Partial asymptotic stability of neutral pantograph stochastic differential equations with Markovian switching

Abstract

In this paper, we investigate the partial asymptotic stability (PAS) of neutral pantograph stochastic differential equations with Markovian switching (NPSDEwMSs). The main tools used to show the results are the Lyapunov method and the stochastic calculus techniques. We discuss a numerical example to illustrate our main results.

1 Introduction

Neutral stochastic delay differential equations with and without Markovian switching have been recently intensively investigated (see [1, 10, 11, 13, 14, 19, 20, 22], and [23]). Many systems are often subject to component repairs or failures, abrupt changes, environmental disturbances, and subsystem interconnections. The pantograph SDEs (PSDEs) have been widely used in electrodynamics and quantum mechanics. In the last decades the stability analysis of stochastic differential equations (SDEs) has received much attention (see [2, 3, 7–9, 15, 18, 25]). In general, due to the characteristics and specifications of SDEs themselves, it is difficult to obtain explicit solutions of equations. Therefore we use the Lyapunov method to study the stability and the asymptotic behavior of solutions. The almost sure polynomial and exponential stabilities were investigated by many researchers (see [2, 3], and [7–9]). The stochastic pantograph differential equations are a kind of stochastic delay differential equations (see [4, 7–9]), also called equations with proportional delay. They play an important role in industrial and mathematical problems. The NPSDEwMS are very well investigated (see [4, 25], and [17]). In [4] the authors proved the existence, uniqueness, and p-moment stability of solutions in the case $p>0$. However, in many dynamical systems, such a stability is usually too strong to be satisfied. Therefore the notion of partial stability (PS) (see [5, 6, 12], and [16]) has been studied, and the Lyapunov method, as an important tool, has been used to investigate the PS in various practically important domains. In the literature, we did not find any result on PAS of NPSDEwMS. Using the technique of stochastic calculus and Lyapunov method, we show a new sufficient condition for the PS of a class of NPSDEwMS.

In [5] and [12] the authors investigated the PAS of the solutions of ordinary SDEs by using an appropriate Lyapunov function satisfying some specific properties. In our paper, we prove the PAS of solutions of NPSDEwMSs. In this sense, our results extend the analysis in [5] and [12] providing the neutral term and the delay in the case of the PSDE with Markovian switching.

Let us outline the framework of this paper. After preliminaries and notations (see Sect. 1), in Sect. 2, we recall some important notions and definitions. In Sect. 3, we establish the PAS for a class of NPSDEwMSs. Finally, in Sect. 4, we present a numerical example to show the applicability of our results.

2 Preliminaries and notations

Let $\{\Omega ,\mathcal{F}, (\mathcal{F}_{s})_{s\geq 0},\mathbb{P}\}$ be a complete probability space with filtration $\{\mathcal{F}_{s}\}_{s\geq 0}$ satisfying the usual conditions, and let $W(s)$ be an m-dimensional Brownian motion defined on this probability space. Let $s\geq s_{0}>0$, let $C([qs_{0}, s_{0}];\mathbb{R}^{n})= \{ \psi :[qt_{0}, s_{0}] \rightarrow \mathbb{R}^{n} \text{such that} \psi \text{is a continuous function} \}$ with the norm $\|\psi \|= \sup_{qs_{0}\leq b\leq s_{0}}|\psi (b)|$, and let $|x|=\sqrt{x^{T}x}$ for $x \in \mathbb{R}^{n}$. If B is a matrix, then its trace norm is denoted by $|B| =\sqrt{\operatorname{Trace}(B^{T}B)}$, and its norm is given by ${\|B\| = \sup_{|x| = 1}|Bx|}$. Denote by $L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$ the set of all $\mathcal{F}_{s_{0}}$-measurable $C([qs_{0}, s_{0}];\mathbb{R}^{n})$-valued random variables $\psi = \{\psi (\theta ) : qs_{0}\leq \theta \leq s_{0}\}$ such that $E\|\psi \|^{p}<\infty$, where $p \in \mathbb{N}^{*}$.

Let $\{m(s), s\geq 0 \}$ be a right-continuous Markov chain on $\{\Omega ,\mathcal{F}, (\mathcal{F}_{s})_{s\geq 0},\mathbb{P}\}$ taking values in a finite state space $\bar{S} = \{1,2,3,\dots ,N\}$, where $\Gamma = (\gamma _{jk} )_{\mathbb{N}\times \mathbb{N}}$ is the generator given by

$$ P \bigl(m(s+\varpi )=k|m(s)=j \bigr)=\textstyle\begin{cases} \gamma _{jk}\varpi + o(\varpi ) & \text{if } j\neq k, \\ 1+\gamma _{jj}\varpi + o(\varpi ) & \text{if } j=k, \end{cases} $$

for $\varpi >0$. Here $\gamma _{jk}\geq 0$ is the transition rate from j to k if $j \neq k$, whereas

$$ \gamma _{jj}=-\sum_{j\neq k }\gamma _{jk}. $$

We suppose that r and W are independent.

Consider the following NPSDEwMS:

$$ \begin{aligned}[b] &d \bigl(z(s)-G \bigl(s,z(qs),m(s) \bigr) \bigr) \\ &\quad = f \bigl(s,z(s),z(qs),m(s) \bigr)\,ds+g \bigl(s,z(s),z(qs),m(s) \bigr) \,dW(s),\quad s\geq s_{0}, \end{aligned} $$

(2.1)

with initial data $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$, i.e.,

$$ z(s)=\zeta (s)\quad \text{for } qs_{0}\leq s \leq s_{0}. $$

(2.2)

Let $u(s) = z(s)-G(s,z(qs), m(s))$, where $G(s,z(qs), m(s))=(G_{1}(s,z(qs), m(s)),G_{2}(s,z(qs), m(s)))^{T}\in \mathbb{R}^{n}$. We assume that

$$\begin{aligned}& f:[s_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times \bar{S}\rightarrow \mathbb{R}^{n}, \qquad g: [s_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times \bar{S}\rightarrow \mathbb{R}^{n \times m},\\& G :[s_{0},+\infty )\times \mathbb{R}^{n}\times \bar{S} \rightarrow \mathbb{R}^{n}. \end{aligned}$$

Let $z= (z_{1},z_{2} )^{T}\in \mathbb{R}^{n}$ be the solution of equation (2.1), where $z_{1}\in \mathbb{R}^{k}$ and $z_{2}\in \mathbb{R}^{p}$, and $k+p=n$.

