(s−t)−αf′(s)ds.\left({D}_{b-}^{\alpha }f)\left(t)=\frac{1}{\Gamma \left(1-\alpha )}\left[\frac{f\left(b)}{{\left(b-t)}^{\alpha }}-\underset{t}{\overset{b}{\int }}{\left(s-t)}^{-\alpha }f^{\prime} \left(s){\rm{d}}s\right].Definition 2.3[22] (Jumarie, 2005): Let σ(t)\sigma \left(t)be a continuous function, then its integration with respect to (dt)α{\left({\rm{d}}t)}^{\alpha }, 0<α≤10\lt \alpha \le 1, is defined by ∫0tσ(s)(ds)α=α∫0t(t−s)α−1σ(s)ds.\underset{0}{\overset{t}{\int }}\sigma \left(s){\left({\rm{d}}s)}^{\alpha }=\alpha \underset{0}{\overset{t}{\int }}{\left(t-s)}^{\alpha -1}\sigma \left(s){\rm{d}}s.Definition 2.4The function λ:R→(0,∞)\lambda :{\mathbb{R}}\to \left(0,\infty )is said to be λ\lambda -type function if the function satisfies the following three conditions: (1)It is continuous and nondecreasing in R{\mathbb{R}}and differentiable in R+{{\mathbb{R}}}^{+},(2)λ(0)=1,λ(∞)=∞\lambda \left(0)=1,\lambda \left(\infty )=\infty and r=supt≥0λ′(t)λ(t)<∞r={\sup }_{t\ge 0}\left[\frac{\lambda ^{\prime} \left(t)}{\lambda \left(t)}\right]\lt \infty ,(3)For any s,t≥0,λ(t)≤λ(s)λ(t−s)s,t\ge 0,\lambda \left(t)\le \lambda \left(s)\lambda \left(t-s).Remark 2.5It is obvious that the functions λ(t)=et,λ(t)=(1+t+)\lambda \left(t)={e}^{t},\lambda \left(t)=\left(1+{t}^{+})and log(1+t+)\log \left(1+{t}^{+})are λ\lambda -type functions since they satisfy the aforementioned three conditions. Next, we give the definition of the almost sure stability with general decay rate based on Definition 2.4.Definition 2.6Let the function λ(t)∈C(R+;R+)\lambda \left(t)\in C\left({{\mathbb{R}}}^{+};\hspace{0.33em}{{\mathbb{R}}}^{+})be a λ\lambda -type function. Then for any initial x0∈Rn{x}_{0}\in {{\mathbb{R}}}^{n}, the trivial solution is said to be almost surely stable with decay λ(t)\lambda \left(t)of order γ\gamma if limsupt→∞log∣x(t,x0)∣logλ(t)≤−γ,a.s.\mathop{\mathrm{limsup}}\limits_{t\to \infty }\frac{\log | x\left(t,{x}_{0})| }{\log \lambda \left(t)}\le -\gamma ,\hspace{1.0em}\hspace{0.1em}\text{a.s.}\hspace{0.1em}Remark 2.7It is obvious that this almost sure λ\lambda -type stability implies the almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability when λ(t)\lambda \left(t)is replaced by et,1+t+,log(1+t+){e}^{t},1+{t}^{+},\log \left(1+{t}^{+}), respectively. Because we have a wide choice for λ\lambda -type functions, thus our results will be more general than some classical results.Let C2,1(Rn×R+×S→R+){C}^{2,1}\left({{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}\to {{\mathbb{R}}}^{+})denote the family of all functions V(x,t,i)V\left(x,t,i)on Rn×R+×S{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}}, which are continuously twice differentiable in xxand once in tt. Define three functions L1V,L2V,L3V:Rn×R+×S→R{L}_{1}V,{L}_{2}V,{L}_{3}V:{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\hspace{0.25em}\times {\mathbb{S}}\to {\mathbb{R}}by L1V(x,t,i)=Vt(x,t,i)+Vx(x,t,i)u(x,t,i)+∑j=1NγijV(x,t,j)+12trace[bT(x,t,i)Vxx(x,t,i)b(x,t,i)],L2V(x,t,i)=∫∣y∣<c[V(x+h(x,y,t,i),t,i)−V(x,t,i)−Vx(x,t,i)h(x,y,t,i)]ν(dy),L3V(x,t,i)=Vx(x,t,i)σ(x,t,i),\begin{array}{rcl}{L}_{1}V\left(x,t,i)& =& {V}_{t}\left(x,t,i)+{V}_{x}\left(x,t,i)u\left(x,t,i)+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}V\left(x,t,j)+\frac{1}{2}\hspace{0.1em}\text{trace}\hspace{0.1em}\left[{b}^{T}\left(x,t,i){V}_{xx}\left(x,t,i)b\left(x,t,i)],\\ {L}_{2}V\left(x,t,i)& =& \mathop{\displaystyle \int }\limits_{| y| \lt c}\left[V\left(x+h\left(x,y,t,i),t,i)-V\left(x,t,i)-{V}_{x}\left(x,t,i)h\left(x,y,t,i)]\nu \left({\rm{d}}y),\\ {L}_{3}V\left(x,t,i)& =& {V}_{x}\left(x,t,i)\sigma \left(x,t,i),\end{array}where Vt(x,t,i)=∂V(x,t,i)∂t{V}_{t}\left(x,t,i)=\frac{\partial V\left(x,t,i)}{\partial t}, Vx(x,t,i)=∂V(x,t,i)∂x1,…,∂V(x,t,i)∂xn{V}_{x}\left(x,t,i)=\left(\frac{\partial V\left(x,t,i)}{\partial {x}_{1}},\ldots ,\frac{\partial V\left(x,t,i)}{\partial {x}_{n}}\right), Vxx(x,t,i)=∂V2(x,t,i)∂xk∂xln×n{V}_{xx}\left(x,t,i)={\left(\frac{\partial {V}^{2}\left(x,t,i)}{\partial {x}_{k}\partial {x}_{l}}\right)}_{n\times n}. By the generalized Itô formula, for t∈[tj−1,tj)t\in {[}{t}_{j-1},{t}_{j})V(x(t),t,r(t))=V(x(tj−1),tj−1,r(tj−1))+∫tj−1tL1V(x(s),s,r(s))ds+∫tj−1tL2V(x(s),s,r(s))ds+∫tj−1tα(t−s)α−1L3V(x(s),s,r(s))ds+G(t),\begin{array}{rcl}V\left(x\left(t),t,r\left(t))& =& V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s+G\left(t),\end{array}where G(t)=∫tj−1tVx(x(s),s,r(s))b(x(s−),s,r(s))dB(s)+∫tj−1t∫∣y∣<c[V(x(s)+h(x(s−),y,s,r(s)),s,r(s))−V(x(s),s,r(s))]N˜(ds,dy).G\left(t)=\underset{{t}_{j-1}}{\overset{t}{\int }}{V}_{x}\left(x\left(s),s,r\left(s))b\left(x\left(s-),s,r\left(s)){\rm{d}}B\left(s)+\underset{{t}_{j-1}}{\overset{t}{\int }}\mathop{\int }\limits_{| y| \lt c}\left[V\left(x\left(s)+h\left(x\left(s-),y,s,r\left(s)),s,r\left(s))-V\left(x\left(s),s,r\left(s))]\tilde{N}\left({\rm{d}}s,{\rm{d}}y).In fact, if a stochastic process is a martingale, then it is a local martingale. Hence, we can easily know that {G(t)}t≥0{\left\{G\left(t)\right\}}_{t\ge 0}is a local martingale.3Main resultBefore we state our main results in this section, the following hypotheses are imposed.Assumption 3.1(Local Lipschitz condition). For arbitrary x1,x2∈Rn{x}_{1},{x}_{2}\in {{\mathbb{R}}}^{n}, and ∣x1∣∨∣x2∣≤n| {x}_{1}| \vee | {x}_{2}| \le n, there is a positive constant Ln{L}_{n}such that ∣u(x1,t,i)−u(x2,t,i)∣∨∣b(x1,t,i)−b(x2,t,i)∣∨∣σ(x1,t,i)−σ(x2,t,i)∣∨∫∣y∣<c∣h(x1,y,t,i)−h(x2,y,t,i)∣ν(dy)≤Ln(∣x1−x2∣2).| u\left({x}_{1},t,i)-u\left({x}_{2},t,i)| \vee | b\left({x}_{1},t,i)-b\left({x}_{2},t,i)| \vee | \sigma \left({x}_{1},t,i)-\sigma \left({x}_{2},t,i)| \vee \mathop{\int }\limits_{| y| \lt c}| h\left({x}_{1},y,t,i)-h\left({x}_{2},y,t,i)| \nu \left({\rm{d}}y)\le {L}_{n}\left(| {x}_{1}-{x}_{2}\hspace{-0.25em}{| }^{2}).Assumption 3.2For any (t,i)∈R×S,u(0,t,i)=b(0,t,i)=σ(0,t,i)=h(0,y,t,i)=0\left(t,i)\in {\mathbb{R}}\times {\mathbb{S}},u\left(0,t,i)=b\left(0,t,i)=\sigma \left(0,t,i)=h\left(0,y,t,i)=0.For the stability analysis, Assumption 3.2 implies that x(t)=0x\left(t)=0is the trivial solution.Assumption 3.3There are several nonnegative number CCand pu{p}_{u}, real number Ku{K}_{u}and βu{\beta }_{u}(1≤u≤U1\le u\le U, for positive integer UU), a Lyapunov function V∈C2,1(Rn×R+×S;R+)V\in {C}^{2,1}\left({{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times {\mathbb{S}};\hspace{0.33em}{{\mathbb{R}}}^{+}), such that (i)lim∣x∣→∞inf0≤t<∞V(x(t),t,i)=∞{\mathrm{lim}}_{| x| \to \infty }{\inf }_{0\le t\lt \infty }V\left(x\left(t),t,i)=\infty .(ii)L1V(x,s,i)+L2V(x,s,i)+α(t−s)α−1L3V(x,s,i)≤C+∑u=1UKu∣x∣pu{L}_{1}V\left(x,s,i)+{L}_{2}V\left(x,s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\le C+{\sum }_{u=1}^{U}{K}_{u}| x\hspace{-0.25em}{| }^{{p}_{u}}.(iii)V(x(tk−)+Ik(x(tk−),tk),tk,r(tk))≤d(x(tk−),tk)+V(x(tk−),tk,r(tk)).V\left(x\left({t}_{k}^{-})+{I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),{t}_{k},r\left({t}_{k}))\le d\left(x\left({t}_{k}^{-}),{t}_{k})+V\left(x\left({t}_{k}^{-}),{t}_{k},r\left({t}_{k})).(iv)∑j=nkd(x(tj−1−),tj−1)≤∑u=1Uβu∫tj−1tk∣x∣puds{\sum }_{j=n}^{k}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\le {\sum }_{u=1}^{U}{\beta }_{u}{\int }_{{t}_{j-1}}^{{t}_{k}}| x\hspace{-0.25em}{| }^{{p}_{u}}{\rm{d}}s.For any x∈Rn,y∈R,0≤s<t,i∈Sx\in {{\mathbb{R}}}^{n},y\in {\mathbb{R}},0\le s\lt t,i\in {\mathbb{S}}.According to the above assumptions, let us state the following existence and uniqueness theorem:Theorem 3.4Assume that Assumptions 3.1–3.3 hold, then for any initial data x0∈Rn{x}_{0}\in {{\mathbb{R}}}^{n}, there is a unique global solution x(t;0,x(0))x\left(t;\hspace{0.33em}0,x\left(0))on t>0t\gt 0to system (1).ProofApplying the standing truncation technique, Assumptions 3.1 and 3.2 admit a unique maximal local solution to system (1)). Let x(t)(t∈[0,ϱ∞))x\left(t)\left(t\in {[}0,{\varrho }_{\infty }))be the maximal local solution to system (1) and ϱ∞{\varrho }_{\infty }be the explosion time. And let a0∈R+{a}_{0}\in {{\mathbb{R}}}^{+}be sufficiently large for ∣x0∣≤a0| {x}_{0}| \le {a}_{0}. For any integer a≥a0a\ge {a}_{0}, define the stopping time τa=inf{t∈[0,ϱ∞);∣x(t)∣≥a},{\tau }_{a}=\inf \left\{t\in {[}0,{\varrho }_{\infty });\hspace{0.33em}| x\left(t)| \ge a\right\},where infϕ=∞\inf \phi =\infty . Obviously, the sequence τa{\tau }_{a}is increasing. So we have a limit τ∞=lima→∞τa{\tau }_{\infty }={\mathrm{lim}}_{a\to \infty }{\tau }_{a}, whence τ∞≤ϱ∞{\tau }_{\infty }\le {\varrho }_{\infty }. If we can show that τ∞=∞{\tau }_{\infty }=\infty a.s., then we have ϱ∞=∞{\varrho }_{\infty }=\infty a.s. Therefore, we only need to devote to prove τ∞=∞{\tau }_{\infty }=\infty a.s., which is equivalent to proving that P(τa≤t)→0P\left({\tau }_{a}\le t)\to 0as a→∞a\to \infty for any t>0t\gt 0. In fact, by the generalized Itô formula, for t∈[tj−1,tj)t\in {[}{t}_{j-1},{t}_{j}), we have (2)E(Iτa≤tV(x(τa),τa,r(τa)))≤EV(x(t∧τa),t∧τa,r(t∧τa))=EV(x(tj−1),tj−1,r(tj−1))+E∫tj−1t∧τaL1V(x(s),s,r(s))ds+E∫tj−1t∧τaL2V(x(s),s,r(s))ds+E∫tj−1t∧τaα(t−s)α−1L3V(x(s),s,r(s))ds.