-Stability in -th Moment of Neutral Impulsive Stochastic Functional Differential Equations with Markovian Switching

Abstract

In this paper, we investigate the -stability in q-th moment for neutral impulsive stochastic functional differential equations with Markovian switching (NISFDEwMS). Moreover, -stability in q-th moment is studied by using the Lyapunov techniques and a new Razumikhin-type theorem to prove our result. Finally, we check the main result by a numerical example.

1. Introduction

The aim of stochastic differential equations is to provide a mathematical model for a differential equation disturbed by a random noise. Consider an ordinary differential equation of the form

An ordinary differential equation is used to describe the evolution of a physical system. If we take into account the random disturbances, we add a noise term, which will be of the form , where denotes a Brownian motion and , for this instant, is a constant which corresponds to the intensity of the noise. Then, we have a stochastic differential equation (SDE) of the form

Moreover, we have the following integral form:

We generalize equation (3) by taking depending on the state of the system to an instant :

In view of the integral in , this equation will depend on the theory of stochastic integral (see [1]).

One of the most important qualitative studies of SDE is the stability theory.

There are a lot of papers about the stability theory of different types of SDE (see [1–3]).

A special case of SDE is the neutral stochastic functional differential equations (NSFDE).

In the literature, there are many research papers on the stability theory of NSFDE (see [4–6]). Recently, stability of SFDE with Markovian switching (SFDEwMS) has received a lot of attention (see [7–11]).

One of the most important classes of SDE is the impulsive SDE (ISDE). In the last decades, the stability theory of ISDE takes much more attention (see [7, 11–13]).

To the best of our knowledge, there are a few papers on the -stability or stability of SDE with general decay rate. In [14–16], the authors established the stability with general decay rate of stochastic functional differential equations with finite and infinite delay.

In the literature, there is no existing paper on the -stability of NISFDEwMS.

In this paper, we study the -stability in -th moment of NISFDEwMS. In view of the other related topics on NISFDEwMS, the main contributions of this paper are to obtain a new technique to study the -stability in a hybrid impulsive neutral stochastic system, to develop a new Razumikhin-type theorem on the Lyapunov function to prove the -stability of the system, and to propose new theories on asymptotic properties in -th moment of NISFDEwMS. In this sense, our results generalize the paper [16] in the case of NISFDEwMS.

The paper is organized as follows. In Section 2, we cite some definitions and basic notions. In Section 3, we investigate the -stability in -th moment of NISFDEwMS. In Section 4, one numerical example is given to show the interest of our main results.

2. Preliminaries and Definitions

The complete probability space is denoted by , where is a filtration satisfying the usual conditions. is an -dimensional Brownian motion defined on the probability space. Let and be the family of continuous functions from to with the norm , where for any . If is a matrix, its trace norm is denoted by , where its norm is given by .

Let , where exists and for all but at most a finite number of points t, with the uniform norm , where and denote the right-hand and left-hand limits of at , respectively. For , let be the family of all function such that .

Let be a right-continuous Markov chain on the probability space taking values in a finite state space with as a generator given bywhile . Here, is the transition rate from to , if , while

We consider the following NISFDEwMS:where is defined by , which is a -valued stochastic process, are impulsive moments satisfying and , and represents the jump in the state at with determining the size of the jump. Furthermore, we assume that

Let be the family of all nonnegative functions on , which are twice continuously differentiable with respect to and once continuously differentiable with respect to .

Define the operator by (see [9])where

Throughout this paper, we assume that , , , and are smooth enough to ensure the global existence and uniqueness of solutions for all . For any , there is a unique solution satisfying (7), which is continuous on the left-hand side and have a finite limit on the right-hand side. We assume that , , and , where , , and , which implies that 0 is an equilibrium solution.

Definition 1. The function is said to be -type function if it satisfies the following conditions:(i)It is continuous and nondecreasing in and continuously differentiable in (ii), , and (iii)For any , ,

Definition 2. System (7) is said to be -stable in -th moment (-s.q.m), if there exists a function satisfying Definition 1 and positive constants and such that, for any ,which is equivalent towhen , it is said to be -stable in mean square.

3. Main Results

Theorem 1. Assume that , constants , , , , , , , , , , and , and a function satisfying Definition 1 such that(i) and .(ii) and satisfying where and those satisfying(iii) satisfying(iv).(v)For any , .

Then, the solution of system (7) is -stable in -th moment, where and .

Proof. For any , let be the solution of (7) and , where . Let be sufficiently small such that . By using Itô formula, we obtainLet , we can derive that, for ,whereLet , then we have for ,By (iii), we obtainLet such thatWe can claim that, for ,By (19), it is clear to see that for . We will prove thatOtherwise, there exists satisfying and , for . By the continuity of in , there exists satisfyingFor and , we obtainThus, we haveCombining (26) with (19) and (ii), for , we haveThen,This is a contradiction. Therefore, (23) holds. By (iii), we can deriveNow, we will prove thatSuppose that (30) does not hold, there exists such thatBy the continuity of in , there exists satisfyingFor and , we haveUsing (19) and (ii), for , we obtainThus, we obtainThis is a contradiction. Then, (30) holds. By induction, we can show that, for ,Hence, we can derive for ,By (i) and (37), we obtainwhich implies thatOn the other hand, for any , we haveThen,Letting , we havewhereThis implies that, for all ,Therefore,as desired.

Theorem 2. Assume that , constants , , , , , , , , , , and, and a function satisfying Definition 1 such that(i) and .(ii) and satisfying where and those satisfying(iii) satisfying(iv).(v)For any , .

Then, the solution of system (7) is -stable in -th moment, where and .

Proof. For any , let be the solution of (7) and , where . Since () is satisfied, we can choose sufficiently small such thatLetFor , we obtainBy (iii), we can deriveLet such thatWe will prove that, for ,Using (51), it is obvious to see thatNow, we will prove thatOtherwise, there exists satisfying and , for . By the continuity of in , there exists satisfyingFor and , we obtainThus,Combining (59) with (51) and (ii), for , we obtainThen,which is a contradiction. Hence, (56) holds. Now, we will prove thatIf it is not satisfied, we have . By the continuity of in , there exists satisfyingFor and , we obtainBy (51) and (ii), for and , we haveThen,This is a contradiction. Using (52), (56), and (62), it follows thatMoreover, we can show thatIndeed, there is satisfyingThere exists for . Thus, for and . Then, by (51) and (ii), for , we obtainwhich implies thatwhich is a contradiction. If there is satisfying . By the continuity of in , there exists satisfyingThus, for and ,Then, forwe haveThis is a contradiction. Hence, we can see that . If this is false, we have . To prove this claim, there are two cases to be presented: Case 1: for , . By (53), (56), and (69), we have for and , Thus, by (51) and (ii), for , Then, which implies a contradiction. Case 2: there is satisfying . Since and by the continuity of in , there is satisfyingFor and , we obtainBy (51) and (ii), for , we haveThen,which is a contradiction. By induction, we can show that, for ,Finally, we can derive that, for , .
Proceeding as the proof of Theorem 1, we have and whereTherefore,as desired.

4. Example

Consider the following NISFDEwMS:wherewhere is arbitrary and . We assume that and such that , is standard one-dimensional Brownian Motion, and is Borel measurable such that , and there exists such thatwhere and

Letthen

Then,

Thus, and .

By the same computation, we have