We will impose the following assumptions on f, g, and G:

($\mathcal{A}_{1}$) For each $l\in \mathbb{N}^{*}$, there exists $k_{l}>0$ such that

$$ \bigl\vert f(s,u,x,j)-f(s,\overline{u},\overline{x},j) \bigr\vert ^{2}\vee \bigl\vert g(s,u,x,j)-g(s,\overline{u}, \overline{x},j) \bigr\vert ^{2} \leq k_{l} \bigl( \vert u- \overline{u} \vert ^{2}+ \vert x-\overline{x} \vert ^{2} \bigr). $$

(2.3)

($\mathcal{A}_{2}$) For all $(s,j)\in [s_{0},+\infty )\times \bar{S}$ and $\varsigma ,x\in \mathbb{R}^{n}$, there exists $\kappa _{j}\in (0,1)$ such that

$$ \bigl\vert G(s,\varsigma ,j)-G(s,x,j) \bigr\vert ^{2}\leq \kappa _{j} \vert \varsigma -x \vert ^{2}. $$

(2.4)

Set $G(s,0,j)=0$ and $\kappa =\max_{j\in \bar{S}}\kappa _{j}$.

Let $C^{1,2} ([qs_{0},+\infty )\times \mathbb{R}^{n}\times \bar{S}; \mathbb{R}^{+} )$ be the set of all nonnegative functions $V(s, z, j)$ on $[qs_{0},+\infty )\times \mathbb{R}^{n}\times \bar{S}$ that are once continuously differentiable with respect to s and twice continuously differentiable with respect to z.

For any $(s,z,v,j)\in [qs_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n} \times S$, $u = z-G(s,v,j)$, by the generalized Itô formula (see [18] and [24]) we have

$$ V \bigl(s,u(s),m(s) \bigr)=V \bigl(s_{0},u(s_{0}),m(s_{0}) \bigr)+ \int _{s_{0}}^{s}\mathcal{L}V \bigl( \tau ,z(\tau ),z(q\tau ),m(\tau ) \bigr)\,d\tau +M(s), $$

where the stochastic process $M(s)$ and the operator $\mathcal{L}V(s,z,v,i):[qs_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times \bar{S}\rightarrow \mathbb{R}$ are defined by

$$\begin{aligned}& M(s)= \int _{s_{0}}^{s}V_{z} \bigl(\tau , u( \tau ), m(\tau ) \bigr)g \bigl(\tau ,z(\tau ),z(q \tau ),m(\tau ) \bigr)\,dW(\tau ), \\& \begin{aligned} \mathcal{L}V(s,z,v,j) &= V_{s}(s, u, j) + V_{z}(s, u, j)f(s,z,v,j) \\ &\quad {}+ \frac{1}{2}\operatorname{Trace} \bigl(g^{T}(s,z,v,j)V_{zz}(s,u, j)g(s,z,v,j) \bigr) \\ &\quad {}+\sum_{k=1}^{N} \gamma _{jk}V(s,u, k), \end{aligned} \\& V_{s}=\frac{\partial V(s,z,j)}{\partial s},\qquad V_{zz}= \biggl( \frac{\partial ^{2} V(s,z,j)}{\partial z_{j}\partial z_{j}} \biggr)_{n \times n}, \\& V_{z}= \biggl(\frac{\partial V(s,z,j)}{\partial z_{1}},\dots , \frac{\partial V(s,z,j)}{\partial z_{n}} \biggr). \end{aligned}$$

($\mathcal{A}_{3}$) There exist functions $\mu _{1}$, $\mu _{2}$, $\mu _{3}$, $\mu _{4}$ in $\mathcal{K}$ and $V\in C^{1,2} (\mathbb{R}_{+}\times \mathbb{R}^{n}\times \bar{S} ; \mathbb{R}_{+} )$ satisfying, for all $(s,z,v,j)\in [s_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n} \times \bar{S}$,

• (i) $\mu _{1} (|z_{1}| )\leq V (s,z,j )\leq \mu _{2} (|z_{1}| )$,

• (ii) $LV ( s,z,v,j )\leq - \mu _{3} (|z_{1}| )+q \mu _{4} (|v_{1}| )$.

3 Main results

We discuss the PS in probability and PAS of equation (2.1).

Definition 3.1

• (i) The solution $z(s)= (z_{1}(s),z_{2}(s) )$ of equation (2.1) is called PS in probability with respect to $z_{1}$ if for all $\eta >0$ and $\lambda \in (0,1)$, there exists $\delta _{0}=\delta _{0}(\lambda ,\eta ,s_{0})>0$ such that

  $$ P \bigl( \bigl\vert z_{1}(s) \bigr\vert < \eta , \forall s\geq s_{0} \bigr) \geq 1-\lambda $$

  whenever $\|\zeta \|<\delta _{0}$.

• (ii) The solution $z(s)= (z_{1}(s),z_{2}(s) )$ of equation (2.1) is called PAS in probability with respect to $z_{1}$ if it is stable in probability with respect to $z_{1}$ and for all $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$, we have

  $$ P \Bigl(\lim_{s\to +\infty }z_{1}(s)=0 \Bigr)=1. $$

Let $\mathcal{K}$ be the set of all continuous nondecreasing functions $\mu :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ such that $\mu (0)=0$ and $\mu (\nu )>0$ for $\nu >0$. For $H>0$, let $S_{H}= \{ z\in \mathbb{R}^{n}, \vert z_{1} \vert < H \}$.

Theorem 3.1

Suppose that there exist a function $V(s,z,j)\in C^{1,2} ([s_{0},+\infty )\times S_{H}\times S; \mathbb{R}_{+} )$ and $\mu \in \mathcal{K}$ such that

• (i) $\mu ( \vert z_{1} \vert )\leq V(s,z,j)$ for all $(s,z)\in [s_{0},+\infty )\times S_{H}$,

• (ii) $\mathcal{L}V(s,z,v,j)\leq 0$ for all $(s,z)\in [s_{0},+\infty )\times S_{H}$.

Then the solution of equation (2.1) is PS in probability with respect to $z_{1}$.

Proof

By Assumptions ($\mathcal{A}_{1}$)–($\mathcal{A}_{3}$) system (2.1) has a unique global solution $z(s)$ for $s\geq s_{0}$ (see [17]).