\begin{array}{l}{\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a})))\\ \hspace{1.0em}\le {\mathbb{E}}V\left(x\left(t\wedge {\tau }_{a}),t\wedge {\tau }_{a},r\left(t\wedge {\tau }_{a}))\\ \hspace{1.0em}={\mathbb{E}}V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))+{\mathbb{E}}\underset{{t}_{j-1}}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{\mathbb{E}}\underset{{t}_{j-1}}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{\mathbb{E}}\underset{{t}_{j-1}}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s.\end{array}By condition (iii), we have V(x(tj−1),tj−1,r(tj−1))=V(x(tj−1−)+Ij−1(x(tj−1−),tk),tj−1−,r(tj−1))≤d(x(tj−1−),tj−1)+V(x(tj−1−),tj−1−,r(tj−1)).\hspace{-35.5em}\begin{array}{rcl}V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))& =& V\left(x\left({t}_{j-1}^{-})+{I}_{j-1}\left(x\left({t}_{j-1}^{-}),{t}_{k}),{t}_{j-1}^{-},r\left({t}_{j-1}))\\ & \le & d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})+V\left(x\left({t}_{j-1}^{-}),{t}_{j-1}^{-},r\left({t}_{j-1})).\end{array}Hence, for all t≥0t\ge 0, (3)E(Iτa≤tV(x(τa),τa,r(τa)))≤EV(x(0),0,r(0))+E∫0t∧τaL1V(x(s),s,r(s))ds+E∫0t∧τaL2V(x(s),s,r(s))ds+E∫0t∧τaα(t−s)α−1L3V(x(s),s,r(s))ds+∑j:0<tj≤tEd(x(tj−1−),tj−1)≤EV(x(0),0,r(0))+Ct+∑u=1UKuE∫0t∧τa∣x(s)∣puds+∑j:0<tj≤tEd(x(tj−1−),tj−1).\begin{array}{rcl}{\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a})))& \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}{L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s+\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\mathbb{E}}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\\ & \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+\mathop{\displaystyle \sum }\limits_{u=1}^{U}{K}_{u}{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s+\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\mathbb{E}}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1}).\end{array}For almost all ω∈Ω\omega \in \Omega , there is an integer m0=m0(ω){m}_{0}={m}_{0}\left(\omega ), for any m≥m0m\ge {m}_{0}and 0≤t∧τa<m0\le t\wedge {\tau }_{a}\lt m, define tkm=max{tk:tk≤t∧τa}.\hspace{-17em}{t}_{{k}_{m}}={\rm{\max }}\left\{{t}_{k}:{t}_{k}\le t\wedge {\tau }_{a}\right\}.Combining with condition (iv), (4)∑j:0<tj≤td(x(tj−1−),tj−1)=∑j=1tkmd(x(tj−1−),tj−1)=∑u=1Uβu∫0tkm∣x(s)∣puds.\sum _{j:0\lt {t}_{j}\le t}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})=\mathop{\sum }\limits_{j=1}^{{t}_{{k}_{m}}}d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})=\mathop{\sum }\limits_{u=1}^{U}{\beta }_{u}\underset{0}{\overset{{t}_{{k}_{m}}}{\int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s.\hspace{8.5em}Substituting (4) to (3), for 0<t∧τa<m0\lt t\wedge {\tau }_{a}\lt m, (5)E(Iτa≤tV(x(τa),τa,r(τa)))≤EV(x(0),0,r(0))+Ct+∑u=1UKuE∫0t∧τa∣x(s)∣puds+∑u=1UβuE∫0tkm∣x(s)∣puds≤EV(x(0),0,r(0))+Ct+∑u=1U(Ku+βu)E∫0t∣x(s)∣puds.\begin{array}{rcl}{\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a})))& \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+\mathop{\displaystyle \sum }\limits_{u=1}^{U}{K}_{u}{\mathbb{E}}\underset{0}{\overset{t\wedge {\tau }_{a}}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s+\mathop{\displaystyle \sum }\limits_{u=1}^{U}{\beta }_{u}{\mathbb{E}}\underset{0}{\overset{{t}_{{k}_{m}}}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s\\ & \le & {\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+\mathop{\displaystyle \sum }\limits_{u=1}^{U}\left({K}_{u}+{\beta }_{u}){\mathbb{E}}\underset{0}{\overset{t}{\displaystyle \int }}| x\left(s){| }^{{p}_{u}}{\rm{d}}s.\end{array}By lim∣x∣→∞inf0≤t<∞V(x(t),t,r(t))=∞{\mathrm{lim}}_{| x| \to \infty }{\inf }_{0\le t\lt \infty }V\left(x\left(t),t,r\left(t))=\infty , let μa=inf∣x∣≥a,0≤t<∞V(x(t),t,r(t)){\mu }_{a}={\inf }_{| x| \ge a,0\le t\lt \infty }V\left(x\left(t),t,r\left(t)), for a≥a0a\ge {a}_{0}. Therefore, we have P(τa≤t)μa≤E(Iτa≤tV(x(τa),τa,r(τa))).P\left({\tau }_{a}\le t){\mu }_{a}\le {\mathbb{E}}\left({I}_{{\tau }_{a}\le t}V\left(x\left({\tau }_{a}),{\tau }_{a},r\left({\tau }_{a}))).\hspace{0.16em}Using the idea of Theorem 3.1 in [8], letting a→∞a\to \infty , by Fatou’s lemma, we can derive (6)0≤P(τ∞≤t)≤lima→∞P(τa≤t)=lima→∞EV(x(0),0,r(0))+Ct+∑u=1U(Ku+βu)E∫0t∣x(s)∣pudsμa=0.\begin{array}{rcl}0\le P\left({\tau }_{\infty }\le t)& \le & \mathop{\mathrm{lim}}\limits_{a\to \infty }P\left({\tau }_{a}\le t)\\ & =& \mathop{\mathrm{lim}}\limits_{a\to \infty }\frac{{\mathbb{E}}V\left(x\left(0),0,r\left(0))+Ct+{\displaystyle \sum }_{u=1}^{U}\left({K}_{u}+{\beta }_{u}){\mathbb{E}}{\displaystyle \int }_{0}^{t}| x\left(s){| }^{{p}_{u}}{\rm{d}}s}{{\mu }_{a}}\\ & =& 0.\end{array}This implies that there exists a unique global solution x(t;0,x(0))x\left(t;\hspace{0.33em}0,x\left(0))for system (1).