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. Themain 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.


Introduction
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 6 considered polynomial stability of stochastic system, which shows that solution tends to zero polynomially.Then in 7 , 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 8 .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 14 built the existenceand-uniqueness theorem of global solutions to stochastic functional differential equations 2 International Journal of Stochastic Analysis 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 pth moment stability and almost surely asymptotic stability with general decay rate of the global solution to stochastic functional differential equations with infinite delay dx t f t, x t , x t dt g t, x t , x t dw t , 1.1 are Borel measurable functionals, and w t is an r-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 0, ∞ .
ii There exists a pair of positive constants p and q such that this global solution has properties lim sup where ψ t is a general decay function defined in the next section, namely, this solution is pth 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 pth 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.

Preliminaries
Throughout this paper, unless otherwise specified, we use the following notation.Let Ω, F, {F t } t 0 , P be a complete probability space with the filtration {F t } t 0 satisfying the usual conditions, that is, it is right continuous and increasing while F 0 contains all P-null sets.w t is an r-dimensional Brownian motion defined on this probability space.
Let R 0, ∞ , R 0, ∞ , and R − −∞, 0 .Let |x| be the Euclidean norm of vector x ∈ R n .If A is a vector or matrix, its transpose is denoted by A T .For a matrix A, its trace norm is denoted by It is is easy to find that functions ψ t e γt and ψ t 1 t γ for any γ, γ > 0 are ψ-type functions.
For any p, q 0 and ϕ ∈ C b , define and C p, q {ϕ ∈ C b : T p,q ϕ < ∞}.Denote by M 0 the family of all probability measures on R − .For any μ ∈ M 0 and ε 0, define We also impose the following standard assumption on coefficients f and g.

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Assumption 2.2.Let f and g satisfy the Local Lipschitz condition.That is, for every integer n 1, there is k n > 0 such that for all t 0 and those x, Let us present the continuous semimartingale convergence theory cf.18 .
Lemma 2.3.Let M t be a real-value local martingale with M 0 0 a.s.Let ζ be a nonnegative F 0 -measurable random variable.If X t is a nonnegative continuous F t -adapted process and satisfies

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 pth 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 13 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 .

3.2
Under Assumption 2.2, there exists a constant q > 0 such that for any ξ ∈ C α, q , where α min 0 i k {α i }, 1.1 almost surely admits a unique global solution x t on 0, ∞ and this solution has the properties 1.2 .
Proof.For sufficiently small q ∈ 0, ε , fix the initial data ξ ∈ C α, q .We divide this proof into the two steps. Step Obviously, σ n is increasing and σ n → σ ∞ ρ e as n → ∞.Thus, to prove ρ e ∞ a.s., it is sufficient to show that σ ∞ ∞ a.s., which is equivalent to the statement that for any t > 0, P σ n t → 0 as n → ∞.
For any t 0, define t n t ∧ σ n .Applying the It o formula to ψ q t V x t yields

3.4
Note that by 2.5 , μ iε ≥ μ iq for q ≤ ε.By the Fubini theorem and a substitution technique, we have

3.5
Noting that ξ ∈ C α, q , we have ξ ∈ C α i , q , which implies that for all i 1, . . ., k, Hence, there exists By 3.4 and 3.7 , we have nP σ n t const E t n 0 ψ q s qφV x s − a|x s | p ds.

3.8
Choosing q sufficiently small such that qφ a, by 3.8 we have nP σ n t const, which implies that P σ n t → 0 as n → ∞.
Step 2 Proof of 1.2 .Define h t ψ q t V x t .

3.9
By the It ô formula and 3.2 ,

Further Result
In Theorem 3.1, it is not convenient to check condition 3.2 since it is not related to coefficients f and g explicitly.To make our theory more applicable, let us impose the following assumption on coefficients f and g.

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We also need the following lemma.

4.4
Proof.Noting that a > cρ, choose the constant a such that cρ < a < a.

4.5
If we can show that for any t ∈ 0, ∞ , F t : a bt α − k i 0 c i t α i 0, then the inequality holds.Let a a − a.This is equivalent to prove that at p bt α p − k i 0 c i t α i p at p .

4.7
For all t ∈ 1, ∞ , there exists F t a bt α − ct α .By a > cρ 0 and b > c, we have F t a bt α − ct α > 0. For all t ∈ 0, 1 , there exists F t F * t : a bt α −ct α 0 .To prove F t 0, we consider three cases of α 0 , respectively.Case 1. α 0 0. By α 0 0, we have F * t a bt α − c and a ∈ c, a .Then there exists F t F * t > 0.
where ρ is defined by Lemma 4.2 except that α 0 is replaced by α 0 ∧ 2β.For any where there exists a positive constant q such that for any initial data ξ ∈ C α 0 ∧ 2β 0 p, q , 1.1 admits a unique global solution x t on 0, ∞ and this solution has the properties 1.2 .