Let $\lambda \in (0,1)$ and $\eta >0$ be arbitrary. We will assume that $\eta < H$. By the continuity of $V(s,z,j)$ and the fact $V(s_{0},0,m(s_{0}))=0$ we can find $\rho =\rho (\lambda ,\eta ,s_{0})>0$ such that

$$ \frac{1}{\lambda }\sup_{z\in S_{\rho }} \bigl(V \bigl(s_{0},z,m(s_{0}) \bigr) \bigr) \leq \mu (\eta ). $$

(3.1)

We can see that $\rho <\eta$. Fix an arbitrary initial condition $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$ such that $\|\zeta \|<\rho$. Let ϑ be the stopping time given by

$$ \vartheta =\inf_{s\geq s_{0}} \bigl\{ z_{1}(s)\notin S_{\eta } \bigr\} . $$

By the Itô formula, for every $s\geq s_{0}$, we have

$$\begin{aligned}& E \bigl(V \bigl(s\wedge \vartheta ,z(s\wedge \vartheta ),m(s\wedge \vartheta ) \bigr) \bigr) \\& \quad =E \bigl(V \bigl(s_{0},z(s_{0}),m(s_{0}) \bigr) \bigr)+E \biggl( \int _{s_{0}}^{s\wedge \vartheta }\mathcal{L}V \bigl(\tau ,z(\tau ),v(\tau ),m(\tau ) \bigr)\,d\tau \biggr). \end{aligned}$$

Using (ii) and equation (3.1), we obtain that

$$ E \bigl(V \bigl(s\wedge \vartheta ,z(s\wedge \vartheta ),m(s \wedge \vartheta ) \bigr) \bigr) \leq E \bigl(V \bigl(s_{0},z(s_{0}),m(s_{0}) \bigr) \bigr) = \lambda \mu (\eta ). $$

(3.2)

Notice that if $\vartheta \leq s$, then

$$ \bigl\vert z_{1}(\vartheta \wedge s) \bigr\vert = \bigl\vert z_{1}(\vartheta ) \bigr\vert =\eta . $$

Then by (i) we have

$$ E \bigl(V \bigl(s\wedge \vartheta ,z(s\wedge \vartheta ),m(s \wedge \vartheta ) \bigr) \bigr) \geq E \bigl( \mathbf{1}_{\{\vartheta \leq s\}}\mu \bigl( \bigl\vert z_{1}(\vartheta ) \bigr\vert \bigr) \bigr)=\mu (\eta )P (\vartheta \leq s ). $$

(3.3)

Using (3.2) and (3.3), we obtain $P (\vartheta \leq s )\leq \lambda$. Letting $s\to +\infty$, we have $P (\vartheta \leq \infty )\leq \lambda$, which implies

$$ P \bigl( \bigl\vert z_{1}(s) \bigr\vert < \eta , \forall s\geq s_{0} \bigr) \geq 1-\lambda , $$

and the proof is completed. □

($\mathcal{A}_{4}$) There exist positive constants $\alpha _{1}$ and p and functions $\mu _{2}$, $\mu _{3}$, $\mu _{4}$ in $\mathcal{K}$ and $V\in C^{1,2} (\mathbb{R}_{+}\times \mathbb{R}^{n}\times \bar{S} ; \mathbb{R}_{+} )$ satisfying, for all $(s,z,v,j)\in [s_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n} \times \bar{S}$,

• (i) $\alpha _{1}|z_{1}|^{p}\leq V (s,z,j )\leq \mu _{2} (|z_{1}| )$,

• (ii) $LV ( s,z,v,j )\leq - \mu _{3} (|z_{1}| )+q \mu _{4} (|v_{1}| )$.

Theorem 3.2

Suppose that assumptions ($\mathcal{A}_{1}$), ($\mathcal{A}_{2}$), and ($\mathcal{A}_{4}$) hold. Let $\mu _{3}$ and $\mu _{4}$ in $\mathcal{K}$ satisfy, for all $(s,z)\in [s_{0},+\infty )\times \mathbb{R}^{n}$,

$$ \mu _{3} \bigl( \vert z \vert \bigr)\geq \mu _{4} \bigl( \vert z \vert \bigr), $$

(3.4)

where ${\mu _{3}-\mu _{4}}$ is an increasing function. Then, for any initial value $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$, the solution of equation (2.1) is PAS in probability with respect to $z_{1}$.

Proof

We will proceed as in the proof of Theorem 3.1 in [23] with necessary changes.

By Theorem 3.1 it is easy to prove that equation (2.1) is stable in probability with respect to $z_{1}$.

Step 1. Fix $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$ and $i_{0}\in \bar{S}$. By the Itô formula, (i), (ii), and (3.4) we have

$$\begin{aligned}& V \bigl(s,u(s),m(s) \bigr) \\& \quad \leq V \bigl(s_{0},u(s_{0}),m(s_{0}) \bigr)+ \int _{s_{0}}^{s}q \mu _{4} \bigl( \bigl\vert z_{1}(q\tau ) \bigr\vert \bigr)\,d\tau - \int _{s_{0}}^{s}\mu _{3} \bigl( \bigl\vert z_{1}( \tau ) \bigr\vert \bigr)\,d\tau +M(s) \\& \quad \leq V \bigl(s_{0},u(s_{0}),m(s_{0}) \bigr)+ \int _{qs_{0}}^{s_{0}}\mu _{4} \bigl( \bigl\vert z_{1}( \tau ) \bigr\vert \bigr)\,d\tau - \int _{s_{0}}^{s} \bigl(\mu _{3} \bigl( \bigl\vert z_{1}(\tau ) \bigr\vert \bigr)-\mu _{4} \bigl( \bigl\vert z_{1}( \tau ) \bigr\vert \bigr) \bigr)\,d\tau \\& \qquad {}+M(s) \\& \quad \leq \mu _{2} \bigl( \bigl\vert u(s_{0}) \bigr\vert \bigr)+\mu _{4} \bigl( \Vert \zeta \Vert \bigr)s_{0}(1-q) - \int _{s_{0}}^{s} \bigl(\mu _{3} \bigl( \bigl\vert z_{1}(\tau ) \bigr\vert \bigr)-\mu _{4} \bigl( \bigl\vert z_{1}(\tau ) \bigr\vert \bigr) \bigr)\,d\tau +M(s), \end{aligned}$$