□Next, in order to obtain sufficient conditions of the almost sure stability with general decay rate, we need the following lemmas.Lemma 3.5[23] Let {Mt}t≥0{\left\{{M}_{t}\right\}}_{t\ge 0}be a local martingale and {Nt}t≥0{\left\{{N}_{t}\right\}}_{t\ge 0}be a locally bounded predictable process, then the stochastic integral ∫0tNsdMs{\int }_{0}^{t}{N}_{s}{\rm{d}}{M}_{s}is also a local martingale.Lemma 3.6(Nonnegative semi-martingale convergence theorem) [10] Assume {At}\left\{{A}_{t}\right\}and {Ut}\left\{{U}_{t}\right\}are two continuous predictable increasing processes vanishing at t=0t=0a.s. and {Mt}\left\{{M}_{t}\right\}is a real-valued continuous local martingale with M0=0{M}_{0}=0a.s. Let X(t)X\left(t)be a nonnegative adapted process and ξ\xi be a nonnegative ℱ0{{\mathcal{ {\mathcal F} }}}_{0}-measurable random variable satisfyingXt≤ξ+At−Ut+Mt,t≥0.{X}_{t}\le \xi +{A}_{t}-{U}_{t}+{M}_{t},\hspace{1.0em}t\ge 0.If A∞≔limt→∞At<∞{A}_{\infty }:= {\mathrm{lim}}_{t\to \infty }{A}_{t}\lt \infty , a.s., then, we havelimt→∞supXt<∞,a.s.\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup {X}_{t}\lt \infty ,\hspace{1em}{a.s.}With these above assumptions and lemmas, we can now state our results about the almost sure stability with general decay rate.Theorem 3.7Let Assumptions 3.1 and 3.2 hold. If there are positive numbers c1,c2,α1,α2,q>2{c}_{1},{c}_{2},{\alpha }_{1},{\alpha }_{2},q\gt 2, such that the functions V(x,t,i)V\left(x,t,i)and LjV(x,t,i)(j=1,2,3){L}_{j}V\left(x,t,i)\left(j=1,2,3)satisfy(i)c1∣x∣2≤V(x,t,i)≤c2∣x∣2.{c}_{1}| x\hspace{-0.25em}{| }^{2}\le V\left(x,t,i)\le {c}_{2}| x\hspace{-0.25em}{| }^{2}.(ii)L1V(x,s,i)+L2V(x,s,i)+α(t−s)α−1L3V(x,s,i)≤−α1∣x∣2−α2∣x∣q.{L}_{1}V\left(x,s,i)+{L}_{2}V\left(x,s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\le -{\alpha }_{1}| x\hspace{-0.25em}{| }^{2}-{\alpha }_{2}| x\hspace{-0.25em}{| }^{q}.(iii)V(x(tk−)+Ik(x(tk−),tk),tk,r(tk))≤d(x(tk−),tk)+V(x(tk−),tk,r(tk)).V\left(x\left({t}_{k}^{-})+{I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),{t}_{k},r\left({t}_{k}))\le d\left(x\left({t}_{k}^{-}),{t}_{k})+V\left(x\left({t}_{k}^{-}),{t}_{k},r\left({t}_{k})).(iv)Σj=nkλε(tj−1)d(x(tj−1−),tj−1)≤α3∫tj−1tkλε(s)∣x∣2ds+α4∫tj−1tkλε(s)∣x∣qds{\Sigma }_{j=n}^{k}{\lambda }^{\varepsilon }\left({t}_{j-1})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\le {\alpha }_{3}{\int }_{{t}_{j-1}}^{{t}_{k}}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{2}{\rm{d}}s+{\alpha }_{4}{\int }_{{t}_{j-1}}^{{t}_{k}}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{q}{\rm{d}}s.If there exists a small enough ε>0\varepsilon \gt 0such that α1−c2εα−α3>0,α2−α4>0{\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3}\gt 0,{\alpha }_{2}-{\alpha }_{4}\gt 0.Therefore, for any initial data x0{x}_{0}, the inequalitylimt→∞suplog∣x(t,x0)∣logλ(t)<−ε2\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup \frac{\log | x\left(t,{x}_{0})| }{\log \lambda \left(t)}\lt -\frac{\varepsilon }{2}holds, that is, the trivial solution of system (1) is almost surely stable with decay λ(t)\lambda \left(t)of order ε2\frac{\varepsilon }{2}.ProofNote that conditions (i)–(iv) are stronger than Assumption 3.3, so there is a unique global solution for system (1). Let λ(t)\lambda \left(t)be a λ\lambda -type function, and applying the generalized Itô formula to λε(t)V(x(t),t,r(t)){\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t)), for t∈[tj−1,tj)t\in {[}{t}_{j-1},{t}_{j})(7)λε(t)V(x(t),t,r(t))=λε(tj−1)V(x(tj−1),tj−1,r(tj−1))+∫tj−1tελ′(s)λ(s)λε(s)V(x(s),s,r(s))ds+∫tj−1tλε(s)L1V(x(s),s,r(s))ds+∫tj−1tλε(s)L2V(x(s),s,r(s))ds+∫tj−1tλε(s)α(t−s)α−1L3V(x(s),s,r(s))ds+Mtj−1t,\hspace{0.8em}\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& =& {\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}\varepsilon \frac{\lambda ^{\prime} \left(s)}{\lambda \left(s)}{\lambda }^{\varepsilon }\left(s)V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\underset{{t}_{j-1}}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s+{M}_{{t}_{j-1}}^{t},\end{array}where Mtj−1t=∫tj−1tλε(s)dG(s){M}_{{t}_{j-1}}^{t}={\int }_{{t}_{j-1}}^{t}{\lambda }^{\varepsilon }\left(s){\rm{d}}G\left(s)is a real-valued continuous local martingale with M0=0{M}_{0}=0by Lemma 3.5. By condition (iii), we have (8)λε(tj−1)V(x(tj−1),tj−1,r(tj−1))=λε(tj−1)V(x(tj−1−)+Ij−1(x(tj−1−),tk),tj−1−,r(tj−1))≤λε(tj−1)[d(x(tj−1−),tj−1)+V(x(tj−1−),tj−1−,r(tj−1))]=λε(tj−1)V(x(tj−1−),tj−1−,r(tj−1))+λε(tj−1)d(x(tj−1−),tj−1).