The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

and Applied Analysis 3 Let C2,1(Rn × [−τ, +∞); R + ) denote the family of all continuous nonnegative functions V(x, t) on Rn × [−τ, +∞), which are continuously twice differentiable in x and once differentiable in t. For each V ∈ C2,1(Rn × [−τ, +∞); R + ), denote an operatorLV from C([−τ, 0]; Rn) × R + to R by LV (φ, t) = V t (φ (0) , t) + V x (φ (0) , t) f (φ, t) + 1 2 trace [gT (φ, t) V xx (φ (0) , t) g (φ, t)] , (9) where V t (x, t) = ∂V(x, t)/∂t, V xx (x, t) = (∂V(x, t)/ ∂x i x j ) n×n , and V x (x, t) = (∂V(x, t)/∂x 1 , . . . , ∂V(x, t)/∂x n ). Then let us recall a number of lemmas. Lemma 5 (cf. [20]). If h(t) is a bounded function on [0,∞) and h(t) ∈ L1(R + ; R + ), then for any β ≥ 1, ∫+∞ 0 hβ(t)dt < ∞. Lemma 6 (cf. [11]). Assume a, b, q > 0, b ≥ q, α > β > 0. If the following condition holds, a b > (α − β) β/(α−β) βα −α/(α−β) , (10) then there exists a ∈ (0, a) satisfying a + bt α − qt β > a, (11) for all t ≥ 0. Lemma 7 (cf. [14]). Assume α, β > 0. For any h(t) ∈ C(Rn; R), if lim sup |t|→∞ (h(t)/|t|α) = 0, then there exists a constant H satisfying sup t∈R n {−β|t| α + h (t)} < H. (12) Lemma 8 (Kolmogorov-Chentsov theorem [21]). Suppose that a stochastic process X(t) on t ≥ 0 satisfies the condition E|X (t) − X (s)| α ≤ D|t − s| 1+β , 0 ≤ s, t < ∞ (13) for some positive constants α, β, and D. Then there exists a continuous modification X(t) of X(t), which has the property that, for every γ ∈ (0, β/α), there is a positive random variable δ(ω) such that P{ω : sup 0<t−s<δ(ω),0≤s,t<∞ 󵄨󵄨󵄨󵄨 X (t, ω) − X (s, ω) 󵄨󵄨󵄨󵄨 |t − s| γ ≤ 2 1 − 2 } = 1. (14) In other words, almost every sample path of X(t) is locally but uniformly Hölder-continuous with exponent γ. 3. Boundedness of SFDEs 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 [2, 4, 5]) or a given non-Lipschitz condition and the linear growth condition (see [22]). However, when the coefficients of the system (5) satisfy the local Lipschitz condition and the polynomial growth condition, the solution of the system (5) may explode at a finite time. So it is necessary to examine the existence and uniqueness of the global solution of the system (5). Here we state the following existence-and-uniqueness result. Lemma 9. If Assumptions 1 and 3 and κ > κ hold, then for any initial data ζ ∈ Cb F0 0]; R ), there is a unique global solution x(t, ζ) of system (5) on t ≥ −τ. Remark 10. This result is the special case of Theorem 3.2 of [15]. Since it is not so easy to see this fact directly, we give the proof in the Appendix. The fact that we write down our Lemma 9 here is to keep our paper completely based on Assumptions 1 and 3. We now show the following asymptotic boundedness of the global solution in the sense of the pth moment and the trajectory with large probability. Theorem 11. If Assumptions 1 and 3 and κ > κ hold, then for any initial data ζ ∈ C and any p ≥ 0, the global solution x(t, ζ) of system (5) is bounded in the sense of pth moment; that is, there exists a constantM p > 0 such that sup −τ≤t<+∞ E|x (t, ζ)|p ≤ M p . (15) Proof. Since κ > κ, the existence and uniqueness of the solution follow from Lemma 9. And there exists at least a sufficiently small positive constant ε satisfying κ > κe. So by the continuity, define ε󸀠󸀠 = sup{ε > 0 : κ > κe}. For the sake of simplicity, write x(t) = x(t, ζ), x t = x t . For any p ≥ 2, applying Itô’s formula to V(x, t) = e|x(t)|, ε ∈ (0, ε], we yield LV (x, t) = e εt (L|x (t)|p + ε|x (t)|p) ≤ e εt [ p 2 |x (t)|p−2 × (2xT (r) f (x r , r) + (p − 1) 󵄨󵄨󵄨g (xr, r) 󵄨󵄨󵄨 2 )