(3.5)

where

$$ M(s)= \int _{s_{0}}^{s}V_{z} \bigl(\tau ,u( \tau ),m(\tau ) \bigr)g \bigl(\tau ,z(\tau ),z(q \tau ),m(\tau ) \bigr)\,dW(\tau ) $$

is a continuous local martingale with $M(s_{0})=0$ a.s. Applying Lemma 2.5 in [17] and taking $\chi =\mu _{2}(|u(s_{0})|)+\mu _{4}(\|\zeta \|)s_{0}(1-q)$, $A(s)=0$, ${N(s)=\int _{s_{0}}^{s} (\mu _{3}(|z_{1}(\tau )|)- \mu _{4}(|z_{1}(\tau )|) )\,d\tau ,}$ and $M(s)=\int _{s_{0}}^{s}V_{z}(\tau ,u(\tau ),m(\tau ))g( \tau ,z(\tau ),z(q\tau ),m(\tau ))\,dW(\tau )$, we have

$$ \limsup_{s\rightarrow +\infty } \bigl(V \bigl(s,u(s),m(s) \bigr) \bigr)< \infty \quad \text{a.s.} $$

(3.6)

Then

$$ \sup_{s_{0}\leq s< \infty }V \bigl(s,u(s),m(s) \bigr)< \infty \quad \text{a.s.} $$

(3.7)

Thus using (3.4), (3.7), and (i) (in Assumption ($\mathcal{A}_{4}$)), we obtain

$$ \sup_{s_{0}\leq s< \infty } \bigl(z_{1}(s)-G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr)< \infty . $$

(3.8)

For $T>0$, by Assumption ($\mathcal{A}_{2}$), for $s_{0}\leq s\leq T$, we have

$$\begin{aligned} \bigl\vert z_{1}(s) \bigr\vert \leq & \bigl\vert z_{1}(s)-G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr\vert + \bigl\vert G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr\vert \\ \leq & \bigl\vert z_{1}(s)-G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr\vert +k \bigl\vert z_{1}(qs) \bigr\vert . \end{aligned}$$

It then follows that

$$\begin{aligned} \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}(s) \bigr\vert \leq &\kappa \sup_{s_{0} \leq s\leq T} \bigl\vert z_{1}(qs) \bigr\vert +\sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}(s)-G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr\vert \\ \leq & \kappa \Vert \zeta \Vert +\kappa \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}(qs) \bigr\vert +\sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}(s)-G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr\vert . \end{aligned}$$

Thus

$$ \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}(s) \bigr\vert \leq \frac{1}{1-\kappa } \Bigl( \kappa \Vert \zeta \Vert +\sup _{s_{0} \leq s\leq T} \bigl\vert z_{1}(s)-G_{1} \bigl(s,z(qs),m(s) \bigr) \bigr\vert \Bigr). $$

Using (3.8) and letting $T\rightarrow \infty$, we have

$$ \sup_{s_{0}\leq s< \infty } \bigl\vert z_{1}(s) \bigr\vert \quad \text{a.s.} $$

(3.9)

Thus taking the expectations of both sides of (3.5) and letting $s\rightarrow +\infty$, we have

$$ E \biggl( \int _{s_{0}}^{+\infty } \bigl(\mu _{3} \bigl( \bigl\vert z_{1}(\tau ) \bigr\vert \bigr) -\mu _{4} \bigl( \bigl\vert z_{1}( \tau ) \bigr\vert \bigr) \bigr)\,d\tau \biggr)< \infty . $$

(3.10)

This implies that

$$ \int _{s_{0}}^{+\infty } \bigl(\mu _{3} \bigl( \bigl\vert z_{1}(\tau ) \bigr\vert \bigr) -\mu _{4} \bigl( \bigl\vert z_{1}( \tau ) \bigr\vert \bigr) \bigr)\,d\tau < \infty \quad \text{a.s.} $$

(3.11)

Step 2. Set $\mu =\mu _{3}-\mu _{4}$ ($\mu \in C(\mathbb{R}_{+},\mathbb{R}_{+})$). By (3.11) we can see that (see [15])

$$ \liminf_{s\rightarrow +\infty } \bigl(\mu \bigl( \bigl\vert z_{1}(s) \bigr\vert \bigr) \bigr)=0\quad \text{a.s.} $$

(3.12)

Now we claim that

$$ \lim_{s\rightarrow +\infty }\mu \bigl( \bigl\vert z_{1}(s) \bigr\vert \bigr)=0\quad \text{a.s.} $$

(3.13)

If (3.13) is false, then

$$ P \Bigl(\limsup_{s\rightarrow +\infty }\mu \bigl( \bigl\vert z_{1}(s) \bigr\vert \bigr)>0 \Bigr)>0. $$

Thus there exists a positive constant λ such that

$$ P (\Gamma _{1} )\geq 3\lambda $$

(3.14)

with $\Gamma _{1}= \{ \limsup_{s\rightarrow +\infty }\mu (|z_{1}(s)|)>2 \lambda \}$. By (3.9) and using the fact $\Vert \zeta \Vert <\infty$, we can find $h=h(\lambda )>0$ sufficiently large such that

$$ P (\Gamma _{2} )\geq 1-\lambda , $$

(3.15)

where $\Gamma _{2}= \{ \sup_{qs_{0}\leq s<\infty } (|z_{1}(s)|<h ) \}$. Using (3.14) and (3.15), we have

$$ P (\Gamma _{1}\cap \Gamma _{2} )\geq 2 \lambda . $$

(3.16)

Now we define the following stopping times:

$$\begin{aligned}& \vartheta _{h} = \inf \bigl\{ s\geq s_{0}, \bigl\vert z_{1}(s) \bigr\vert \geq h \bigr\} , \\& \vartheta _{1} = \inf \bigl\{ s\geq s_{0}, \mu \bigl( \bigl\vert z_{1}(s) \bigr\vert \bigr)\geq 2 \lambda \bigr\} , \\& \vartheta _{2k} = \inf \bigl\{ s\geq \vartheta _{2k-1}, \mu \bigl( \bigl\vert z_{1}(s) \bigr\vert \bigr) \leq \lambda \bigr\} ,\quad k=1,2,3,\dots , \\& \vartheta _{2k+1} = \inf \bigl\{ s\geq \vartheta _{2k}, \mu \bigl( \bigl\vert z_{1}(s) \bigr\vert \bigr) \geq 2\lambda \bigr\} ,\quad k=1,2,3,\dots . \end{aligned}$$

By the definitions of $\Gamma _{1}$ and $\Gamma _{2}$ and (3.12) we can see that if $\omega \in \Gamma _{1}\cap \Gamma _{2}$, then

$$ \vartheta _{k}< \infty \quad \text{and}\quad \vartheta _{h}=\infty \quad \forall k\in \mathbb{N}^{*}. $$

(3.17)