\begin{array}{rcl}{\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}),{t}_{j-1},r\left({t}_{j-1}))& =& {\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}^{-})+{I}_{j-1}\left(x\left({t}_{j-1}^{-}),{t}_{k}),{t}_{j-1}^{-},r\left({t}_{j-1}))\\ & \le & {\lambda }^{\varepsilon }\left({t}_{j-1})\left[d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})+V\left(x\left({t}_{j-1}^{-}),{t}_{j-1}^{-},r\left({t}_{j-1}))]\\ & =& {\lambda }^{\varepsilon }\left({t}_{j-1})V\left(x\left({t}_{j-1}^{-}),{t}_{j-1}^{-},r\left({t}_{j-1}))+{\lambda }^{\varepsilon }\left({t}_{j-1})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1}).\end{array}\hspace{0.25em}Hence, for all t≥0t\ge 0, we have (9)λε(t)V(x(t),t,r(t))=V(x(0),0,r(0))+∫0tελ′(s)λ(s)λε(s)V(x(s),s,r(s))ds+∫0tλε(s)L1V(x(s),s,r(s))ds+∫0tλε(s)L2V(x(s),s,r(s))ds∫0tλε(s)α(t−s)α−1L3V(x(s),s,r(s))ds+∑j:0<tj≤tλε(tj−1)d(x(tj−1−),tj−1)+M0t.\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& =& V\left(x\left(0),0,r\left(0))+\underset{0}{\overset{t}{\displaystyle \int }}\varepsilon \frac{\lambda ^{\prime} \left(s)}{\lambda \left(s)}{\lambda }^{\varepsilon }\left(s)V\left(x\left(s),s,r\left(s)){\rm{d}}s+\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{1}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s){L}_{2}V\left(x\left(s),s,r\left(s)){\rm{d}}s\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x\left(s),s,r\left(s)){\rm{d}}s\\ & & +\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\lambda }^{\varepsilon }\left({t}_{j-1})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})+{M}_{0}^{t}.\end{array}For almost all ω∈Ω\omega \in \Omega , there is an integer m0=m0(ω){m}_{0}={m}_{0}\left(\omega ), for any m≥m0m\ge {m}_{0}and 0≤t<m0\le t\lt m, define tkm=max{tk:tk≤t},\hspace{1.22em}{t}_{{k}_{m}}={\rm{\max }}\left\{{t}_{k}:{t}_{k}\le t\right\},\hspace{9.1em}while 0<t<m0\lt t\lt m, (10)∑j:0<tj≤tλε(tj)d(x(tj−1−),tj−1)=∑j=1tkmλε(tj)d(x(tj−1−),tj−1)≤α3∫0tkmλε(s)∣x∣2ds+α4∫0tkmλε(s)∣x∣qds.≤α3∫0tλε(s)∣x∣2ds+α4∫0tλε(s)∣x∣qds.\begin{array}{rcl}\displaystyle \sum _{j:0\lt {t}_{j}\le t}{\lambda }^{\varepsilon }\left({t}_{j})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})& =& \mathop{\displaystyle \sum }\limits_{j=1}^{{t}_{{k}_{m}}}{\lambda }^{\varepsilon }\left({t}_{j})d\left(x\left({t}_{j-1}^{-}),{t}_{j-1})\\ & \le & {\alpha }_{3}\underset{0}{\overset{{t}_{{k}_{m}}}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{{t}_{{k}_{m}}}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{q}{\rm{d}}s.\\ & \le & {\alpha }_{3}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\hspace{-0.25em}{| }^{q}{\rm{d}}s.\end{array}\hspace{3.25em}By condition (ii) and the definition of λ\lambda -type function, we have (11)λε(t)V(x(t),t,r(t))≤V(x(0),0,r(0))+εα∫0tλε(s)V(x(s),s,r(s))ds−α1∫0tλε(s)∣x(s)∣2ds−α2∫0tλε(s)∣x(s)∣qds+α3∫0tλε(s)∣x(s)∣2ds+α4∫0tλε(s)∣x(s)∣qds+M0t.\hspace{-43em}\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& \le & V\left(x\left(0),0,r\left(0))+\varepsilon \alpha \underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)V\left(x\left(s),s,r\left(s)){\rm{d}}s-{\alpha }_{1}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s\\ & & -{\alpha }_{2}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{\alpha }_{3}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{M}_{0}^{t}.\end{array}Moreover, (12)λε(t)V(x(t),t,r(t))≤V(x(0),0,r(0))+c2εα∫0tλε(s)∣x(s)∣2ds−α1∫0tλε(s)∣x(s)∣2ds−α2∫0tλε(s)∣x(s)∣qds+α3∫0tλε(s)∣x(s)∣2ds+α4∫0tλε(s)∣x(s)∣qds+M0t≤V(x(0),0,r(0))−(α1−c2εα−α3)∫0tλε(s)∣x(s)∣2ds−(α2−α4)∫0tλε(s)∣x(s)∣qds+M0t.\hspace{0.2em}\begin{array}{rcl}{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))& \le & V\left(x\left(0),0,r\left(0))+{c}_{2}\varepsilon \alpha \underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s-{\alpha }_{1}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s\\ & & -{\alpha }_{2}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{\alpha }_{3}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s+{\alpha }_{4}\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{M}_{0}^{t}\\ & \le & V\left(x\left(0),0,r\left(0))-\left({\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3})\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{2}{\rm{d}}s-\left({\alpha }_{2}-{\alpha }_{4})\underset{0}{\overset{t}{\displaystyle \int }}{\lambda }^{\varepsilon }\left(s)| x\left(s){| }^{q}{\rm{d}}s+{M}_{0}^{t}.\end{array}Consequently, by inequality α1−c2εα−α3>0,α2−α4>0{\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3}\gt 0,{\alpha }_{2}-{\alpha }_{4}\gt 0and (12) imply λε(t)V(x(t),t,r(t))≤V(x(0),0,r(0))+M0t.{\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))\le V\left(x\left(0),0,r\left(0))+{M}_{0}^{t}.Applying Lemma 3.6, we have limt→∞supλε(t)V(x(t),t,r(t))<∞,a.s.