Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching

Abstract: The aim of this work is to study the asymptotic stability of the time-changed stochastic delay differential equations (SDDEs) with Markovian switching. Some sufficient conditions for the asymptotic stability of solutions to the time-changed SDDEs are presented. In contrast to the asymptotic stability in existing articles, we present the new results on the stability of solutions to time-changed SDDEs, which is driven by time-changed Brownian motion. Finally, an example is given to demonstrate the effectiveness of the main results.


Introduction
The research on stochastic differential equations (SDEs) plays an important role in modeling dynamic system areas, such as physics, economics and finance, biological and so forth. Recently, the qualitative study of the solution of SDEs has received much attention. Particularly, the stability of SDEs has been considered widely by many researchers [1][2][3][4]. It is well known that time delay is unavoidable in practice, then the corresponding stochastic delay differential equations (SDDEs) are used more widely in systems. It considers the effects of past behaviors imposed to the current status. The stability results of SDDEs we have mentioned here can be found in [5][6][7][8]. The delay term has main influence on the stability of SDDEs. It could be regarded as a perturbation to the stable systems, or may be the delay part has a stabilizing effect as well [8]. Jump system is a new type of SDE with Markovian switching [9][10][11][12]. In practice, the system can switch from one mode to another randomly, and the switching between the modes is governed by a Markov process. SDDE with Markovian switching is a kind of hybrid system, including both the logical switching mode and the state of system. It is used widely in many applied areas such as neural networks, traffic control model, and so on.
Very recently, Chlebak et al. [13] considered a sub-diffusion process in Hilbert space and the associated fractional Fokker-Planck-Kolmogorov equations. The process is connected with a limit process arising from continuous-time random walks. In fact, the limit process is a time-changed Lévy process, which is the first hitting time process of certain stable subordinator (see [14,15] for details). The existence and stability of SDEs driven by time-changed Brownian motion attracted lot of attention. Wu [16] established the timechanged Itô formula of time-changed SDE, and the stability analysis is investigated. Subsequently, Nane and Ni [17,18] established the Itô formula for time-changed Lévy noise, then discussed the asymptotic stability and path stability for the solution of time-changed SDEs with jump, respectively. And in [19], we considered the exponential stability for the time-changed stochastic functional differential equations with Markov switching.
Motivated strongly by the above, in this paper, we will study the stability of time-changed SDDEs with Markovian switching. By applying the time-changed Itô formula and Lyapunov function, we present the LaSalle-Type theorem [6,12] of the time-changed SDDEs with Markovian switching. More precisely, we consider the following SDDEs driven by time-changed Brownian motions:

x t ρ t E r t x t x t τ t f t E r t x t x t τ E g t E r t x t x t τ B
, where ρ f g , , are appropriately specified later.
In the remaining parts of this paper, further needed concepts and related background are presented in Section 2. In Section 3, the main stability results of the time-changed SDDEs with Markovian switching are given. Finally, an example is given to illustrate the effectiveness of the main results.

Preliminary
Throughout this paper, let P Ω, , , is right continuous and F contains all the P-null sets in F). Let U t t 0 { ( )} ≥ be a right continuous with left limit (RCLL) increasing Lévy process that is called a subordinator. In particular, a β-stable subordinator is a strictly increasing process denoted by U t β ( ) and characterized by Laplace transform For an adapted β-stable subordinator U t β ( ), define its generalized inverse as which is called the first hitting time process. And E t is continuous since U t β ( ) is strictly increasing. Let B t be a standard Brownian motion independent of E t , define the filtration as ∨ denotes the σ-algebra generated by the union of σ-algebras σ 1 and σ 2 . It concludes that the time-changed Brownian motion B Et is a square integrable martingale with respect to the filtration E t 0 If w is a real-valued function on n , then its kernel is expressed by We also denote by L , 1 ( ) + + the family of all functions r : Let r t t , 0 ( ) ≥ be a right continuous Markov chain on the probability space taking values in a finite state space S N 1, 2, , , , , , , , , , In this paper, the following hypothesis is imposed on the coefficients ρ f , and g.  c x x  y y   , , , ,  , , , ,  , , , ,  , , , ,  , , , , , , , , for all t 0 ≥ , i S ∈ and x y x y , , , , , , 0, 0 , , , 0, 0 , , , 0, 0 : 0 .
F denotes the class of RCLL and Et F -adapted process.
We will need the useful semimartingale convergence theorem, which is cited here as a lemma.
X ω exists and is finite and