Since $\vartheta _{2k}<\infty$ whenever $\vartheta _{2k-1}<\infty$, by (3.10) we obtain that

$$\begin{aligned}& \lambda \sum_{k=1}^{\infty }E \bigl( \mathbf{1}_{\{\vartheta _{2k-1}< \infty ,\vartheta _{h}=\infty \}} (\vartheta _{2k}-\vartheta _{2k-1} ) \bigr) \\& \quad \leq \sum_{k=1}^{\infty }E \biggl( \mathbf{1}_{\{ \vartheta _{2k-1}< \infty ,\vartheta _{2k}< \infty ,\vartheta _{h}= \infty \}} \int _{\vartheta _{2k-1}}^{\vartheta _{2k}}\mu \bigl( \bigl\vert z_{1}( \tau ) \bigr\vert \bigr)\,d\tau \biggr) \\& \quad \leq E \biggl( \int _{s_{0}}^{+\infty }\mu \bigl( \bigl\vert z_{1}(\tau ) \bigr\vert \bigr)\,d\tau \biggr) \\& \quad < \infty . \end{aligned}$$

(3.18)

In fact, by assumption ($\mathcal{A}_{1}$) there exists $k_{h}>0$ such that

$$ \bigl\vert g(s,z,v,j) \bigr\vert ^{2}\vee \bigl\vert f(s,z,v,j) \bigr\vert ^{2} \leq k_{h} $$

whenever $(s,j)\in [s_{0},+\infty )\times \bar{S}$ and $|z|\vee |v|\leq h$. Using the Hölder and Doob martingale inequalities, we have that for $k=1,2,3,\dots$ and $T>0$,

$$\begin{aligned}& E \Bigl(\mathbf{1}_{\{\vartheta _{h}\wedge \vartheta _{2k-1}< \infty \}}\sup_{s_{0}\leq s\leq T} \bigl\vert z_{1} \bigl(\vartheta _{h}\wedge ( \vartheta _{2k-1}+s) \bigr)-z_{1}( \vartheta _{h}\wedge \vartheta _{2k-1}) \bigr\vert ^{2} \Bigr) \\& \quad \leq 2 E \biggl(\mathbf{1}_{\{\vartheta _{h}\wedge \vartheta _{2k-1}< \infty \}}\sup _{s_{0}\leq s\leq T} \biggl\vert \int _{\vartheta _{h} \wedge \vartheta _{2k-1}}^{\vartheta _{h}\wedge (\vartheta _{2k-1}+s)} f \bigl(\tau ,z(\tau ),z(q\tau ),m(\tau ) \bigr)\,d\tau \biggr\vert ^{2} \biggr) \\& \qquad {} +2 E \biggl(\mathbf{1}_{\{\vartheta _{h}\wedge \vartheta _{2k-1}< \infty \}}\sup _{s_{0}\leq s\leq T} \biggl\vert \int _{\vartheta _{h} \wedge \vartheta _{2k-1}}^{\vartheta _{h}\wedge (\vartheta _{2k-1}+s)} g \bigl(\tau ,z(\tau ),z(q\tau ),m(\tau ) \bigr)\,dW(\tau ) \biggr\vert ^{2} \biggr) \\& \quad \leq 2 TE \biggl(\mathbf{1}_{\{\vartheta _{h}\wedge \vartheta _{2k-1}< \infty \}} \int _{\vartheta _{h}\wedge \vartheta _{2k-1}}^{\vartheta _{h} \wedge (\vartheta _{2k-1}+T)} \bigl\vert f \bigl(\tau ,z(\tau ),z(q\tau ),m( \tau ) \bigr) \bigr\vert ^{2}\,d\tau \biggr) \\& \qquad {} +8E \biggl(\mathbf{1}_{\{\vartheta _{h}\wedge \vartheta _{2k-1}< \infty \}} \int _{\vartheta _{h}\wedge \vartheta _{2k-1}}^{\vartheta _{h} \wedge (\vartheta _{2k-1}+T)} \bigl\vert g \bigl(\tau ,z(\tau ),z(q\tau ),m( \tau ) \bigr) \bigr\vert ^{2}\,d\tau \biggr) \\& \quad \leq 2k_{h}T (T+4 ). \end{aligned}$$

(3.19)

We know that if μ is a continuous function in $\mathbb{R}^{n}$, then it is uniformly continuous in $\overline{B}_{h}= \{ z\in \mathbb{R}^{n} : \vert z \vert \leq h \}$. Thus we can choose sufficiently small $\varphi =\varphi (\lambda )>0$ such that

$$ \bigl\vert \mu (z)-\mu (v) \bigr\vert < \frac{\lambda }{2}\quad \text{whenever } z,v \in \overline{B_{h}}, \vert z-v \vert < \varphi . $$

(3.20)

Set $T=T(\lambda ,\varphi ,h)>0$ sufficiently small such that ${\frac{2k_{h}T(T+4)}{\varphi ^{2}}<\lambda }$. By (3.19) we have

$$ P \Bigl( \{ \vartheta _{h}\wedge \vartheta _{2k-1}< \infty \} \cap \Bigl\{ \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1} \bigl(\vartheta _{h} \wedge (\vartheta _{2k-1}+s) \bigr) -z_{1}(\vartheta _{h}\wedge \vartheta _{2k-1}) \bigr\vert \geq \varphi \Bigr\} \Bigr)< \lambda . $$

We can see that

$$ \{ \vartheta _{h}=\infty , \vartheta _{2k-1}< \infty \} = \{ \vartheta _{h}\wedge \vartheta _{2k-1}< \infty , \vartheta _{h}= \infty \} \subset \{ \vartheta _{h} \wedge \vartheta _{2k-1}< \infty \} . $$

Then we obtain

$$ P \Bigl( \{ \vartheta _{2k-1}< \infty , \vartheta _{h}= \infty \} \cap \Bigl\{ \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}( \vartheta _{2k-1}+s) -z_{1}(\vartheta _{2k-1}) \bigr\vert \geq \varphi \Bigr\} \Bigr)< \lambda . $$

