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ηStability for stochastic functional differential equation driven by timechanged Brownian motion
Journal of Inequalities and Applications volume 2024, Article number: 60 (2024)
Abstract
This manuscript focuses on a class of stochastic functional differential equations driven by timechanged Brownian motion. By utilizing the Lyapunov method, we capture some sufficient conditions to ensure that the solution for the considered equation is ηstable in the pth moment sense. Subsequently, we present some new criteria of the ηstability in mean square by using timechanged Itô formula and proof by contradiction. Finally, we provide some examples to demonstrate the effectiveness of our main results.
1 Introduction
At present, the timechanged semimartingale theory has attracted much attention because of its widespread applications in cell biology, hydrology, physics, and economics [19]. Since Kobayashi [10] investigated stochastic calculus of the timechanged semimartingales, there have been a lot of authors working on the stochastic differential equations with the timechanged Brownian motion or Lévy processes. For example, we refer to [7, 9, 13, 20] for the numerical approximation scheme and to [2, 18] for the averaging principle. Particularly, an increasing number of experts devoted themselves to research the stability in significant senses for various SDEs with the timechanged semimartingales. For example, see [21, 22] for the stability in probability; [16, 17] for the moment stability and path stability; [25–27] for the asymptotic stability; and [12, 28] for the exponential stability.
Meanwhile, the ηstability is a valuable extension of certain wellknown stability types such as polynomial, exponential, and logarithmic stability, etc. The ηstability with respect to the deterministic systems has attracted much attention from experts within a short time because it leads to a new understanding on the longtime behavior of the solution. For instance, see Choi et al. [3] for the linear dynamic equations; Damak et al. [5] for the boundedness and ηstability of the perturbed equations; Ghanmi [8] for the practical ηstability; Xu and Liu [23] and Xu et al. [24] for the ηstability of the numerical solutions of the pantograph equations; Damak et al. [6] for the converse theorem on practical ηstability of nonlinear differential equations. Aslo Damak [4] and Mihit [15] worked on the ηstability of some evolution equations in Banach spaces by using some Gronwalltype inequalities.
However, according to the literature we reviewed, there is little literature on the ηstability for stochastic systems. Employing the Lyapunov’s method, Caraballo et al. [1] studied the ηstability for neutral stochastic pantograph differential equations driven by Lévy noise, and Li et al. [11] explored the ηstability for stochastic Volterra–Levin equations. In our paper, we try to make a study of the hstability of the following functional SDE:
where \(E_{t}\) is defined as the inverse of the βstable subordinator with index \(0<\beta <1\), \(y_{t}=\{y(t+\vartheta ):\vartheta \in [r,0]\}\) is treated as a \(C([r,0];\mathbb{R}^{d})\)valued process. Giving some coefficient conditions ensuring that the solution of (1.1) is hstable in the pth moment by using Lyapunov’s technique is our first major research aim.
Effectively, it is difficult to look for a Lyapunov’s function (functional) for time changed stochastic systems. Meanwhile, the obtained conditions captured by making use of Lyapunov’s function are generally shown on the basis of some differential inequalities, matrix inequalities, and so on. There calculations are complicated and difficult to test. The second aim of our paper is to study some new explicit conditions to ensure that the solution of (1.1) possesses the ηstability in mean square under some hypotheses. In the proof, our method takes advantage of the Itô formula and involves a proof by contradiction.
2 Preliminary
Let \((\Omega ,\mathcal {F},\{\mathcal {F}_{t}\}_{t\geq 0},\mathbb{P})\) be a complete probability space with a filtration \(\{\mathcal {F}_{t}\}_{t\geq 0}\) satisfying the usual conditions. Assume that \(\{D(t), t\geq 0\}\) is a càdlàg nondecreasing Lévy process, which is named a subordinator, that starts at 0. Particularly, \(D(t)\) is called the βstable subordinator denoted by \(D_{\beta}(t)\) if it is strictly increasing with the following Laplace transform:
Define the generalized inverse of \(D_{\beta}(t)\) as
which is well known as the initial hitting time process. The timechange process \(E_{t}\) is nondecreasing and continuous. Define the special filtration as
where \(B_{v}\) is the standard Brownian motion and the notation \(\sigma _{1}\vee \sigma _{2}\) denotes the σalgebra generated by the union of σalgebras \(\sigma _{1}\) and \(\sigma _{2}\). By the results in [14], we can deduce that \(B_{E_{t}}\) is a square integrable martingale with respect to the filtration \(\mathcal {G}_{t}=\mathcal {F}_{E_{t}}\).
