Research Article | Open Access
General Decay Stability for Stochastic Functional Differential Equations with Infinite Delay
So far there are not many results on the stability for stochastic functional differential equations with infinite delay. The main aim of this paper is to establish some new criteria on the stability with general decay rate for stochastic functional differential equations with infinite delay. To illustrate the applications of our theories clearly, this paper also examines a scalar infinite delay stochastic functional differential equations with polynomial coefficients.
Stability is one of the central problems for both deterministic and stochastic dynamic systems. Due to introduction of stochastic factors, stochastic stability mainly includes almost sure stability and the moment stability. In a series of papers (see [1–5]), Mao et al. examined the moment exponential stability and almost sure exponential stability for various stochastic systems.
In many cases we may find that the Lyapunov exponent equals zero, namely, the equation is not exponentially stable, but the solution does tend to zero asymptotically. By this phenomenon, Mao  considered polynomial stability of stochastic system, which shows that solution tends to zero polynomially. Then in , he extended these two classes of stability into the general decay stability.
In general, time delay and system uncertainty are commonly encountered and are often the source of instability (see ). Many studies focused on stochastic systems with delay. Especially, infinite delay systems have received the increasing attention in the recent years since they play important roles in many applied fields (cf. [7, 9–13]). Under the Lipschitz condition and the linear growth condition, Wei and Wang  built the existence-and-uniqueness theorem of global solutions to stochastic functional differential equations with infinite delay. There is also some other literature to consider stochastic functional differential equations with infinite delay and we here only mention [15–17].
However, to the best knowledge of the authors, there are not many results on the stability with general decay rate for stochastic functional equations with infinite delay. It is therefore interesting to consider the stability of infinite delay stochastic systems. The main aim of this paper is to establish some new criteria for th moment stability and almost surely asymptotic stability with general decay rate of the global solution to stochastic functional differential equations with infinite delay
where , and are Borel measurable functionals, and is an -dimensional Brownian motion. Without the linear growth condition, we will show that (1.1) has the following properties.(i)This equation almost surely admits a global solution on .(ii)There exists a pair of positive constants and such that this global solution has properties where is a general decay function defined in the next section, namely, this solution is th moment and almost surely asymptotically stable with general decay rate.
In the next section, we introduce some necessary notation and definitions. Section 3 gives the main result of this paper by establishing a new criteria for th moment stability and almost surely asymptotic stability with general decay rate for the global solution of (1.1). To make our results more applicable, Section 4 gives the further result. To illustrate the application of our result, Section 5 considers a scalar stochastic functional differential equation with infinite delay in detail.
Throughout this paper, unless otherwise specified, we use the following notation. Let be a complete probability space with the filtration satisfying the usual conditions, that is, it is right continuous and increasing while contains all -null sets. is an -dimensional Brownian motion defined on this probability space.
Let , , and . Let be the Euclidean norm of vector . If is a vector or matrix, its transpose is denoted by . For a matrix , its trace norm is denoted by . Denote by the family of all bounded continuous functions from to with the norm , which forms a Banach space. In this paper, always represents some positive constants whose precise value is not important. If is an -valued stochastic process on , for any , define . denotes the family of continuously twice differentiable -valued functions defined on For any define an operator by
If is a solution of (1.1), for any applying the Itô formula yields
Let us introduce the following -type function, which will be used as the decay function.
Definition 2.1. The function is said to be the -type function if it satisfies the following conditions:(i)it is continuous and nondecreasing in and differentiable in ,(ii) and ,(iii), where ,(iv) for any and , .
It is is easy to find that functions and for any are -type functions.
For any and , define
and . Denote by the family of all probability measures on . For any and define
We also impose the following standard assumption on coefficients and .
Assumption 2.2. Let and satisfy the Local Lipschitz condition. That is, for every integer , there is such that for all and those , with .
Let us present the continuous semimartingale convergence theory (cf. ).
Lemma 2.3. Let be a real-value local martingale with a.s. Let be a nonnegative -measurable random variable. If is a nonnegative continuous -adapted process and satisfies then and is almost surely bounded, namely, , a.s.
3. Main Results
In this section, we establish the stability result with general decay rate for (1.1). This result includes the global existence and uniqueness of the solution, the th moment stability, and almost surely asymptotic stability with general decay rate.
In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of this equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [18, 19]) or a given non-Lipschitz condition and the linear growth condition (cf. [20, 21]). These show that the linear growth condition plays an important role to suppress the potential explosion of solutions and guarantee existence of global solutions. References [16, 22] extended these two classes conditions to infinite delay cases. However, many well-known infinite delay systems such that the Lotka-Volterra (see ) do not satisfy the linear growth condition. It is therefore necessary to examine the global existence of the solution for (1.1).