Using (3.16) and (3.17), we deduce

$$\begin{aligned}& P \Bigl( \{ \vartheta _{2k-1}< \infty , \vartheta _{h}= \infty \} \cap \Bigl\{ \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}( \vartheta _{2k-1}+s) -z_{1}(\vartheta _{2k-1}) \bigr\vert < \varphi \Bigr\} \Bigr) \\& \quad =P \bigl( \{ \vartheta _{2k-1}< \infty , \vartheta _{h}= \infty \} \bigr) \\& \qquad {}- P \Bigl( \{ \vartheta _{2k-1}< \infty , \vartheta _{h}=\infty \} \cap \Bigl\{ \sup_{s_{0}\leq s\leq T} \bigl\vert z_{1}(\vartheta _{2k-1}+s) -z_{1}( \vartheta _{2k-1}) \bigr\vert \geq \varphi \Bigr\} \Bigr) \\& \quad >2\lambda -\lambda =\lambda . \end{aligned}$$

Therefore by (3.20) we have

$$ P \Bigl( \{ \vartheta _{2k-1}< \infty , \vartheta _{h}=\infty \} \cap \Bigl\{ \sup_{s_{0}\leq s\leq T} \bigl\vert \mu \bigl(z_{1}( \vartheta _{2k-1}+s) \bigr) -\mu \bigl(z_{1}(\vartheta _{2k-1}) \bigr) \bigr\vert < \lambda \Bigr\} \Bigr)>\lambda . $$

(3.21)

Set $\overline{M}_{k}= \{ \sup_{s_{0}\leq s\leq T} \vert \mu (z_{1}( \vartheta _{2k-1}+s)) -\mu (z_{1}(\vartheta _{2k-1})) \vert < \lambda \}$. Notice that if $\omega \in \{ \vartheta _{2k-1}<\infty , \vartheta _{h}= \infty \} \cap \overline{M}_{k}$, then

$$ \vartheta _{2k}(\omega )-\vartheta _{2k-1}(\omega )\geq T. $$

By (3.18) and (3.21) we can derive that

$$\begin{aligned} \infty >& \lambda \sum_{k=1}^{\infty }E \bigl(\mathbf{1}_{\{ \vartheta _{2k-1}< \infty , \vartheta _{h}=\infty \}} ( \vartheta _{2k}-\vartheta _{2k-1} ) \bigr) \\ \geq &\lambda \sum_{k=1}^{\infty }E \bigl( \mathbf{1}_{\{\vartheta _{2k-1}< \infty , \vartheta _{h}=\infty \}\cap \overline{M}_{k}} ( \vartheta _{2k}-\vartheta _{2k-1} ) \bigr) \\ \geq &\lambda T\sum_{k=1}^{\infty }P \bigl( \{\vartheta _{2k-1}< \infty , \vartheta _{h}=\infty \}\cap \overline{M}_{k} \bigr) \\ \geq &\lambda T\sum_{k=1}^{\infty } \lambda =\infty , \end{aligned}$$

which is impossible. Then (3.13) holds.

Step 3. By (3.9) and (3.13) there is $\Omega _{0}\subset \Omega$ with $P (\Omega _{0} )=1$ such that for all $\omega \in \Omega _{0}$,

$$ \lim_{s\rightarrow +\infty }\mu \bigl( \bigl\vert z_{1}(s,\omega ) \bigr\vert \bigr)=0, \quad \text{and}\quad \sup_{s_{0}\leq s\leq \infty } \bigl\vert z_{1}(s,\omega ) \bigr\vert < \infty . $$

(3.22)

Now we must show that

$$ \lim_{s\rightarrow +\infty }z_{1}(s,\omega )=0 \quad \forall \omega \in \Omega _{0}. $$

(3.23)

If we suppose that (3.23) is false, then there is $\hat{\omega }\in \Omega _{0}$ such that ${\lim_{s\rightarrow +\infty }\sup |z_{1}(s,\hat{\omega })|>0}$. Thus there exist subsequences $\{ z_{1}(s_{k},\hat{\omega }) \} _{k\geq 0}$ of $\{ z_{1}(s,\hat{\omega }) \} _{s\geq s_{0}}$ satisfying $\vert z_{1}(s_{k},\hat{\omega }) \vert >\bar{\alpha }$ for some $\bar{\alpha }>0$ and all $k\geq 0$. Since $\{ z_{1}(s_{k},\hat{\omega }) \} _{k\geq 0}$ is bounded, we can find an increasing subsequence $\{ \hat{s}_{k} \} _{k\geq 0}$ such that $\{ z_{1}(\hat{s}_{k},\omega ) \} _{k\geq 0}$ converges to some $\bar{z}\in \mathbb{R}^{n}$ such that $|\bar{z}|>\bar{\alpha }$. Therefore ${\mu (|\bar{z}| )=\lim_{k\rightarrow \infty }\mu (|z_{1}(s_{k},\omega )| )>0}$. However, by (3.22) we have $\mu (|\bar{z}| )=0$, a contradiction.

Consequently, the solution of system (2.1) is asymptotically stable in probability with respect to $z_{1}$. □

4 Asymptotic instability of NPSDEwMS

We will state a theorem about the asymptotic instability with respect to all variables of NPSDEwMS.

Definition 4.1

The solution $z(s)= (z_{1}(s),z_{2}(s) )$ of equation (2.1) is called asymptotically unstable in probability if it is unstable in probability or for all $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$,

$$ P \Bigl(\lim_{s\to +\infty }z_{1}(s)\neq 0 \Bigr)=1. $$

Theorem 4.1

Suppose that there exist a function $V\in C^{1,2} (\mathbb{R}_{+}\times \mathbb{R}^{n}\times \bar{S} ; \mathbb{R}_{+} )$ and $\mu _{1}$, $\mu _{2}$, $\mu _{3}$, and $\mu _{4}$ in $\mathcal{K}$ such that for all $(s,z,v,j)\in [s_{0},+\infty )\times \mathbb{R}^{n}\times \mathbb{R}^{n} \times \bar{S}$,

• (i) $\mu _{1} (|z| )\leq V (s,z,j )\leq \mu _{2} (|z| )$,

• (ii) $\mathcal{L}V ( s,z,v,j )\geq - \mu _{3} (|z| )+q \mu _{4} (|v| )$.

Then for any initial value $\zeta \in L^{p}_{\mathcal{F}_{s_{0}}}([qs_{0}, s_{0}];\mathbb{R}^{n})$, the solution of equation (2.1) is asymptotically unstable in probability.

Proof

The proof is similar to that of Theorem 4.3 in [6]. □

5 Example and numerical solution

We now give a numerical example to illustrate the application of our results.