Let \(r>0\) and \(C:=C([r,0];\mathbb{R}^{d})\) denote the family of all continuous functions ϕ from \([r,0]\) to \(\mathbb{R}^{d}\) with the norm \(\\phi \_{C}=\sup_{r\leq s\leq 0}\phi (s)\).
Based on [26], we put forward the following hypotheses for ensuring the existence and uniqueness of a solution for (1.1):

(H1)
\(f,u:\mathbb{R}_{+}\times \mathbb{R}_{+}\times C\rightarrow \mathbb{R}^{d}\) and \(g:\mathbb{R}_{+}\times \mathbb{R}_{+}\times C\rightarrow \mathbb{R}^{d \times k}\) are some measurable functions and there is a positive constant K such that for all \(t_{1},t_{2}\geq 0\) and \(x,y\in C\),
$$\begin{aligned} & \bigl\vert f(t_{1},t_{2},x)f(t_{1},t_{2},y) \bigr\vert \vee \bigl\vert u(t_{1},t_{2},x)u(t_{1},t_{2},y) \bigr\vert \vee \bigl\Vert g(t_{1},t_{2},x)g(t_{1},t_{2},y) \bigr\Vert \\ &\quad \leq K \Vert xy \Vert _{C}. \end{aligned}$$ 
(H2)
If \(y(t)\) is a càdlàg \(\mathcal {G}_{t}\)adapted process, then
$$ f(t,E_{t},y_{t}),u(t,E_{t},y_{t}),g(t,E_{t},y_{t}) \in \mathcal {L}( \mathcal {G}_{t}), $$where \(\mathcal {L}(\mathcal {G}_{t})\) denotes the class of càdlàg \(\mathcal {G}_{t}\)adapted processes.
To establish ηstability, we also demand the following assumption:
Referring to [26], we conclude that (1.1) has a unique \(\mathcal {G}_{t}\)adapted solution process \(y(t)\) under the assumptions (H1) and (H2). Furthermore, equation (1.1) has a trivial solution when the initial value is \(\xi \equiv 0\).
Definition 2.1
A positive function η on \(\mathbb{R}_{+}\) is called an ηtype function if the following assumptions are fulfilled:

(i)
It is nondecreasing and continuously differentiable in \(\mathbb{R}_{+}\).

(ii)
\(\eta (0)=1\), \(\lim_{t\rightarrow \infty}\eta (t)=\infty \), and \(J=\sup_{t>0} \frac{\eta '(t)}{\eta (t)} <\infty \).

(iii)
For all \(u\geq 0\) and \(v\geq 0\), one has \(\eta (u+v)\leq \eta (u)\eta (v)\).
Definition 2.2
A solution \(y(t,\varphi )\) of (1.1) is called ηstable in the pth moment sense if, for any initial data φ, there are positive constants \(\delta >0\) and \(K>0\) such that for each \(t\geq 0\),
In particular, if p is equal to 2, \(y(t,\varphi )\) is said to be ηstable in mean square.
Remark 2.1
We remark that hstability coincides with some known stability types when h are some special functions. In fact, if \(h(t)=e^{t}\), then ηstability is consistent with exponential stability; if \(\eta (t)=1+t\), then ηstability is consistent with polynomial stability, and if \(\eta (t)=\operatorname{ln}(e+t)\), then ηstability is consistent with logarithmic stability.
Remark 2.2
There exists an ηtype function which tends to infinity faster than \(e^{t}\). For instance, \(\eta (t)=(1+t)e^{t}\) is an ηtype function and \(\lim_{t\rightarrow +\infty}\frac{(1+t)e^{t}}{e^{t}}=+\infty \).