\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup {\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))\lt \infty ,\hspace{1em}\hspace{0.1em}\text{a.s.}\hspace{0.1em}\hspace{0.2em}Thus, there exists a positive constant HHsuch that for any t>0t\gt 0, c1λε(t)∣x(t)∣2≤λε(t)V(x(t),t,r(t))≤H.{c}_{1}{\lambda }^{\varepsilon }\left(t)| x\left(t){| }^{2}\le {\lambda }^{\varepsilon }\left(t)V\left(x\left(t),t,r\left(t))\le H.\hspace{0.25em}Moreover, λε(t)∣x(t)∣2≤Hc1<∞.{\lambda }^{\varepsilon }\left(t)| x\left(t){| }^{2}\le \frac{H}{{c}_{1}}\lt \infty .\hspace{7.62em}Hence, limt→∞suplog∣x(t,x0)∣logλ(t)<−ε2.□\hspace{14.5em}\mathop{\mathrm{lim}}\limits_{t\to \infty }\sup \frac{\log | x\left(t,{x}_{0})| }{\log \lambda \left(t)}\lt -\frac{\varepsilon }{2}.\hspace{19em}\square 4Almost surely stable with decay λ(t)\lambda \left(t)We find that conditions (i) and (ii) of Theorem 3.7 are somewhat inconvenient in applications since they are not related to the coefficient u,b,σ,hu,b,\sigma ,hexplicitly. In this section, we will give the visualized conditions for the coefficient u,b,σ,hu,b,\sigma ,hto study the λ\lambda -type stability.We first recall some definitions and preliminaries.Definition 4.1A square matrix A={aij}N×NA={\left\{{a}_{ij}\right\}}_{N\times N}is called a ZZ-matrix if its off-diagonal entries are less than or equal to zero, namely aij≤0{a}_{ij}\le 0for i≠ji\ne j.Definition 4.2((M-matrix) [24]) Let AAbe a N×NN\times Nreal ZZ-matrix. And the matrix AAis also a nonsingular MM-matrix if it can be expressed in the form A=sI−BA=sI-Bwhile all the elements of B=(bij)B=\left({b}_{ij})are nonnegative and s≥ρ(B)s\ge \rho \left(B), where IIis an identity matrix and ρ(B)\rho \left(B)the spectral radius of BB.Noting that if A={aij}N×NA={\left\{{a}_{ij}\right\}}_{N\times N}is an MM-matrix, then it has positive diagonal entries and nonpositive off-diagonal entries, that is, aii≥0{a}_{ii}\ge 0while aij≥0,i≠j.{a}_{ij}\ge 0,i\ne j.Lemma 4.3([24]) If A={aij}N×NA={\left\{{a}_{ij}\right\}}_{N\times N}is a ZZ-matrix, then the following statements are equivalent: (1)AAis a nonsingular MM-matrix.(2)Every real eigenvalue of AAis positive.(3)All of the principle minors of AAare positive.(4)A−1{A}^{-1}exist and its elements are all nonnegative.Assumption 4.4For each i∈S,x∈Rni\in {\mathbb{S}},x\in {{\mathbb{R}}}^{n}and 0≤s<t0\le s\lt t, there exists a positive integer q>2q\gt 2, suppose the following conditions hold. (1)There exist constants αi1,αi2{\alpha }_{i1},{\alpha }_{i2}such that xT(u(x,s,i)+∫∣y∣<ch(x,y,s,i)ν(dy))≤−αi1∣x∣2+αi2∣x∣q.{x}^{T}\left(u\left(x,s,i)+\mathop{\int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y))\le -{\alpha }_{i1}| x\hspace{-0.25em}{| }^{2}+{\alpha }_{i2}| x\hspace{-0.25em}{| }^{q}.(2)There exist constants βi1,βi2{\beta }_{i1},{\beta }_{i2}such that xT(σ(x,s,i)α(t−s)α−1−∫∣y∣<ch(x,y,s,i)ν(dy))≤βi1∣x∣2−βi2∣x∣q.{x}^{T}\left(\sigma \left(x,s,i)\alpha {\left(t-s)}^{\alpha -1}-\mathop{\int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y))\le {\beta }_{i1}| x\hspace{-0.25em}{| }^{2}-{\beta }_{i2}| x\hspace{-0.25em}{| }^{q}.(3)There exist constants ηi1,ηi2{\eta }_{i1},{\eta }_{i2}such that ∣b(x,s,i)∣2≤ηi1∣x∣2+ηi2∣x∣q.\hspace{-13.8em}| b\left(x,s,i){| }^{2}\le {\eta }_{i1}| x\hspace{-0.25em}{| }^{2}+{\eta }_{i2}| x\hspace{-0.25em}{| }^{q}.(4)There exist constants ρi1,ρi2{\rho }_{i1},{\rho }_{i2}such that ∫∣y∣<c∣h(x,y,s,i)∣2ν(dy)≤ρi1∣x∣2+ρi2∣x∣q.\mathop{\int }\limits_{| y| \lt c}| h\left(x,y,s,i){| }^{2}\nu \left({\rm{d}}y)\le {\rho }_{i1}| x\hspace{-0.25em}{| }^{2}+{\rho }_{i2}| x\hspace{-0.25em}{| }^{q}.Remark 4.5Note that q>2q\gt 2in Assumption 4.4 which means that we can allow the coefficients u,b,σ,hu,b,\sigma ,hof system (1) to be high-order nonlinear.Assumption 4.6According to the definition of MM-matrix, let A≔−diag(−2α11+2β11+η11+ρ11,…,−2αN1+2βN1+ηN1+ρN1)−Γ{\mathcal{A}}:= -{\rm{diag}}\left(-2{\alpha }_{11}+2{\beta }_{11}+{\eta }_{11}+{\rho }_{11},\ldots ,-2{\alpha }_{N1}+2{\beta }_{N1}+{\eta }_{N1}+{\rho }_{N1})-\Gamma be a nonsingular MM-matrix, where Γ=(γij)N×N\Gamma ={\left({\gamma }_{ij})}_{N\times N}is given in Section 2. Moreover, from the fourth equivalent condition in Lemma 4.3, we obtain (θ1,θ2,…,θN)T≔A−11→>0,{\left({\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{N})}^{T}:= {{\mathcal{A}}}^{-1}\overrightarrow{1}\gt 0,where 1→=(1,1,…,1)T.\overrightarrow{1}={\left(1,1,\ldots ,1)}^{T}.That is, θi>0{\theta }_{i}\gt 0for all i∈Si\in {\mathbb{S}}.Theorem 4.7Let Assumptions 3.1–3.2, 4.4 and 4.6 hold. Those constants satisfy the following inequalities0<a̲<a¯,0\lt \underline{a}\lt \overline{a},wherea̲=mini∈S[θi(−2αi2+2βi2−ηi2−ρi2)],a¯=maxi∈S[θi(−2αi2+2βi2−ηi2−ρi2)].