Let $W(s)$ be a three-dimensional Brownian motion. Let $m(s)$ be a right-continuous Markov chain taking values in $\bar{S} = \{ 1,2,3 \}$ with $\Gamma = (\gamma _{jk} )_{1\leq j,k\leq 3}$ given by

$$ \Gamma = \begin{pmatrix} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{pmatrix}. $$

Moreover, we assume that $W(s)$ and $m(s)$ are independent. Consider the following NPSDEwMS:

$$ \textstyle\begin{cases} d (z_{1}(s)-G (s,z_{1}(qs),m(s) ) ) \\ \quad = f_{1} (s,z(s),z(qs),m(s) )\,ds+g_{1} (s,z(s),z(qs),m(s) )\,dW_{1}(s), \\ d (z_{2}(s)-G (s,z_{2}(qs),m(s) ) ) \\ \quad = f_{2} (s,z(s),z(qs),m(s) )\,ds+g_{2} (s,z(s),z(qs),m(s) )\,dW_{2}(s), \\ d (z_{3}(s)-G (s,z_{3}(qs),m(s) ) ) \\ \quad = f_{3} (s,z(s),z(qs),m(s) )\,ds+g_{3} (s,z(s),z(qs),m(s) )\,dW_{3}(s), \end{cases} $$

(5.1)

with initial data $\zeta (s)$. Moreover, for $(s,z,v,j)\in [s_{0},+\infty )\times \mathbb{R}^{3}\times \mathbb{R}^{3} \times \bar{S}$, let

$$\begin{aligned}& G(s,z,j)=\textstyle\begin{cases} \frac{1}{5}z &\text{if } j=1, \\ \frac{1}{6}z &\text{if } j=2, \\ \frac{1}{9}z &\text{if } j=3, \end{cases}\displaystyle \qquad f_{1}(s,z,v,j)=\textstyle\begin{cases} - (z_{1}+\frac{1}{5}v_{1} ) &\text{if } j=1, \\ - (z_{1}+\frac{1}{6}v_{1} ) &\text{if } j=2, \\ - (z_{1}+\frac{1}{9}v_{1} ) &\text{if } j=3, \end{cases}\displaystyle \\& f_{2}(s,z,v,j)=\textstyle\begin{cases} -\frac{1}{3} (z_{1}-\frac{1}{5}v_{1} )^{2} (z_{2}- \frac{1}{5}v_{2} ) &\text{if } j=1, \\ -\frac{1}{3} (z_{1}-\frac{1}{6}v_{1} )^{2} (z_{2}- \frac{1}{6}v_{2} ) &\text{if } j=2, \\ -\frac{1}{3} (z_{1}-\frac{1}{9}v_{1} )^{2} (z_{2}- \frac{1}{9}v_{2} ) &\text{if } j=3, \end{cases}\displaystyle \\& f_{3}(s,z,v,j)=\textstyle\begin{cases} -2 (z_{3}+\frac{1}{5}v_{3} ) &\text{if } j=1, \\ -2 (z_{3}+\frac{1}{6}v_{3} ) &\text{if } j=2, \\ -\frac{11}{2} (z_{3}+\frac{1}{9}v_{3} ) &\text{if } j=3, \end{cases}\displaystyle \qquad g_{1}(s,z,v,j)=\textstyle\begin{cases} \frac{1}{\sqrt{5}}v_{2} &\text{if } j=1, \\ \frac{1}{\sqrt{6}}v_{2} &\text{if } j=2, \\ \frac{1}{3}v_{2} &\text{if } j=3, \end{cases}\displaystyle \\& g_{2}(s,z,v,j)=\textstyle\begin{cases} \sqrt{\frac{2}{3}} (z_{1}-\frac{1}{5}v_{1} ) (z_{2}- \frac{1}{5}v_{2} ) &\text{if } j=1, \\ \sqrt{\frac{2}{3}} (z_{1}-\frac{1}{6}v_{1} ) (z_{2}- \frac{1}{6}v_{2} ) &\text{if } j=2, \\ \sqrt{\frac{2}{3}} (z_{1}-\frac{1}{9}v_{1} ) (z_{2}- \frac{1}{9}v_{2} ) &\text{if } j=3, \end{cases}\displaystyle \qquad g_{3}(s,z,v,j)=\textstyle\begin{cases} \frac{2}{\sqrt{5}}v_{3} &\text{if } j=1, \\ \sqrt{\frac{2}{3}}v_{3} &\text{if } j=2, \\ \frac{2}{3}v_{3}&\text{if } j=3. \end{cases}\displaystyle \end{aligned}$$

Let $V(s,z,j)=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}$ for $j\in \bar{S}$. Then for $j=1$, we have

$$\begin{aligned} \mathcal{L}V(s,z,v,1) =& -2 \biggl(z_{1}^{2}- \frac{1}{25}v_{1}^{2} \biggr)+ \frac{1}{5}v_{2}^{2} -4 \biggl(z_{3}^{2}- \frac{1}{25}v_{3}^{2} \biggr)+ \frac{4}{5}v_{3}^{2} \\ =&-2z_{1}^{2}+\frac{2}{25}v_{1}^{2}-4z_{3}^{2}+ \frac{24}{25}v_{3}^{2}+ \frac{1}{5}v_{2}^{2} \\ \geq &-4 \bigl(z_{1}^{2}+z_{2}^{2}+z_{3}^{2} \bigr)+\frac{2}{25} \bigl(v_{1}^{2}+v_{2}^{2}+v_{3}^{2} \bigr) \\ =&-4 \vert z \vert ^{2}+\frac{2}{25} \vert v \vert ^{2}. \end{aligned}$$

For $j=2$, it follows that

$$\begin{aligned} \mathcal{L}V(s,z,v,2) =& -2 \biggl(z_{1}^{2}- \frac{1}{36}v_{1}^{2} \biggr)+ \frac{1}{6}v_{2}^{2} -4 \biggl(z_{3}^{2}- \frac{1}{36}v_{3}^{2} \biggr)+ \frac{2}{3}v_{3}^{2} \\ =&-2z_{1}^{2}+\frac{1}{18}v_{1}^{2}+ \frac{1}{6}v_{2}^{2}-4z_{3}^{2}+ \frac{7}{9}v_{3}^{2} \\ \geq &-4 \bigl(z_{1}^{2}+z_{2}^{2}+z_{3}^{2} \bigr)+\frac{1}{18} \bigl(v_{1}^{2}+v_{2}^{2}+v_{3}^{2} \bigr) \\ =&-4 \vert z \vert ^{2}+\frac{1}{18} \vert v \vert ^{2}. \end{aligned}$$