3 Main results
Let \(V\in C^{1,1,2}(\mathbb{R}_{+}\times \mathbb{R}_{+}\times \mathbb{R}^{d}, \mathbb{R})\), \(V_{t_{1}}(t_{1},t_{2},y)= \frac{\partial V(t_{1},t_{2},y)}{\partial t_{1}}\), \(V_{t_{2}}(t_{1},t_{2},y)= \frac{\partial V(t_{1},t_{2},y)}{\partial t_{2}}\), \(V_{y}(t_{1}, t_{2},y)= ( \frac{\partial V(t_{1},t_{2},y)}{\partial y_{1}},\dots , \frac{\partial V(t_{1},t_{2},y)}{\partial y_{d}} )\) and \(V_{yy}(t_{1},t_{2},y)= ( \frac{\partial ^{2}V(t_{1},t_{2},y)}{\partial y_{i}\partial y_{j}} )_{d\times d}\) be continuous for all \((t_{1},t_{2},y)\in \mathbb{R}_{+}\times \mathbb{R}_{+}\times \mathbb{R}^{d}\). By Itô formula (see [21]), from (1.1) we have
where
and
Theorem 3.1
Let the hypotheses (H1) and (H2) hold. Assume that there is \(V\in C^{1,1,2}(\mathbb{R}_{+}\times \mathbb{R}_{+}\times \mathbb{R}^{d}, \mathbb{R}_{+})\) such that for any \((t,E_{t},y(t))\in \mathbb{R}_{+}\times \mathbb{R}_{+}\times \mathbb{R}^{d}\),

(i)
\(c_{1}y(t)^{p}\leq V(t,E_{t},y(t))\leq c_{2}y(t)^{p}\),

(ii)
\(J_{1}V(t,E_{t},y(t))\leq \lambda V(t,E_{t},y(t))\),

(iii)
\(J_{2}V(t,E_{t},y(t))\leq 0\),
hold for the solution \(y(t)\) of (1.1), where p, \(c_{1}\), \(c_{2}\), and λ are some positive constants. If for \(\delta \in (0,\lambda /J)\), we can show
then the trivial solution of (1.1) is ηstable in mean square.
Proof
Let \(\delta \in (0,\lambda /J)\). Applying Itô formula to \(\eta ^{\delta}(t)V(t_{1},t_{2},y(t))\), for every \(t\geq 0\), we obtain
By condition (iii), we get
Notice that
Then, taking expectations on both sides of (3.4), we have
By using Definition 2.1(ii), one has
Since \(\delta \in (0,\lambda /J)\), according to condition (ii), one has
Furthermore, from condition (i), we can obtain
Hence
The proof is complete. □
Remark 3.1
The authors of [28] showed that the solution of (1.1) without time delay is the pth moment exponentially stable under (H1) and (H2) when the conditions (i)–(iii) are satisfied. Thus, it follows from Remark 2.1 that our Theorem 3.1 generalizes Theorem 4.1 of [28].
Remark 3.2
The authors of [26] showed that the solution for (1.1) with Markovian switching is the pth moment exponentially stable under (H1) and (H2) when the conditions (i)–(iii) are satisfied. We remark that the solution of (1.1) with Markovian switching is hstable in the pth moment sense under (H1) and (H2) when the corresponding conditions (i)–(iii) hold, which implies that our Theorem 3.1 expands Theorem 3.1 of [26] by using Remark 2.1.
Next, we want to make use of Theorem 3.1 to establish the following corollary.
Corollary 3.1
Let the assumptions (H1)–(H2) hold. If there is a positive constant \(\lambda >0\) such that, for all \(t\geq 0\) and the solution \(x(t)\) of (1.1), one has
and
then the trivial solution of (1.1) is ηstable in mean square.
Proof
Let \(V(t,E_{t},y(t))=y(t)^{2}\). One can obviously check that condition (i) in Theorem 3.1 holds for \(p=2\), \(c_{1}=c_{2}=1\), and
Thus, (3.5) and (3.6) respectively imply that the conditions (ii) and (iii) hold. The proof is complete. □
And now we are going to study some new conditions ensuring the ηstability for (1.1). At this time, we need to introduce some functions. Let \(\kappa (\vartheta ,t):[r,0]\times \mathbb{R}_{+}\rightarrow \mathbb{R}^{d}\) be increasing in ϑ for all \(t\in \mathbb{R}_{+}\). Besides, we also assume that \(\kappa (\theta ,t)\) is normalized to be continuous from the left in ϑ on \([r,0]\). Let
be a locally bounded Borelmeasurable function in t for each \(\varphi \in C([r,0];\mathbb{R}^{d})\). In our case, the integral in (3.4) is the Riemann–Stieltjes integral.
