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Local stability and parameter dependence of mild solutions for stochastic differential equations
Advances in Difference Equations volume 2014, Article number: 276 (2014)
For nonlinear stochastic equations with parameter λ in a Hilbert space, we show the existence and uniqueness of mild solutions. Provided that f satisfies a locally Lipschitz condition and g is a uniformly Lipschitz function, some sufficient conditions for p () moment locally exponential stability as well as almost surely exponential stability of mild solutions are obtained under a sufficiently small initial value ξ. Meanwhile, we also consider parameter dependence of stable mild solutions for the stochastic system if f, g are sufficiently small Lipschitz perturbations in the parameter λ.
Stochastic differential equation has attracted a great attention from both theoretical and applied disciplines since it has been successfully applied to problems in mechanics, economics, physics and several fields in engineering. For details, see [1–5] and the references therein. In recent years, existence, uniqueness, stability, invariant measures and other quantitative and qualitative properties of solutions to stochastic partial differential equations have been extensively considered by many authors. For example, in 1978, Haussmann  studied asymptotic stability of Itô equations in infinite dimensions. Caraballo  extended the results from Haussmann to stochastic partial differential equations with delay. In , Mao considered exponential stability in the mean square sense for the strong solutions of linear stochastic differential equations. Caraballo and Real  considered the stability for the strong solutions of semilinear stochastic evolution equations based on the ideas in . Govindan [10, 11] studied the existence and stability of mild solutions for stochastic partial differential equations by the comparison theorem. Caraballo and Liu  discussed the exponential stability for mild solutions of stochastic partial differential equations with delays by employing the well-known Gronwall inequality and stochastic analysis technique under the Lipschitz condition. Liu and Mao  established Razuminkhin-type exponential stability for mild solution of stochastic partial functional differential equations. Da Prato et al.  and Dawson  developed this topic by the semigroup approach . In , Taniguchi et al. discussed the existence, uniqueness and asymptotic behavior of solutions to stochastic partial functional differential equations in a Hilbert space.
Recently, Burton  has successfully extended the fixed point theorem to study the stability for deterministic systems. By employing the contraction mapping principle and stochastic analysis, Luo  has obtained some sufficient conditions which ensure exponential stability in p () moment and almost sure exponential stability for mild solutions of stochastic partial differential equations with delays. Following the ideas of Luo, Sakthivel and Luo investigated the existence and asymptotic stability in the p th moment of mild solutions of nonlinear impulsive stochastic differential equations with infinite delay in .
Our main objective is to obtain local stability of mild solutions for stochastic parameter systems, with two novelties when compared with the former work in this area:
We consider th moment locally exponential stability as well as almost surely exponential stability of mild solutions under sufficiently small nonlinear perturbations and sufficiently small initial value.
We first discuss parameter dependence of stable mild solutions under sufficiently small Lipschitz perturbation in the parameter space Y.
The rest of this paper is organized as follows. In Section 2, stochastic differential equations with parameter are presented together with some definitions of mild solutions. In Section 3, existence, uniqueness, stability and parameter dependence of mild solutions are derived. An example is given to illustrate our results in Section 4. At last, we present some conclusions of our paper in Section 5.
Let H be a real separable Hilbert space with the inner product and the norm , and let K be another real separable Hilbert space with the inner product and the norm . denotes the space of bounded operators from K to H. Let be a complete probability space equipped with a complete family of right-continuous increasing sub σ-algebras satisfying .
Let , , be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over . Set
where () are nonnegative real numbers and () is a complete orthonormal basis in K. Let be an operator defined by with a finite trace . Then the above K-valued stochastic process is called a Q-Wiener process.
Let and define
If , then φ is called a Q-Hilbert-Schmidt operator, where denotes the space of all Q-Hilbert-Schmidt operators .
We denote by () the collection of all strongly-measurable, p-integrable H-valued variables with the norm , where E is defined by .
We consider the following stochastic differential equations with parameter in a Hilbert space:
where ξ is an H-valued random variable, A is the infinitesimal generator of a semigroup of bounded linear operators , , , are all Borel measurable, Y is a Banach space (which is the parameter space).
Now, we recall the mild solutions for Eq. (1) as follows.
Definition 2.1 A stochastic process , , is called a mild solution of Eq. (1) if
is adapted to , ;
has a càdlàg path on almost surely;
for arbitrary ,(2)
We assume that the following conditions hold:
A is the infinitesimal generator of a semigroup of bounded linear operators , , and there exist , such that for ;
there exist constants , such that for , , ,
there exists such that for , , ,
, where .