\begin{array}{rcl}\underline{a}& =& \mathop{\min }\limits_{i\in {\mathbb{S}}}\left[{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})],\\ \overline{a}& =& \mathop{\max }\limits_{i\in {\mathbb{S}}}\left[{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})].\end{array}Then for any given initial data x0{x}_{0}, there is a unique global solution x(t;x0)x\left(t;\hspace{0.33em}{x}_{0})and the trivial solution is almost surely stable with decay λ(t)\lambda \left(t).ProofLet function V:Rn×R+×S→R+V:{{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{+}\times S\to {{\mathbb{R}}}^{+}by V(x,s,i)=θi∣x∣2V\left(x,s,i)={\theta }_{i}| x\hspace{-0.25em}{| }^{2}, choose two positive numbers c1=mini∈S{θi}{c}_{1}={\min }_{i\in {\mathbb{S}}}\left\{{\theta }_{i}\right\}, c2=maxi∈S{θi}{c}_{2}={\max }_{i\in {\mathbb{S}}}\left\{{\theta }_{i}\right\}, such that c1∣x∣2≤V(x,s,i)≤c2∣x∣2.{c}_{1}| x\hspace{-0.25em}{| }^{2}\le V\left(x,s,i)\le {c}_{2}| x\hspace{-0.25em}{| }^{2}.By the definition of operator LjV{L}_{j}V, j=1,2,3j=1,2,3in Section 2, we obtain ∑j=12LiV(x(s),s,i)+α(t−s)α−1L3V(x,s,i)=2θixTu(x,s,i)+θi∣b(x,s,i)∣2+∑j=1Nγijθj∣x∣2+2θixTσ(x,s,i)α(t−s)α−1+θi∫∣y∣<c[∣x+h(x,y,s,i)∣2−∣x∣2]ν(dy)−∫∣y∣<c2θixTh(x,y,s,i)ν(dy)≤2θixT[u(x,s,i)+∫∣y∣<ch(x,y,s,i)ν(dy)]+θi∣b(x,s,i)∣2+∑j=1Nγijθj∣x∣2+θi∫∣y∣<c∣h(x,y,s,i)∣2ν(dy)+2θixT[σ(x,s,i)α(t−s)α−1−∫∣y∣<ch(x,y,s,i)ν(dy)].\begin{array}{l}\mathop{\displaystyle \sum }\limits_{j=1}^{2}{L}_{i}V\left(x\left(s),s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\\ \hspace{1.0em}=2{\theta }_{i}{x}^{T}u\left(x,s,i)+{\theta }_{i}| b\left(x,s,i){| }^{2}+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}+2{\theta }_{i}{x}^{T}\sigma \left(x,s,i)\alpha {\left(t-s)}^{\alpha -1}\\ \hspace{1.0em}\hspace{1.0em}+{\theta }_{i}\mathop{\displaystyle \int }\limits_{| y| \lt c}\left[| x+h\left(x,y,s,i){| }^{2}-| x\hspace{-0.25em}{| }^{2}]\nu \left({\rm{d}}y)-\mathop{\displaystyle \int }\limits_{| y| \lt c}2{\theta }_{i}{x}^{T}h\left(x,y,s,i)\nu \left({\rm{d}}y)\\ \hspace{1.0em}\le 2{\theta }_{i}{x}^{T}\left[u\left(x,s,i)+\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y)]+{\theta }_{i}| b\left(x,s,i){| }^{2}+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}\\ \hspace{1.0em}\hspace{1.0em}+{\theta }_{i}\mathop{\displaystyle \int }\limits_{| y| \lt c}| h\left(x,y,s,i){| }^{2}\nu \left({\rm{d}}y)+2{\theta }_{i}{x}^{T}\left[\sigma \left(x,s,i)\alpha {\left(t-s)}^{\alpha -1}-\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x,y,s,i)\nu \left({\rm{d}}y)].\end{array}\hspace{10.5em}By Assumption 4.4, we have (13)∑j=12LjV(x(s),s,i)+α(t−s)α−1L3V(x,s,i)≤2θi(−αi1∣x∣2+αi2∣x∣q)+2θi(βi1∣x∣2−βi2∣x∣q)+θi(ηi1∣x∣2+ηi2∣x∣q)+θi(ρi1∣x∣2+ρi2∣x∣q)+∑j=1Nγijθj∣x∣2=(−2θiαi1+2θiβi1+θiηi1+θiρi1)∣x∣2+∑j=1Nγijθj∣x∣2−(−2θiαi2+2θiβi2−θiηi2−θiρi2)∣x∣q=[θi(−2αi1+2βi1+ηi1+ρi1)+∑j=1Nγijθj]∣x∣2−θi(−2αi2+2βi2−ηi2−ρi2)∣x∣q.\begin{array}{l}\mathop{\displaystyle \sum }\limits_{j=1}^{2}{L}_{j}V\left(x\left(s),s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\\ \hspace{1.0em}\le 2{\theta }_{i}\left(-{\alpha }_{i1}| x\hspace{-0.25em}{| }^{2}+{\alpha }_{i2}| x\hspace{-0.25em}{| }^{q})+2{\theta }_{i}\left({\beta }_{i1}| x\hspace{-0.25em}{| }^{2}-{\beta }_{i2}| x\hspace{-0.25em}{| }^{q})+{\theta }_{i}\left({\eta }_{i1}| x\hspace{-0.25em}{| }^{2}+{\eta }_{i2}| x\hspace{-0.25em}{| }^{q})+{\theta }_{i}\left({\rho }_{i1}| x\hspace{-0.25em}{| }^{2}+{\rho }_{i2}| x\hspace{-0.25em}{| }^{q})+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}\\ \hspace{1.0em}=\left(-2{\theta }_{i}{\alpha }_{i1}+2{\theta }_{i}{\beta }_{i1}+{\theta }_{i}{\eta }_{i1}+{\theta }_{i}{\rho }_{i1})| x\hspace{-0.25em}{| }^{2}+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}| x\hspace{-0.25em}{| }^{2}-\left(-2{\theta }_{i}{\alpha }_{i2}+2{\theta }_{i}{\beta }_{i2}-{\theta }_{i}{\eta }_{i2}-{\theta }_{i}{\rho }_{i2})| x\hspace{-0.25em}{| }^{q}\\ \hspace{1.0em}=\left[{\theta }_{i}\left(-2{\alpha }_{i1}+2{\beta }_{i1}+{\eta }_{i1}+{\rho }_{i1})+\mathop{\displaystyle \sum }\limits_{j=1}^{N}{\gamma }_{ij}{\theta }_{j}]| x\hspace{-0.25em}{| }^{2}-{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})| x\hspace{-0.25em}{| }^{q}.\end{array}By Assumption 4.6, we have (14)∑j=12LjV(x(s),s,i)+α(t−s)α−1L3V(x,s,i)≤−∣x∣2−θi(−2αi2+2βi2−ηi2−ρi2)∣x∣q≤−∣x∣2−a̲∣x∣q.