Theorem 3.2
Let \(\zeta (\cdot ):\mathbb{R}_{+}\rightarrow \mathbb{R}\) be a locally bounded Borelmeasurable function. If for any \(t\in \mathbb{R}_{+}\) and \(\varphi \in C([r,0];\mathbb{R}^{d})\), one has
and
then the solution of (1.1) is ηstable in the mean square sense if there exists \(\beta >0\) such that for any \(t\in \mathbb{R}_{+}\),
Proof
To prove the above conclusion, we will divide the value range of J into two, namely \(J\in (0,1]\) and \(J\in (1,+\infty )\). The specific proof process is as follows:
Case 1. Suppose \(J\in (0,1]\). Fix \(K>1\) and let \(\varphi \in C([r,0];\mathbb{R}^{d})\) be such that \(\mathbb{E}\\varphi \_{C}^{2}>0\). We denote \(y(t):=y(t,\varphi )\), \(t\geq r\), where \(y(t,\varphi )\) is the solution of (2.1). Denote \(Y(t):=\mathbb{E}y(t)^{2}\), \(t\in \mathbb{R}_{+}\) and \(Z(t):=K\mathbb{E}\\varphi \_{C}^{2}\eta ^{\beta}(t)\), \(t\geq 0\). For convenience, we define \(\eta (t)=\eta (0)=1\) for \(t\in [r,0]\). Next, we can conclude that \(Y(t)< Z(t)\) for \(t\in [r,0]\) since \(K>1\) and \(\mathbb{E}\\varphi \_{C}^{2}>0\). We shall show
To the contrary, we assume that there exists \(t_{1}>0\) such that \(Y(t_{1})>Z(t_{1})\). Let \(t^{\ast}:=\inf \{t>0:Y(t)>Z(t)\}\). By continuity of \(Y(t)\) and \(Z(t)\), we have
and
for some \(t_{n}\in (t^{\ast},t^{\ast}+\frac{1}{n})\), \(n\in \mathbb{N}\).
Choosing \(0<\delta <\beta \) and applying Itô formula to \(V(t,y)=\eta ^{\delta}(t)y(t)^{2}\), one has
Utilizing the standard property of the Itô integral, one has
From (3.8), (3.9), and the Fubini theorem, we get
Let \(K_{1}:=K\mathbb{E}\\xi \^{2}\). Since \(\kappa (u,\vartheta )\) is nondecreasing in ϑ on \([r,0]\), from (3.11) we know that
for any \(u\leq t^{\ast}\).
If \(u+\vartheta \leq 0\), then \(u\leq \vartheta \). Since η is increasing in \(\mathbb{R}_{+}\), we have
If \(u+\vartheta \geq 0\), by Definition 2.1(iii), we also have
So, we get for each \(t^{\ast}\geq s\) that
Then, combining (3.10) and (3.13), we get for any \(u\leq t^{\ast}\),
Noticing that \(J\in (0,1]\), we have \(\eta (u)\leq \eta '(u)\). Since \(\delta \beta <0\), we obtain
Due to the fact that \(K>1\), we have
Thus, \(\mathbb{E}y(t^{\ast})^{2}< K\mathbb{E}\\varphi \_{C}^{2}\eta ^{ \beta}(t^{\ast})\), which conflicts with (3.12). Hence,
Consequently, the solution of (1.1) is ηstable in the mean square sense.
Case 2. Suppose \(J\in (1,+\infty )\). Choose \(0<\delta <\frac{\beta}{J}\). By a simple calculation, we know that \(0<\frac{J\delta \beta}{J(\delta \beta )}<1\). Fix \(K>\frac{J(\delta \beta )}{J\delta \beta}>1\) and let \(\varphi \in C([r,0];\mathbb{R}^{d})\) be such that \(\mathbb{E}\\varphi \_{C}^{2}>0\). In a similar manner, we can also show that for each \(t\in \mathbb{R}_{+}\),
To the contrary, we assume that there exists \(t_{1}>0\) such that \(X(t_{1})>Z(t_{1})\). Let \(t^{\ast}:=\inf \{t>0: Z(t)< Y(t)\}\). By continuity of \(Y(t)\) and \(Z(t)\),
and
for some \(t_{n}\in (t^{\ast},t^{\ast}+\frac{1}{n})\), \(n\in \mathbb{N}\).
Applying the Itô formula to \(V(t,y)=\eta ^{\delta}(t)y(t)^{2}\), we get for all \(t\leq t^{\ast}\),
Noticing that \(J\in (1,+\infty )\), we have \(\eta (s)\leq \frac{1}{J}\eta '(s)\). Since \(J\delta \beta <0\), we get
Noticing that \(0<\frac{J\delta \beta}{J(\delta \beta )}<1\) and using the fact that \(K>\frac{J(\delta \beta )}{J\delta \beta}\), one has
Thus, \(\mathbb{E}y(t^{\ast})^{2}< K\mathbb{E}\\varphi \_{C}^{2}\eta ^{ \beta}(t^{\ast})\), which conflicts with (3.15). Hence,
Consequently, the solution of (1.1) is hstable in the mean square sense.
