Stability Criteria of Solutions for Stochastic Set Differential Equations

The existence and uniqueness results on solutions of set stochastic differential equation were studied in [1]. In this paper, we present the stability criteria for solutions of stochastic set differential equation.


Introduction
Recently, the field of stochastic differential equations (SDEs) has been studying in a very abstract method.Instead of considering the behaviours of one solution of (SDEs), one studies its set-valued solution.Instead of studying a (SDEs), some study stochastic differential inclusion (SDIs) (see e.g.[2][3][4] and references therein), stochastic fuzzy differential equations (SFDEs), (see e.g.[5][6] and references therein) stochastic set differential equations (SSDEs) (see e.g.[7][8][9][10] and references therein), stochastic set differential equations with selector (see [11][12][13]).Latest, the existence and uniqueness of solutions to the stochastic set differential equations were studied in [1].We remark that the problems of properties of stochastic set solution are still open.
We organize this paper as follows: In Section 2, we recall some basic concepts and notations which are useful in next sections.In Section 3, we study some kinds of stability properties such as stable, asymptotically stable, exponentially stable by Lyapunov and some other stability criterion.In Section 4, we give the examples and further research of this paper.

Preliminaries
We recall some notations and concepts presented in detail in recent series works of V. Lakshmikantham et al (see [14]).Let The Hausdorff metric (2.1) satisfes the properties below: 1) ) Given a complete probability space with a filtration   be an -adapted one dimensional Wiet  ner process defined on and (Ω, , ) with     is one-dimensional "white noise", i.e., the time derivative of the Wiener process.In [1], authors considered the initial valued problem (IVP) for a set stochastic differential equation (SSDE) as follows is continuous mapping with respect to the metric ; , we have the following definitions: 1) For every ,  , , , ,  Ω, , ; , ocesses (see [7]) Let set-valued stochastic pr Ω, , ; we have the following confirms: where is any solution to Equation (2.2).
satisfy the following hypotheses: where .

Main Results
In this section, we study some kinds of stability properties such as stable, asymptotically stable, exponentially stable by Lyapunov and some other stability criteria such as equi, uniform and equi-asymptotical stabilities for SSDE.
Next, we present some results about (S1)-(S6) of solution with using the Lyapunov-like functions.
Theorem 3.1.Suppose that the positive Lyapunov-like satisfies the following conditions: 2) The Dini derivative where   Now for small , by our assumption it follows that 0 by using the Lipschitz condition give (1).Thus and t X is any solution of SSDE Equation (2.1), we find that We therefore have the scalar differential inequality which yields, as before, the estimate 0 0 where , , l t t k is a maximal solution of ODE (3.1).This proof is complete.
Assume that for SSDE Equation (2.2) exists the Lyapunov like function which satisfies the conditions of Theorem 3.1.
If there exist the positive functions are strictly increasing such that:     0; , : , , t g t V t X  0, then (S1) holds.Futhermore, there exists 1 0 b) If there exist the positive functions are strictly increasing and ( , ), ( ) Proof.Let 0    and be given, choosing If this is not true, there would exists a stochastic set solution t X of SSDE Equation (2.2) and such that with    .By using Corollary 3.1 and a/1, we have This contradiction proves that (S1) holds.Next, we have to prove that: and number such that:  with this we have (S3).If this is not true, there would exists a stochastic set solution t X of SSDE Equation (2.2) such that, , , This contradiction proves that (S3) holds.
The affirmation for (S5) is proved analogous to the proof of the affirmations for (S1), (S3).
Next, we have to prove that (S2) holds: 0   0 0, t   T the affirmation for (S1) holds, that means the affirmation for (S2) holds.
Next, we have to prove that (S4) holds.According to assumption b) of Theorem 3.2 The affirmation for (S6) is proved analogous to the proof of the affirmations for (S2), (S4 The proof for (S7) is proved analogous to the proof of the affirmations for (S4).

Some Applications of Stochastic Set Differential Equations
For example, in a finance market we consider some stock price at time denoted by t t X which is a random variable defined on the probability space .Owing to the quick fluctuation of the stock price from time to time or to the existence of missing data, we may not precisely know the price

 
, ,P     t X  .A possible model for this situation would be to give the upper and the lower prices (i.e. a margin for the error in the observation).Then we obtain an nterval  , which is a special kind of a set-valued random variable, ontains not only randomness but also impreciseness, and we assume   t X  is certainly in this interval.
For example different, in environmental of the insurance premium, the risks is considered a main material of this industry.Beside that, the risks are random factors and associating with premiums, so insurance premiums should be built on the basis of risks to price insurance which could compensate and balance the damage occurs to their business costs.Otherwise, the risks are some kinds different and levels of influence are different, so they could influence to levels of price of the insurance premium.
Hence, we may not precisely know the price of the insurance premium such that be beneficial to company of the insurance and customers.Then, in special the case we assume   t X  is certainly in this interval which admissible prices.
Its graphical representation can be seen in Figure 1.From here it is easily verifiable stability criteria of solution to Equation (4.1).

Further Research
In the future, we will concentrate all our efforts on other properties of this kind of equation discussed in our paper, such as on the existence of extremal solutions for SSDEs (2.2).Beside that, set-valued stochastic differential equations and their solutions seem to be a starting point for distance between A and B is defined by and separable with respect to H d .We define the magnitude of a nonempty subset A as,

t for all 0 Lemma 3 . 1 . 1 )
According to the Definitions 3.1 and Definition 3.2, we can say that t  The stochastic set of SSDE E

Example 4 . 1 .
(Stock prices) Let t X denote the price of a stock at time , where .e. interval-valued).We can model the evolution of t X and the relative change of price, evolves according to the SSDE under the form the volatility of the stock.Since coefficients in Equation (4.1) satisfy the conditions in Corollary 2.2, there is a unique solution of Equation (4.1).This means that for   0, t T  SSDE (4.1) satisfies the following interval-valued stochastic differential equation
and We can slove Equation (4.3) and Equation (4.4) by classic methods.Thus, the solutions of Equation (4.3) and Equation (4.4) respecttively are H. VU ET AL.Copyright © 2012 SciRes.AM 359 further development in the theory of control for SSDEs.Below we present the main idea, we consider the setvalued stochastic control differential equations (SSCDEs) under the form : admissible control, feedback control and contraction control.The problems of the existence and properties of solutions to SSCDEs Equation (4.5) is still open.