Main results and discussion
In this section, we aim to establish the stability results of the system equation.
To prove this result, let us present an existence lemma at first.
Under the conditions of Theorem 3.1, for any initial data x θ τ θ ξ C τ : where σ ∞ is the explosion time [15,21]. So we only need to show that σ = ∞ ∞ a.s. For any k 1 ≥ , define the following stopping time By using conditions (3.1) and (3.2), we can see that Now, let us prove our main results.
Proof of Theorem 3.1. We divide the proof into three steps.
Step 1. For any ξ and i 0 we write x t i ξ x t ; ,       Step 2. If we set w w w We now claim that If it is false, then Since E t is continuous, by means of (3.6) and the continuity of both the solution x t ( ) and the function V t E x r t , , , t ( () ) [6,12], we can see that Since the initial data ξ is bounded, we can find a positive number h sufficiently large, which depends on ε, satisfying ξ θ h | ( )| < for all τ θ 0 − ≤ ≤ almost surely, while It is easy to see from (3.10) and (3.12) that In what follows, we define a sequence of stopping times,  On the other hand, by the hypothesis (H 1 ), there exists a constant K h such that ρ t t x y i f t t x y i g t t x y i K , , , , ,

T I ρ s E x s x s τ r s s T I f s E x s x s τ r s
it is a contradiction. So (3.9) must be hold.
Step 3. We first show that w Ker( ) ≠ ∅. Note from (3.9) and (3.11) that there exists a Ω Ω 0 ⊂ with P Ω 1 If it is not true, there exist some ω Ω 0 ∈ such that , that is, w z 0 ( ) > . However, from (3.19) we can see that and V t E x i lim inf , , , . 4 Controllability of linear stochastic differential system Let us consider the following linear SDE with delay: Here u is an t F-measurable and p -value control law. For each r t i S ki are all n n × constant matrices and D i is an n p × matrix. The aim of this study is to design a delay-independent feedback controller with the form u t ( ) = H r t x t ( ( )) ( ), such that the following closed-loop system of (4.1) x  Then M NR N 0

t A r t x t C r t x t τ D r t H r t x t t F r t x t G r t x t τ E M r t x t N r t x t τ B E
Here, as usual, by R R T = we mean R is a symmetric matrix while by R 0 > or R 0 < we mean R is a positivedefinite or negative matrix, respectively.) it follows that x QCy x QCy y C Qx y y x QCC Qx 2 .
Let w x w x min i S i 11 11 ( ) ( ) = ∈ , clearly w x w x 11 12 Let w x w x min i S i 21 21 ( ) ( ) = ∈ , w x w x max i S i 22 22 ( ) ( ) = ∈ , then it is clear that w x w x 21 22 ( ) ( ) ≥ for x 0 ≠ . Therefore, the system is almost surely asymptotically stable with the controller designed above. □ Remark. If the controller u t ( ) applies to the time-changed term E d t , then system (4.1) becomes the following system: In this case, we can also consider the asymptotic stability by means of the LMIs just renew P i 1 and P i 2 as follows: In what follows, we shall give an example to show the aforementioned results.

Conclusions
The SDDEs driven by time-changed Brownian motions is a new research area for recent years. In this work, we have considered the asymptotic stability of the time-changed SDDEs with Markovian switching, by expanding the time-changed Itô formula and the time-changed semi-martingale convergence theorem. Our result generalizes that of SDDEs in the literature. Due to the more construction of SDDEs with time change than the usual SDDEs, our result is not a trivial generalization. Stability of the time-changed stochastic delay differential equation  627