Under Assumptions 2.2 and 4.1, we can see from the proofs of Theorems 3.1 and 4.1 that the positive lower-boundedness of the functions and play important roles, while condition (4) and the definition of are mainly used to ensure the positive lower-boundedness of the functions and . So we wonder whether there exist more general assumptions and new conditions to guarantee these results. This is equivalent to finding more general assumptions and new conditions such that system (1) can still have the previous results, namely there exists a unique global solution almost surely, and the solution is asymptotically stable and, moreover, exponentially stable.
To make our theory more applicable, we replace the polynomial growth condition (Assumption 2.2) by the following general assumption.
Assumption 5.1 There are two functions , and three probability measures on with (), as well as some constants , and , such that
(21)
while for all ,
(22)
Remark 6 Condition (21) is known as radial unboundedness in the literature [20].
Remark 7 Compared with [22], our condition (22), different from their condition (5.4), emphasizes that is dominated by a polynomial function of W. What is more, the probability measures can also be weakened to any right-continuous nondecreasing functions (see [17]).
In the same way as [22], we can prove the following lemma.
Lemma 5.1 If Assumptions 2.1, 5.1 and hold, then for any initial data , there is a unique global solution of system (1) on .
Now we examine the asymptotic stability and the exponential stability of system (1).
Theorem 5.1 If Assumptions 2.1, 5.1 and the following condition (23) hold,
(23)
except that (22) is replaced by
(24)
then for any initial data , there is a unique global solution of system (1) on , and has the following properties:
-
(i)
If is uniformly continuous about t on , then
(25)
-
(ii)
If is uniformly continuous about t on , then
(26)
where .
Proof The proof is similar to Theorem 3.1. From Lemma 5.1 and condition (23), there exists a unique global solution. For the sake of simplicity, write . Applying Itô’s formula to , we have
(27)
where , , .
Let . By the same technique as function in Theorem 3.1, there exists a constant satisfying . Hence, we have
By virtue of the fact that
for , , , respectively, and using Itô’s formula, we have
(28)
where is a local martingale with the initial value . Applying the nonnegative semi-martingale convergence theorem (see Lemma 2.5), we obtain that
(29)
Hence, from Lemma 2.1 and the uniform continuity of , we get the required result (25)
Next, we prove the result (26). From (28), we get that
(30)
This implies
(31)
Hence, from Lemma 2.1 and the uniform continuity of , we have
which is the required result (26). □
Theorem 5.2 If Assumptions 2.1, 5.1 and condition (23) hold, except that (22) is replaced by (24), then for any initial data , there is a unique global solution of system (1) on , and has the following properties:
where is a positive constant which does not depend on ζ.
Proof The proof is similar to Theorem 4.1. From condition (23), we obtain that there exists at least a sufficiently small positive constant ε satisfying , . So, by the continuity, define
From Lemma 5.1 and condition (23), there exists a unique global solution. For the sake of simplicity, write . Applying Itô’s formula to , , we have
(32)
where , , .
Let . Noting the definition of and by the same technique as the function in Theorem 3.1, there exists a constant satisfying . Therefore, we have
By virtue of the fact that
for , , , respectively, and using Itô’s formula, we have
(33)
where is a local martingale with the initial value . Applying the nonnegative semi-martingale convergence theorem (see Lemma 2.5), we obtain that
(34)
By the same technique as (16) in Theorem 3.1, letting , we claim that
Next, we prove the other result. From (33), we get that
(35)
where =