Introduction
Stochastic modeling plays an important role in many branches of sciences and industries.Since Itô introduced his stochastic calculus, stochastic delay or functional differential equations (SDDEs or SFDEs) have been used successfully to model those systems which depend not only on the present history of the state but also on the past ones (see, e.g., [1][2][3][4][5]).Stability and boundedness are two of the most important topics in the study of SDDEs or SFDEs in modern control theory.Many researchers have done a lot of works for these two topics (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18]).
In general, a SFDE has the form dx () = f (x  , ) d + g (x  , ) d () on  ≥ 0 with initial data  ∈   F 0 ([−, 0];   ), where f : ([−, 0];   ) ×  + →   and g : ([−, 0];   ) ×  + →  × (the notations used here will be illustrated in Section 2).Most of the existing stability criteria of SFDEs require the coefficients of corresponding systems to satisfy the local Lipschitz condition and the linear growth condition or the one-side linear growth condition (see, e.g., [2,4,5]).However, many SDDEs or SFDEs can not be dominated by the linear growth condition, such as stochastic population system, Lotka-Volterra systems, and system (2) as follows: dx () = (−4x 3 () − 3x () + 2 2  1 (x  )) d with initial data  ∈   So it is necessary to consider the cases of the nonlinear growth condition.Recently, Liu et al. [10] study the asymptotic stability of nonlinear stochastic differential equations (SDEs) with polynomial growth condition, and they also develop their results to the case of SDDEs [11].In this paper, we mainly establish some new results on the almost sure asymptotic stability and the boundedness in the sense of the pth moment and the trajectory with large probability of SFDEs with polynomial growth condition, which imply the results in [10,11].
Here we would like to mention the work of Luo et al. [12].It proposes a generalized theory for the asymptotic stability  ( () ,  + ) d μ () −  ( (0) , ) +  ∫ 0 −  ( () ,  + ) d () , (3) where μ and  are probability measures on [−, 0],  1 ≥ 0,  2 >  3 ≥ 0,  ∈ (0, 1).However as to system (2) Since the above L(, ) includes the positive term |(0)| 10/3 , it does not satisfy (3).So their work does not imply ours.Also Shen et al. [13] use the LaSall technique to study the almost sure asymptotical stability of SFDEs under different settings.The organization of this paper is as follows: Section 2 describes some necessary notations and lemmas; the existence of the global solution and the bounedness of SFDEs are stated in Section 3; sufficient conditions are proposed for the almost sure asymptotic stability in Section 4; to show the applications of our results, some illustrative examples are given in the final section.
To get our main results, we firstly put forward the following hypothesis.
However, to ensure the unique maximal local solution is in fact the global solution, we need to impose the following additional polynomial growth condition.