The proof is complete. □
Corollary 3.2
Let \(\Gamma (\cdot ,\cdot ):[r,0]\times \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\), \(\zeta _{i}(\cdot ),r_{i}(\cdot ):\mathbb{R}_{+}\rightarrow \mathbb{R}\), \(i=0,1,2,\dots ,m\) with \(0:=r_{0}(t)\leq r_{1}(t)\leq r_{2}(t)\leq \cdots \leq r_{m}(t)\leq r\), \(t\in \mathbb{R}_{+}\), be locally bounded Borelmeasurable functions. If for each \(t\in \mathbb{R}_{+}\) and \(\varphi \in C([r,0];\mathbb{R}^{d})\) one has
and
then the solution of (1.1) is ηstable in the mean square sense if there exists \(\beta >0\) such that for each \(t\in \mathbb{R}_{+}\),
Proof
Define the following functions for \(t\geq 0\), \(u\in [r,0]\):
By the properties of the Riemann–Stieltjes integrals, for any \(\varphi (\cdot )\in C([r,0];\mathbb{R}^{d})\), we have
Then for any \(t\in \mathbb{R}_{+}\), \(\varphi (\cdot )\in C([r,0];\mathbb{R}^{d})\),
Hence, (3.16) implies that (3.6) holds, and (3.18) implies that (3.10) holds. According to Theorem 3.1, we can immediately derive our desired result. The proof is complete. □
Corollary 3.3
Let ζ be a constant and \(\nu (\cdot ):[r,0]\rightarrow \mathbb{R}_{+}\) an increasing function. If for any \(t\in \mathbb{R}_{+}\) and \(\varphi \in C([r,0];\mathbb{R}^{d})\), one has
and
then the solution of (1.1) is ηstable in the mean square sense if
Proof
By (3.21) and the continuity of \(\eta (t)\), we can show that for a sufficiently small \(\beta >0\) one has the following inequality:
Since \(\nu (\cdot )\) is increasing, we obtain
which implies that (3.10) holds. The proof is complete. □
From Corollaries 3.2 and 3.3, we can immediately obtain the following Corollary 3.4.
Corollary 3.4
Let \(r_{i}(\cdot ):\mathbb{R}_{+}\rightarrow \mathbb{R}\), \(i=0,1,2,\dots , m\) with \(0:=r_{0}(t)\leq r_{1}(t)\leq r_{2}(t)\leq \cdots \leq r_{m}(t)\leq r\), \(t\in \mathbb{R}_{+}\), be locally bounded Borelmeasurable functions. Assume that there exist constants \(\zeta _{i}\), \(i=0,1,2,\dots , m\) and a Borelmeasurable function \(\mu :[r,0]\rightarrow \mathbb{R}_{+}\) such that for any \(t\in \mathbb{R}_{+}\) and \(\varphi \in C([r,0];\mathbb{R}^{d})\),
and
Then, the solution of (1.1) is hstable in the mean square sense if
Remark 3.3
In fact, the assumptions (3.8) and (3.9) are generalizations of some existing conditions. According to the information we have found in the reported literature, even for deterministic differential equations, the assumptions (3.8) and (3.9) have not been used to research the ηstability in mean square of stochastic systems. Our results are of innovative value and they provide advantage when studying applications of “mixed” delay timechanged SDEs, including the point, variable, and distributed delay.
Remark 3.4
The conditions of Theorems 3.1 and 3.2 highlight the dominant role of the drift term “dt” in the study of the ηstability for a timechanged system. In the meantime, it indicates that “\(dE_{t}\)” and “\(dB_{E_{t}}\)” are relatively less important.
Now, we intend to present an example to explain the statement in Remark 3.4. We consider the following two timechanged SDEs:
and
By Corollary 3.4, we conclude that the timechanged equation (3.25) is hstable if \(2c+ d^{2}\leq 0\), while we cannot conclude that the timechanged equation (3.26) is hstable no matter what c and d are.