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Proof.Define V x |x| p for p > 2. Applying 2.1 gives

4.15
By 4.1 and the Young inequality,

4.16
Recall the following elementary inequality: for any λ j 0 and x j ∈ R, j 0, 1, . . ., n, applying the H ölder inequality yields 4.17 By 4.2 and 4.17 , applying the Young inequality and the H ölder inequality, we have

4.18
Substituting 4.16 and 4.18 into 4.15 yields where whose expression is similar to 3.1 and . By 4.13 , we obtain that c 0 > 0, c 0 > 0, and c j 0 > 0 for all 0 j l.By 4.11 and 4.13 , we have b 0 > c 0 .By 4.12 and 4.13 , we obtain a 0 > ρc 0 .Choose sufficiently small ε such that , applying Theorem 3.1 yields that there exists q > 0, such that for any ξ ∈ C α 0 ∧ 2β 0 p, q , the desired assertions hold.The proof is completed.

A Scalar Case
To illustrate the application of our result, this section considers a scalar stochastic functional differential equations dx t n r 1 x r t u r t x r t u r t

5.4
It is obvious that f t, x, ϕ and g t, x, ϕ satisfy the local Lipschtiz condition.By 5.4 , 5.1 can be rewritten as 1.1 .
Choose the Ψ-type function ψ t which shows that μ ∈ M ε .By 5.2 and the Young inequality, we have that

5 . 1 where
for r 1, 2, . . ., n and k 1, 2, . . ., m, u r t , v k t ∈ C R , for 0 r < r s n and 0 k < k l m, u rs t, θ , v kl t, θ ∈ C R × R − , n 3 is an odd number, m 2, and 2m n 1.In this section, 0≤r<r s≤n : n r 0 n s 0 with r s ≤ n and 0≤k<r l≤m has similar International Journal of Stochastic Analysis 13 explanation.Assume u 1 t −a 1 < 0, u n t −a n < 0, |u r t | a r , where 2 r n − 1,|u rs t, θ | a rs 2ε 1 − θ −1−2ε, where 0 r < r s n, 5.2 |v k t | b k , where 1 k m, |v kl t, θ | b kl 2ε 1 − θ −1−2ε , where 0 k < k l m, 5.3 in which a r , a rs , b k , b kl are nonnegative constants and ε > 0. Define f t, x, ϕ n r 1

By 2m n 1 ,In
we have 2 m−1 n−1, which implies 2β α.It is easy to see that σ, σ, λ, λ, and λ are positive, and σ i , σ i , λ j , λ j , are nonnegative, where 0 i n − 2, 0 j m − 2.By the parameters in Theorem 4.3, we can compute Assumption 4.1, the parameter σ is positive, so it is required that family of all bounded continuous functions ϕ from R − to R d with the norm ϕ sup −∞<θ 0 |ϕ θ |, Banach space.In this paper, const always represents some positive constants whose precise value is not important.If x t is an R d -valued stochastic process on R, for any t 0, define x t x t θ {x t θ :θ ∈ R − }.C 2 R d ,R denotes the family of continuously twice differentiable R-valued functions defined on R d .For any V x ∈ C 2 R d , R , define an operator LV : R × R d × C b → R by 1 existence and uniqueness of the global solution .Under Assumption 2.2, 1.1 has a unique maximal local solution x t on 0, ρ e see 21 , where ρ e is the explosion time.If we can show ρ e ∞, a.s., then x t is actually a global solution.Let n 0 be a positive integer such that sup θ 0 |ξ θ | < n 0 .For each integer n n 0 , define the stopping time 23 By 4.23 and Lemma 4.2, there exists a constant a ∈ 0, a such that a|x| p a|x| p b ε |x| α p − c ε |x| 2β p − 5.12 International Journal of Stochastic AnalysisTo apply Theorem 4.3, it is necessary to test that 4.11 -4.13 are satisfied.This requires that Thus, we have the following corollary from Theorem 4.3.Let conditions 5.2 , 5.3 , 5.13 , and 5.14 be satisfied, where W 1 , W 2 , and W 3 are given in 5.12 .For any p ∈ 2, p 1 ∧ p 2 , where p 1 and p 2 are given in 5.15 , there exist q > 0, for any ξ ∈ C p, q , 5.1 has a unique global solution x t x t, ξ , and this solution has properties