\mathop{\sum }\limits_{j=1}^{2}{L}_{j}V\left(x\left(s),s,i)+\alpha {\left(t-s)}^{\alpha -1}{L}_{3}V\left(x,s,i)\le -| x\hspace{-0.25em}{| }^{2}-{\theta }_{i}\left(-2{\alpha }_{i2}+2{\beta }_{i2}-{\eta }_{i2}-{\rho }_{i2})| x\hspace{-0.25em}{| }^{q}\le -| x\hspace{-0.25em}{| }^{2}-\underline{a}| x\hspace{-0.25em}{| }^{q}.By Assumptions 3.1–3.2, conditions (iii)–(iv) of Assumption 3.3, inequality (14) and Theorem 3.4, there is a unique global solution for any initial data. By inequality (14) and Theorem 3.7, the trivial solution is almost surely λ\lambda -type stable.□5ExampleWe consider two scalar stochastic fractional hybrid differential equations driven by Lévy noise with impulsive effects, and 2-state Markovian switching such that our results have been illustrated simply.Example 5.1(15)dx(t)=u(x(t−),t,r(t))dt+b(x(t−),t,r(t))dB(t)+σ(x(t−),t,r(t))(dt)α+∫∣y∣<ch(x(t−),y,t,r(t))N˜(dt,dy),t≠tk,t≥0,Δx(tk)=Ik(x(tk−),tk),k∈N,\left\{\begin{array}{l}{\rm{d}}x\left(t)=u\left(x\left(t-),t,r\left(t)){\rm{d}}t+b\left(x\left(t-),t,r\left(t)){\rm{d}}B\left(t)+\sigma \left(x\left(t-),t,r\left(t)){\left({\rm{d}}t)}^{\alpha }+\mathop{\displaystyle \int }\limits_{| y| \lt c}h\left(x\left(t-),y,t,r\left(t))\widetilde{N}\left({\rm{d}}t,{\rm{d}}y),\hspace{1em}t\ne {t}_{k},t\ge 0,\\ \Delta x\left({t}_{k})={I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k}),\hspace{1em}k\in {\mathbb{N}},\end{array}\right.where α=0.5\alpha =0.5, c=1c=1, and Lévy measure ν\nu satisfies ν(dy)=∣y∣dy\nu \left({\rm{d}}y)=| y| {\rm{d}}y, r(t)r\left(t)is a Markov chain in the state space S=1,2{\mathbb{S}}=1,2with the generator Γ\Gamma Γ=−112−2.\Gamma =\left(\begin{array}{cc}-1& 1\\ 2& -2\\ \end{array}\right).Let u(x,t,1)=−6x−3x2,u(x,t,2)=−5x−1.5x2,b(x,t,1)=22x+22x32,b(x,t,2)=0.5x+0.5x32,σ(x,t,1)=−2x,σ(x,t,2)=−2x3,h(x,y,t,1)=32x32y−2x,h(x,y,t,2)=−x.\begin{array}{ll}u\left(x,t,1)=-6x-3{x}^{2},& u\left(x,t,2)=-5x-1.5{x}^{2},\\ b\left(x,t,1)=\frac{\sqrt{2}}{2}x+\frac{\sqrt{2}}{2}{x}^{\tfrac{3}{2}},& b\left(x,t,2)=0.5x+0.5{x}^{\tfrac{3}{2}},\\ \sigma \left(x,t,1)=-2x,& \sigma \left(x,t,2)=-2{x}^{3},\\ h\left(x,y,t,1)=\frac{\sqrt{3}}{2}{x}^{\tfrac{3}{2}}y-2x,& h\left(x,y,t,2)=-x.\end{array}By computing, we have α11=8,α12=−3,α21=6,α22=−1.5,β11=2,β12=0,β21=1,β22=0,η11=1,η12=1,η21=0.5,η22=0.5,ρ11=4,ρ12=0.375,ρ21=1,ρ22=0\begin{array}{rlll}{\alpha }_{11}=8,& {\alpha }_{12}=-3,& {\alpha }_{21}=6,& {\alpha }_{22}=-1.5,\\ {\beta }_{11}=2,& {\beta }_{12}=0,& {\beta }_{21}=1,& {\beta }_{22}=0,\\ \hspace{0.075em}{\eta }_{11}=1,& {\eta }_{12}=1,& {\eta }_{21}=0.5,& {\eta }_{22}=0.5,\\ {\rho }_{11}=4,& {\rho }_{12}=0.375,& {\rho }_{21}=1,& {\rho }_{22}=0\end{array}and A=8−1−210.5,A−1=18210.5128θ1=0.1402,θ2=0.1220.\begin{array}{rcll}{\mathcal{A}}& =& \left(\begin{array}{cc}8& -1\\ -2& 10.5\\ \end{array}\right),& \hspace{0.8em}{{\mathcal{A}}}^{-1}=\frac{1}{82}\left(\begin{array}{cc}10.5& 1\\ 2& 8\\ \end{array}\right)\\ {\theta }_{1}& =& 0.1402,& \hspace{0.75em}{\theta }_{2}=0.1220.\end{array}Then we take Lyapunov functions V(x,t,1)=0.1402∣x(t)∣2,V(x,t,2)=0.1220∣x(t)∣2,V\left(x,t,1)=0.1402| x\left(t){| }^{2},\hspace{1.0em}V\left(x,t,2)=0.1220| x\left(t){| }^{2},by Theorem 4.7, we derive that a̲=0.305,a¯=0.6484.\underline{a}=0.305,\hspace{1em}\overline{a}=0.6484.Let Ik(x(tk−),tk)=22−1x+22x(tk−tk−1)12,λε(t)=e0.01t.{I}_{k}\left(x\left({t}_{k}^{-}),{t}_{k})=\left(\frac{\sqrt{2}}{2}-1\right)x+\frac{\sqrt{2}}{2}x{\left({t}_{k}-{t}_{k-1})}^{\tfrac{1}{2}},\hspace{1.0em}{\lambda }^{\varepsilon }\left(t)={e}^{0.01t}.By a simple calculation, we have d(x(tk−1−),tk−1)=0.7x2(tk−tk−1),Σj=nke0.01tj−10.7x2(tj−tj−1)≤0.7∫tn−1tke0.01s∣x∣2ds,d\left(x\left({t}_{k-1}^{-}),{t}_{k-1})=0.7{x}^{2}\left({t}_{k}-{t}_{k-1}),\hspace{1.0em}\underset{j=n}{\overset{k}{\Sigma }}{e}^{0.01{t}_{j-1}}0.7{x}^{2}\left({t}_{j}-{t}_{j-1})\le 0.7\underset{{t}_{n-1}}{\overset{{t}_{k}}{\int }}{e}^{0.01s}| x\hspace{-0.25em}{| }^{2}{\rm{d}}s,which verifies the condition of Theorem 3.7 with α1=1,α2=0.305,c2=0.1402,ε=0.01,α=1,α3=0.7,α4=0.{\alpha }_{1}=1,\hspace{1em}{\alpha }_{2}=0.305,\hspace{1em}{c}_{2}=0.1402,\hspace{1em}\varepsilon =0.01,\hspace{1em}\alpha =1,\hspace{1em}{\alpha }_{3}=0.7,\hspace{1em}{\alpha }_{4}=0.Moreover, α1−c2εα−α3=0.2986>0,α2−α4=0.305>0.{\alpha }_{1}-{c}_{2}\varepsilon \alpha -{\alpha }_{3}=0.2986\gt 0,\hspace{1em}{\alpha }_{2}-{\alpha }_{4}=0.305\gt 0.Therefore, system (5.1) is almost surely stable with decay λ(t)\lambda \left(t)of order ε2\frac{\varepsilon }{2}.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: stability; hybrid stochastic fractional differential equations; Lévy noise; general decay; impulsive effects; 60G51; 60H10; 34A37

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