4 Some examples
Example 4.1
Consider the following functional stochastic differential equation driven by the timechanged Brownian motion:
with \(y_{0}(\cdot )=\xi \in C([r,0];\mathbb{R}^{d})\). Assume that there exists a continuous function \(\zeta (\cdot ):[r,0]\rightarrow \mathbb{R}_{+}\) such that for all \(t\geq 0\), \(y\in \mathbb{R}^{d}\), and \(\varphi \in C([r,0];\mathbb{R}^{d})\),
and for all \(t_{1},t_{2}>0\),
According to [26], we can draw the conclusion that the null solution of (4.1) is mean square exponentially stable if
Notice that (4.2) and (4.3) mean that
By Corollary 3.4, the null solution of (4.1) is ηstable in mean square if (4.4) holds and
Notice that
owing to Hölder inequality. So, (4.6) is weaker than (4.5).
Example 4.2
In order to further explain the applicability of our main conclusions, we carefully consider the following scalar linear stochastic differential equation with variable delay:
where c, d, \(h_{1}\) are continuous functions on \(\mathbb{R}_{+}\) and \(h_{1}(t)\leq r\) for some \(r>0\).
Let
for \(t\geq 0\), \(\varphi \in C([r,0];\mathbb{R})\). Then, for all \(t\geq 0\), \(\varphi \in C([r,0];\mathbb{R})\), one has
Next, referring to Corollary 3.2, one can conclude that if there is a positive constant λ such that for each \(t\geq 0\),
and for any \(t_{1},t_{2}>0\),
then the null solution of (4.7) is ηstable in the mean square sense.
On the other hand, due to continuity,
means that (4.9) holds with some sufficiently small \(\zeta >0\). Hence, as long as (4.10) and (4.11) hold, one can derive that the null solution of (4.7) is hstable in the mean square sense.
Example 4.3
Lastly, we intend to investigate the following distributed delay equation for \(t\geq 0\):
where \(\kappa (t)\) has bounded variation on \([r,0]\) and \(c(t)\) is a continuous function.
Denote \(\zeta (t):=c(t)\mathrm{Var}_{[r,0]}\kappa (\cdot )\), \(t\geq 0\). According to [26], the null solution of (4.12) is asymptotically stable in mean square provided that for all \(t,s>0\),
and
In fact, one can also be certain that the null solution of (4.12) is hstable in mean square if (4.13) and (4.14) hold. Let
for \(t\geq 0\), \(\varphi \in C([r,0];\mathbb{R})\). Define \(V(u):=\mathrm{Var}_{[r,u]}\kappa (\cdot )\), \(u\in [r,0]\). Then \(V(u)\) is increasing on \([r,0]\). We obtain by the properties of the Riemann–Stieltjes integral
Thus,
According to Theorem 3.2, the null solution of (4.12) is ηstable in mean square if (4.13) is satisfied and there is a positive constant \(\delta >0\) such that for all \(t\geq 0\),
It follows from (4.14) that for each \(t\in \mathbb{R}_{+}\),
Setting \(\delta \in (0,\frac{\zeta}{2})\) sufficiently small, we can immediately derive that \(\frac{1}{2}(\eta ^{\delta}(r)1)V(0)<\frac{\zeta}{2}\), and, for each \(t\in \mathbb{R}_{+}\), we get
Noticing that \(V(t)\) is increasing, we can immediately know that \(\int _{r}^{0}\eta ^{\delta}(\vartheta )\,d[V(\vartheta )]\leq \eta ^{ \delta}(r)V(0)\). So, one obtains, for any \(t\in \mathbb{R}_{+}\),
5 Conclusion
In this paper, by using the timechanged Itô formula and proof by contradiction, we gained some new criteria of the ηstability in mean square for the stochastic functional differential equation driven by timechanged Brownian motion. Three concrete examples were given to illustrate the validity of our main conclusions. Hopefully, in the future, we can continue our study of the ηstability in mean square for other special stochastic equations.
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This work was partially supported by the NNSF of China (No. 11901058, 62076039) and the Natural Science Foundation of Hubei Province (No. 2021CFB543).
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Zhi Li: Writing Original draft preparation. Yaru Zhang: Resion Yue Wang: Stability. Liping Xu: is the corresponding author is responsible for ensuring that the descriptions are accurate and agreed by all authors. Xianping He: Writing Reviewing and Editing.
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He, X., Zhang, Y., Wang, Y. et al. ηStability for stochastic functional differential equation driven by timechanged Brownian motion. J Inequal Appl 2024, 60 (2024). https://doi.org/10.1186/s1366002403128y
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DOI: https://doi.org/10.1186/s1366002403128y