For $j=3$, we deduce

$$\begin{aligned} \mathcal{L}V(s,z,v,3) =& -2 \biggl(z_{1}^{2}- \frac{1}{81}v_{1}^{2} \biggr)+ \frac{1}{9}v_{2}^{2} -11 \biggl(z_{3}^{2}- \frac{1}{81}v_{3}^{2} \biggr)+ \frac{4}{9}v_{3}^{2} \\ =&-2z_{1}^{2}+\frac{2}{81}v_{1}^{2}+ \frac{1}{9}v_{2}^{2}-11z_{3}^{2}+ \frac{47}{81}v_{3}^{2} \\ \geq &-11 \bigl(z_{1}^{2}+z_{2}^{2}+z_{3}^{2} \bigr)+\frac{2}{81} \bigl(v_{1}^{2}+v_{2}^{2}+v_{3}^{2} \bigr) \\ =&-11 \vert z \vert ^{2}+\frac{2}{81} \vert v \vert ^{2}. \end{aligned}$$

Thus for $j\in \bar{S}$, we obtain

$$ \mathcal{L}V(s,z,v,3)\geq -11 \vert z \vert ^{2}+ \frac{2}{81} \vert v \vert ^{2}. $$

(5.2)

Therefore by Theorem 4.1, system (5.1) is asymptotically unstable with respect to all variables.

For $j\in \bar{S}$, we define $V_{1}$ by

$$ V_{1}(s,z,j)=\textstyle\begin{cases} z_{3}^{2} & \text{if } j=1,2, \\ \frac{1}{2}z_{3}^{2} & \text{if } j=3. \end{cases} $$

For $j=1$, we have

$$\begin{aligned} \mathcal{L}V_{1}(s,z,v,1) =& -4 \biggl(z_{3}^{2}- \frac{1}{25}v_{3}^{2} \biggr)+ \frac{4}{5}v_{3}^{2}-\frac{1}{2} \biggl(z_{3}-\frac{1}{5}v_{3} \biggr)^{2} \\ =&-4z_{3}^{2}+\frac{24}{25}v_{3}^{2}- \frac{1}{2}z_{3}^{2}+ \frac{1}{5}z_{3}v_{3}- \frac{1}{50}v_{3}^{2} \\ =&-\frac{9}{2}z_{3}^{2}+ \frac{47}{50}v_{3}^{2}+\frac{1}{5}z_{3}v_{3} \\ \leq &-\frac{9}{2}z_{3}^{2}+ \frac{47}{50}v_{3}^{2}+ \frac{z_{3}^{2}}{100}+v_{3}^{2} \\ =&-\frac{449}{100}z_{3}^{2}+ \frac{97}{50}v_{3}^{2} \\ =&-4.49z_{3}^{2}+1.94v_{3}^{2}. \end{aligned}$$

For $j=2$, we derive

$$\begin{aligned} \mathcal{L}V_{1}(s,z,v,2) =& -4 \biggl(z_{3}^{2}- \frac{1}{36}v_{3}^{2} \biggr)+ \frac{2}{3}v_{3}^{2}-\frac{1}{2} \biggl(z_{3}-\frac{1}{6}v_{3} \biggr)^{2} \\ =&-4z_{3}^{2}+\frac{7}{9}v_{3}^{2}- \frac{1}{2}z_{3}^{2}+\frac{1}{6}z_{3}v_{3}- \frac{1}{72}v_{3}^{2} \\ =&-\frac{9}{2}z_{3}^{2}+ \frac{55}{72}v_{3}^{2}+\frac{1}{6}z_{3}v_{3} \\ \leq &-\frac{9}{2}z_{3}^{2}+ \frac{55}{72}v_{3}^{2}+ \frac{z_{3}^{2}}{144}+v_{3}^{2} \\ =&-\frac{648}{144}z_{3}^{2}+ \frac{127}{72}v_{3}^{2} \\ =&-4.5z_{3}^{2}+1.76v_{3}^{2}. \end{aligned}$$

For $j=3$, we deduce

$$\begin{aligned} \mathcal{L}V_{1}(s,z,v,3) =& -\frac{11}{2} \biggl(z_{3}^{2}- \frac{1}{81}v_{3}^{2} \biggr)+\frac{2}{9}v_{3}^{2}+ \biggl(z_{3}- \frac{1}{9}v_{3} \biggr)^{2} \\ =&-\frac{11}{2}z_{3}^{2}+ \frac{11}{162}v_{3}^{2}+\frac{2}{9}v_{3}^{2}+z_{3}^{2}- \frac{2}{9}z_{3}v_{3}+\frac{1}{81}v_{3}^{2} \\ \leq &-\frac{9}{2}z_{3}^{2}+ \frac{49}{162}v_{3}^{2}+ \frac{z_{3}^{2}}{9}+ \frac{v_{3}^{2}}{9} \\ =&-\frac{79}{18}z_{3}^{2}+ \frac{67}{162}v_{3}^{2} \\ =&-4.38z_{3}^{2}+0.41v_{3}^{2}. \end{aligned}$$

Then for $j\in \bar{S}$, it follows that

$$ \mathcal{L}V_{1}(s,z,v,j)\leq -4.38z_{3}^{2}+(0.5) (3.88)v_{3}^{2}. $$

Consequently, by Theorem 3.2 system (5.1) is asymptotically stable with respect to $z_{3}$ with $\mu _{1}(|z_{3}|)=4.38z_{3}^{2}$ and $\mu _{2}(|v_{3}|)=3.88v_{3}^{2}$.

For system (5.1), we conduct a simulation using the Euler–Maruyama scheme with step size 0.001, $q=0.35$, $s_{0}=1$, and the linear initial function $\zeta (s)= (s,-s,s-1 )$ for $0.35\leq s\leq 1$. Next, we provide the simulations for system (5.1). In Fig. 1, we show the stability of the component $z_{3}$ by simulation of its trajectories. In Fig. 2, we illustrate the instability of the components $z_{1}$ and $z_{2}$.

Figure 1

Simulations of the trajectory of $z_{3}(s)$ in system (5.1) with $\zeta _{3}(s)=s-1$ for $s\in [0.35,5\times 10^{4}]$

Figure 2

Simulations of the trajectories of the components $z_{1}(s)$ and $z_{2}(s)$ with $\zeta _{1}(s)=s$ and $\zeta _{2}(s)=-s$ on $[0.35,5\times 10^{4}]$

The simulation results clearly show that the trajectories of the corresponding stochastic system converge asymptotically to the equilibrium state for any given initial values, thus verifying the effectiveness of theoretical results.