It is well known for stochastic differential equations that the linear growth condition for global solutions may be replaced by the use of the Lyapunov functions [23, 24]. By this idea, this paper establishes the existence-and-uniqueness theorem for (1.1).
For , let and probability measures . Define as
where is defined by (2.5). Then the following theorem follows.
Theorem 3.1. Assume that there exist positive constants , , , , and probability measures , where , such that for any and , the function satisfies Under Assumption 2.2, there exists a constant such that for any , where , (1.1) almost surely admits a unique global solution on and this solution has the properties (1.2).
Proof. For sufficiently small , fix the initial data . We divide this proof into the two steps.
Step 1 (existence and uniqueness of the global solution). Under Assumption 2.2, (1.1) has a unique maximal local solution on (see ), where is the explosion time. If we can show , a.s., then is actually a global solution. Let be a positive integer such that . For each integer , define the stopping time Obviously, is increasing and as Thus, to prove a.s., it is sufficient to show that a.s., which is equivalent to the statement that for any , as .
For any , define . Applying the It formula to yields Note that by (2.5), for . By the Fubini theorem and a substitution technique, we have Noting that , we have , which implies that for all , Hence, there exists By (3.4) and (3.7), we have Choosing sufficiently small such that , by (3.8) we have , which implies that as .
Step 2 (Proof of (1.2)). Define By the Itô formula and (3.2), where is a continuous local martingale with . Similar to (3.7), there exists By (3.10), (3.12), noting that , By Lemma 2.3, we have which implies the required assertions.
4. Further Result
In Theorem 3.1, it is not convenient to check condition (3.2) since it is not related to coefficients and explicitly. To make our theory more applicable, let us impose the following assumption on coefficients and .
Assumption 4.1. There exist positive constants , , , , , and nonnegative constants , , , , , , , , , such that for any , , where , , , and .
We also need the following lemma.
Lemma 4.2. Let . Assume that , are nonnegative constants such that , and where then, there is such that for all ,
Proof. Noting that choose the constant such that
If we can show that for any , then the inequality
holds. Let This is equivalent to prove that
For all , there exists . By and , we have .
For all , there exists . To prove , we consider three cases of , respectively.
Case 1. . By , we have and . Then there exists .Case 2. . By , we have and . Noting , we obtain .Case 3. . Without the loss of generality, we assume that . Obviously, on the derivative function has a unique null point . We can compute that Since and , we know that By (4.8) and (4.9), we obtain that . Then we have that for any , . The proof is completed.
For the purpose of simplicity, we introduce the following notations:
Then the following theorem follows.
Theorem 4.3. Let Assumptions 2.2 and 4.1 hold. Assume that where is defined by Lemma 4.2 except that is replaced by . For any where there exists a positive constant such that for any initial data , (1.1) admits a unique global solution on and this solution has the properties (1.2).
Proof. Define for . Applying (2.1) gives By (4.1) and the Young inequality, Recall the following elementary inequality: for any and , , applying the Hölder inequality yields By (4.2) and (4.17), applying the Young inequality and the Hölder inequality, we have Substituting (4.16) and (4.18) into (4.15) yields where whose expression is similar to (3.1) and in which Let . Note that , , , , . By (4.13), we obtain that , and for all By (4.11) and (4.13), we have By (4.12) and (4.13), we obtain Choose sufficiently small such that By (4.23) and Lemma 4.2, there exists a constant such that By (4.19), (4.21), and (4.24), we therefore have which implies that condition (3.2) is satisfied. By (4.20), (4.25), and the fact that and , applying Theorem 3.1 yields that there exists , such that for any , the desired assertions hold. The proof is completed.
5. A Scalar Case
To illustrate the application of our result, this section considers a scalar stochastic functional differential equations
where for and , for and is an odd number, and . In this section, with and has similar explanation. Assume
in which , , , are nonnegative constants and . Define
Choose the -type function . Let . It is obvious that and
which shows that .
By (5.2) and the Young inequality, we have that
which shows that condition (4.1) holds with
By (5.3) and the Young inequality, we get that
which shows that condition (4.2) holds with
By we have which implies . It is easy to see that , , , , and are positive, and , , , are nonnegative, where
By the parameters in Theorem 4.3, we can compute
In Assumption 4.1, the parameter is positive, so it is required that
Thus, we have the following corollary from Theorem 4.3.
Corollary. Let conditions (5.2), (5.3), (5.13), and (5.14) be satisfied, where , , and are given in (5.12). For any where and are given in (5.15), there exist , for any (5.1) has a unique global solution and this solution has properties
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