Boundedness of SFDEs
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 [2,4,5]) or a given non-Lipschitz condition and the linear growth condition (see [22]).However, when the coefficients of the system (5) satisfy the local Lipschitz condition and the polynomial growth condition, the solution of the system ( 5) may explode at a finite time.So it is necessary to examine the existence and uniqueness of the global solution of the system (5).Here we state the following existence-and-uniqueness result.
Remark 10.This result is the special case of Theorem 3.2 of [15].Since it is not so easy to see this fact directly, we give the proof in the Appendix.The fact that we write down our Lemma 9 here is to keep our paper completely based on Assumptions 1 and 3.
We now show the following asymptotic boundedness of the global solution in the sense of the pth moment and the trajectory with large probability.Theorem 11.If Assumptions 1 and 3 and  >  hold, then for any initial data  ∈  and any  ≥ 0, the global solution x(, ) of system (5) is bounded in the sense of th moment; that is, there exists a constant   > 0 such that Proof.Since  > , the existence and uniqueness of the solution follow from Lemma 9.And there exists at least a sufficiently small positive constant  satisfying  >   .So by the continuity, define   = sup{ > 0 :  >   }.For the sake of simplicity, write x() = x(, ), x  = x   .For any  ≥ 2, applying Itô's formula to (x, ) =   |x()|  ,  ∈ (0,   ], we yield where Noting that  >   and |x()| ≥ 0 for any  ≥ 0, by Lemma 7 and the same technique as And in view of the fact that for By virtue of the boundedness of  1 (),  2 (), there is a constant Ψ > 0 such that  1 () ∨  2 () ≤ Ψ, which implies that where  = (( From the boundedness of initial data  ∈ , we claim that, for any  ≥ 2, there exists a constant   > 0 such that sup −≤<+∞ |x(, )|  ≤   .When  ∈ (0, 2), using the Lyapunov inequality, we claim that sup From Theorem 11 and the Chebyshev inequality, we get the following proposition about the asymptotic boundedness of the global solution in the sense of the trajectory with large probability.
Further we continue to discuss the asymptotic boundedness of the norm of x  in system (5) in the sense of the th moment and the trajectory with large probability.Theorem 13.If Assumptions 1 and 3 and  >  hold, then for any initial data  ∈ , the norm of x  in system (5) is bounded in the sense of th moment; that is, there exists a constant   > 0 such that the global solution x(, ) of system (5) has the property Proof.For the sake of simplicity, write x() = x(, ).From Theorem 11, set sup −≤<∞ |x()|  ≤   .For  ≥ 2,  ≥ 0, and  ∈ [0, ], using Itô's formula, we compute that where where Let by the same technique as (A.3) in the Appendix, we get that there exists a positive constant  such that (|x()|) ≤ .By virtue of the boundedness of  1 ,  2 , assuming  1 () ∨  2 () ≤ Ψ, we have Abstract and Applied Analysis By virtue of the boundedness of sup −≤<∞ |x()|  , we have In the same way as Proposition 12, we get the following proposition.
Proposition 15.If Assumptions 1 and 3 and  >  hold, then for any initial data  ∈ , the norm of x  in system (5) is stochastically ultimately bounded; namely, for any   ∈ (0, 1), there exists a constant   =   (  ) > 0 such that the global solution x(, ) of system (5) has the property

Almost Sure Asymptotic Stability of SFDEs
In this section, we aim to study the almost sure asymptotic stability of system (5).The following theorem establishes new criteria on the almost sure asymptotic stability.
Theorem 16.If Assumptions 1 and 3 and the following condition (35) hold, then for any initial data  ∈ , there is a unique global solution x(, ) of system (5) on  ≥ −, and x(, ) is almost surely asymptotically stable; that is, where  = ( Proof.The existence and uniqueness of the global solution follow from Lemma 9 directly.For the sake of simplicity, write x() = x(, ).Applying Itô's formula to (x, ) = |x()| 2 , we yield where and we have used the elemental inequality: for any ,  ∈ , 0 <  < 1, By using Lemma 6, we get that there exists a constant  > 0 satisfying inf ≥0 (|x()|) > .Then choose  2 which is sufficiently close to 1 such that We therefore have In view of the fact that

and
where () = ∫  0 2x  ()g(x  , )d(), which is a local martingale with the initial value (0) = 0. From Lemma 5, we get Applying the nonnegative semimartingale convergence theorem (see [23]), we obtain that lim sup To obtain our main result, we need to claim that almost every sample path of |x( Remark 18. From the proof above, Assumptions 1 and 3 are enough to guarantee the asymptotic stability of system (5).And the coefficients of system (2) do not satisfy the conditions which are similar to Assumptions 2 of [10] or Assumptions 3 of [11].So compared with [10,11], the three conditions of guaranteeing the asymptotic stability are weakened to the two conditions by this paper.

Example
In this section, we will discuss some examples to illustrate our results.
Example 1.Let us return to the SFDE (2).We can compute that Abstract and Applied Analysis 9 So we obtain that where   is some positive constant.
− |x()| −2 |g(x  , )| 2 d can be obtained from the boundedness of  1 and sup −≤<∞ |x()|  above and  2 is a constant which is not necessary to know exactly.Substituting it into (30), we yield So we claim that there exists a   > 0 such that sup 0≤<∞ ‖x  ‖  <   for any  ≥ 2.Remark 14.Clearly, the key of the proof is the upperboundedness of function (|x()|), which depends on the condition  > .And the theorem will play an important role to ensure the almost sure asymptotic stability of the solution.