η -Stability for stochastic functional differential equation driven by time-changed Brownian motion

This manuscript focuses on a class of stochastic functional diﬀerential equations driven by time-changed Brownian motion. By utilizing the Lyapunov method, we capture some suﬃcient conditions to ensure that the solution for the considered equation is η -stable in the p th moment sense. Subsequently, we present some new criteria of the η -stability in mean square by using time-changed Itô formula and proof by contradiction. Finally, we provide some examples to demonstrate the eﬀectiveness of our main results


Introduction
At present, the time-changed 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 time-changed semimartingales, there have been a lot of authors working on the stochastic differential equations with the time-changed 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 time-changed semimartingales.For example, see [21,22] for the stability in probability; [16,17] for the moment stability and path stability; [25][26][27] for the asymptotic stability; and [12,28] for the exponential stability.
Meanwhile, the η-stability is a valuable extension of certain well-known stability types such as polynomial, exponential, and logarithmic stability, etc.The η-stability 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 long-time behavior of the solution.For instance, see Choi et al. [3] for the linear dynamic equations; Damak et al. [5] for the boundedness and η-stability of the perturbed equations; Ghanmi [8] for the practical η-stability; Xu and Liu [23] and Xu et al. [24] for the η-stability 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 η-stability of some evolution equations in Banach spaces by using some Gronwall-type 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 η-stability for neutral stochastic pantograph differential equations driven by Lévy noise, and Li et al. [11] explored the η-stability for stochastic Volterra-Levin equations.In our paper, we try to make a study of the h-stability of the following functional SDE: where E t is defined as the inverse of the β-stable subordinator with index 0 < β < 1, Giving some coefficient conditions ensuring that the solution of (1.1) is h-stable 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 η-stability in mean square under some hypotheses.In the proof, our method takes advantage of the Itô formula and involves a proof by contradiction.

Preliminary
Let ( , F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions.Assume that {D(t), t ≥ 0} is a càdlàg nondecreasing Lévy process, which is named a subordinator, that starts at 0. Particularly, D(t) is called the β-stable subordinator denoted by D β (t) if it is strictly increasing with the following Laplace transform: Define the generalized inverse of D β (t) as which is well known as the initial hitting time process.The time-change process E t is nondecreasing and continuous.Define the special filtration as where B v is the standard Brownian motion and the notation σ 1 ∨ σ 2 denotes the σ -algebra generated by the union of σ -algebras σ 1 and σ 2 .By the results in [14], we can deduce that B E t is a square integrable martingale with respect to the filtration G t = F E t .
Based on [26], we put forward the following hypotheses for ensuring the existence and uniqueness of a solution for (1.1): (H1) f , u : R + × R + × C → R d and g : R + × R + × C → R d×k are some measurable functions and there is a positive constant K such that for all t 1 , t 2 ≥ 0 and x, y ∈ C, where L(G t ) denotes the class of càdlàg G t -adapted processes.To establish η-stability, we also demand the following assumption: Referring to [26], we conclude that (1.1) has a unique 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 ξ ≡ 0.
Definition 2.1 A positive function η on R + is called an η-type function if the following assumptions are fulfilled: Definition 2.2 A solution y(t, ϕ) of (1.1) is called η-stable in the pth moment sense if, for any initial data ϕ, there are positive constants δ > 0 and K > 0 such that for each t ≥ 0,
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 h-stable 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.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 η-stability for (1.1).At this time, we need to introduce some functions.Let κ(ϑ, t) : [-r, 0] × R + → R d be increasing in ϑ for all t ∈ R + .Besides, we also assume that κ(θ , t) is normalized to be continuous from the left in ϑ on [-r, 0].Let then the solution of (1.1) is η-stable in the mean square sense if there exists β > 0 such that for any t ∈ R + ,
then the solution of (1.1) is η-stable in the mean square sense if there exists β > 0 such that for each t ∈ R + , Proof Define the following functions for t ≥ 0, u ∈ [-r, 0]: By the properties of the Riemann-Stieltjes integrals, for any 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.
then the solution of (1.1) is η-stable in the mean square sense if Proof By (3.21) and the continuity of η(t), we can show that for a sufficiently small β > 0 one has the following inequality: 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.
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 η-stability in mean square of stochastic systems.Our results are of innovative value and they provide advantage when studying applications of "mixed" delay time-changed 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 η-stability for a time-changed 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 time-changed SDEs:  where c, d, h 1 are continuous functions on R + and h 1 (t) ≤ r for some r > 0.
Let  [26], the null solution of (4.12) is asymptotically stable in mean square provided that for all t, s > 0, -2G(t, s) + H 2 (t, s) ≤ 0 In fact, one can also be certain that the null solution of (4.12) is h-stable in mean square if (4.13) and (4.According to Theorem 3.2, the null solution of (4.12) is η-stable in mean square if (4.13) is satisfied and there is a positive constant δ > 0 such that for all t ≥ 0,

Conclusion
In this paper, by using the time-changed Itô formula and proof by contradiction, we gained some new criteria of the η-stability in mean square for the stochastic functional differential equation driven by time-changed 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 η-stability in mean square for other special stochastic equations.
be locally bounded Borel-measurable functions.Assume that there exist constants ζ i , i = 0, 1, 2, . . ., m and a Borel-measurable function μ : [-r, 0] → R + such that for any t ∈ R + and ϕ ∈ C([-r, 0]; R d ), By Corollary 3.4, we conclude that the time-changed equation (3.25) is h-stable if 2c + d 2 ≤ 0, while we cannot conclude that the time-changed equation (3.26) is h-stable no matter what c and d are.