For example, if
for some functions , and there exist constants , , , such that for , ,
for each , , then conditions (ii) and (iii) hold for λ, μ in a small neighborhood of zero.
Set , . Then we have the following definitions of locally exponential behavior for mild solutions.
Definition 2.2 Let be an integer, mild solution of Eq. (1) is said to be p th moment locally exponentially stable if there exist , such that
for any initial value , and .
Definition 2.3 The mild solution of Eq. (1) is said to be almost surely exponentially stable if for any initial value and for any , there exists such that
Lemma 2.1 
For any and for arbitrary -valued predictable process ,
3 Stability and parameter dependence
In this section, we present and prove our main results. We consider the set .
Theorem 3.1 Assume that (i)-(v) hold. Then, for any sufficiently small and for any , given , there exists a unique stochastic process satisfying (2). Moreover, the following properties hold:
There exists such that(3)
is pth moment locally exponentially stable, that is, there exists such that(4)
for any and ;
There exists such that(5)
for any and .
Proof By (iv), it follows that there exists such that (3) holds. We consider the space χ of all ℱ-adapted processes such that
for any ;
the norm satisfies .
We can easily see that χ is a complete metric space. Define an operator by
for any . We first verify the continuity of in χ. Let , , and be sufficiently small, then
as . By Lemma 2.1, we have
as , where . Thus
as , which implies that is continuous in χ on . For each and , by the Hölder inequality,
where ∗ represents the transpose, and the Lyapunov inequality
and (ii), we have
By (iii), we obtain
Then, by (13), (14) and Lemma 2.1, we have
where . Taking δ sufficiently small so that , the operator becomes a contraction. In addition, we can obtain
This shows that . Thus, there exists a unique function such that . By the above inequality, we have , which means that (4) holds. Writing , we have
for any . Thus
It follows that
Using the Lyapunov inequality
from (ii), (iii) and (4), we get
where . Therefore
which yields (5). This completes the proof of the theorem. □
Remark 3.1 From the proof of Theorem 3.1, if we take δ sufficiently small so that , the mild solutions (1) with δ-domain initial value are p moment locally exponentially stable. In this case, the property of p moment stability is local.
Theorem 3.2 Assume that (i)-(v) hold. Then, for any sufficiently small, mild solution of (1) is almost surely exponentially stable, that is, there exists such that
for any and . Furthermore,
for any , and .
Proof For any fixed positive real number (),
By (i) and (4), we have
It follows from (ii) and (4) that
This implies that
From (iii) and (4), we have the following estimation:
Substituting (28), (30) and (32) into (27), we have
where . As is an arbitrarily given real number, let such that
Consequently, from the Borel-Cantelli lemma, there exists such that for any ,
It remains to verify that (26) holds. The argument is similar to the above proof of (27). For any (),
From (i) and (5), we obtain
By (4), (5) and (ii), applying inequalities (12) and (13), we get
This implies that
It follows from (4), (5) and (iii) that
Substituting (37), (39) and (41) into (36), we have
where . We can proceed to the remaining proof in a similar manner. □
In this section, an example is provided to illustrate our results.
Example 4.1 Consider the following stochastic partial equation:
where is a standard one-dimensional Brownian motion, is a parameter. Let with the domain
then , , which implies that . It is easy to see that (ii)-(iii) hold and , , . Taking , we may show that . Then the mild solution of Eq. (43) has locally stable behavior and parameter dependence for sufficiently small.
This paper is devoted to locally stable behavior and parameter dependence for stochastic differential equations. Our results are derived under sufficiently small Lipschitz perturbation in the parameter system. Meanwhile, a lot of stochastic inequality techniques are used to estimate the bound of mild solution. The future research topics would be extending locally stable behavior and parameter dependence to the stochastic delayed parameter system.
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The author would like to thank the referee and the associate editor for their very helpful suggestions.
The author declares that they have no competing interests.
The author contributed to the manuscript. The author read and approved the final manuscript.
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Pan, L. Local stability and parameter dependence of mild solutions for stochastic differential equations. Adv Differ Equ 2014, 276 (2014). https://doi.org/10.1186/1687-1847-2014-276
- mild solutions
- exponential stability
- parameter dependence