ABSTRACT

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.

1. 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. [2-4] and references therein), stochastic fuzzy differential equations (SFDEs), (see e.g. [5-6] and references therein) stochastic set differential equations (SSDEs) (see e.g. [7-10] and references therein), stochastic set differential equations with selector (see [11-13]). 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.

2. Preliminaries

We recall some notations and concepts presented in detail in recent series works of V. Lakshmikantham et al (see [14]). Let denote the collection of all nonempty compact convex subsets of. Given, the Hausdorff distance between A and B is defined by

(2.1)

and—the zero points set in. It is known that is a complete metric space and

is a complete and separable with respect to. We define the magnitude of a nonempty subset as,

The Hausdorff metric (2.1) satisfes the properties below:

1)

2)

3)

4)

for all and.

If and, then

Given a complete probability space with a filtration

satisfying the usual conditions. Let

be an—adapted one dimensional Wiener process defined on and, 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

(2.2)

where,

is measurable multifunction and Aumann integrably bounded.

is measurable multifunction and integrably bounded, is an -measurable multifunction.

Definition 2.1. (see [1]) Let a set-valued stochastic process satisfy:

1) for every ;

2)

is continuous mapping with respect to the metric;

3) for every:

where is an—measurable multifunction. Then is solution of (2.2).

Definition 2.2. Let set-valued stochastic processes, we have the following definitions:

1) For every,

2) For every,

Using the properties of the Hausdorff distance one can formulate the following results

Lemma 2.1.

1)       if

then

2) if

then

3) If and reconstants, then

Corollary 2.1. (see [7]) Let set-valued stochastic processes we have the following confirms:

1);

2);

3);

4).

Definition 2.3. A solution to Equation (2.2) is unique if for every:

where is any solution to Equation (2.2).

Assume that satisfy the following hypotheses:

(H1) For every set the mappings

: are nonanticipating multifunctions.

(H2) There exists a constant, such that

(H3) There exists a constant, such that

(H4) There exists a function, such that

where.

(H5) There exists a function, such that

where.

Corollary 2.2. (see [1], Theorem 7) Assume be an—measurable multifunction and F, G satisfy (H1)-(H3), then SSDE (2.2) has a unique solution and satisfies estimate

Corollary 2.3. Assume be an—measurable multifunction and F, G satisfy (H1), (H4)-(H5), then SSDE (2.2) has a unique solution and satisfies estimate

3. 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.

Definition 3.1. The trivial stochastic set solution of SSDE Equation (2.2) is said to be

(LS) Lyapunov stable, if for each and

there exist a, such that

implies.

(ALS) Asymptotical Lyapunov stable, if it is (LS) and

.

(ELS) Exponent Lyapunov stable, if there exist , such that:

Definition 3.2. The trivial stochastic set solution of SSDE Equation (2.2) is said to be:

(S1) Equi-stable, if for each, and there exists such that implies that,;

(S2) Uniformly stable, if in (S1) is independent of;

(S3) Quasi-equi-asymptotically stable, if for each, there exists and

such that implies, for all;

(S4) Quasi-uniformly-asymptotically stable, if and in (S3) are independent of;

(S5) Equi-asymptotically stable, if (S1) and (S3) hold simultaneously;

(S6) Uniformly asymptotically stable, if (S2) and (S4) hold simultaneously;

(S7) Exponent-asymptotically stable, if exist such that for all

Lemma 3.1. According to the Definitions 3.1 and Definition 3.2, we can say that 1) The stochastic set solution of SSDE E (2.2) is (S1) if and only if it is (LS) that means (S1) (LS).

2) (S6) (ALS).

3) (S7) (ELS).

4) (S6) or (ALS) (S6).

5) (S6) (S4).

Thus we have to prove (S1), (S6) and (S7).

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 function satisfies the following conditions:

1)where is Lipschitz constant, for all,

,;

2) The Dini derivative

where,;

If is any solution of SSDE Equation

(2.2) Such that, then we have

where is a maximal solution of ordinary differential equation (ODE)

(3.1)

Proof. Let be any solution of SSDE Equation (2.2) existing on. We define the function so that.

Now for small, by our assumption it follows that

by using the Lipschitz condition give (1). Thus

Since

and is any solution of SSDE Equation (2.1), we find that

We therefore have the scalar differential inequality which yields, as before, the estimate where is a maximal solution of ODE (3.1). This proof is complete.

Corollary 3.1. If the Lyapunov-like function satisfies conditions in Theorem 3.1 then we have the estimate:

Next, putting

Theorem 3.2. Assume that for SSDE Equation (2.2) exists the Lyapunov like function which satisfies the conditions of Theorem 3.1.

a) If there exist the positive functions are strictly increasing such that:

1)

and, then (S1) holds.

Futhermore, there exists such that

2) If, then (S3) holds.

3) If, then (S5) holds.

b) If there exist the positive functions are strictly increasing and such that:

1)

and, then (S2) holds.

Futhermore, there exists such that.

2) If, then (S4) holds.

3) If then (S6) holds.

Proof. Let and be given, choosing such that with this we have (S1).

If this is not true, there would exists a stochastic set solution of SSDE Equation (2.2) and such that

with. By using Corollary 3.1 and a/1, we have

and condition

as result, yield:

This contradiction proves that (S1) holds.

Next, we have to prove that: there exists a and number such that:

implies for

. Let and.

Choosing such that with this we have (S3). If this is not true, there would exists a stochastic set solution of SSDE Equation

(2.2) such that, andwhere, for

By using assumption (a/2) of this theorem shows that, and yields:

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:

By implies

and,.

Thus for all and 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

1)

2)

For all, we have

As a result,

and (S4) holds.

The affirmation for (S6) is proved analogous to the proof of the affirmations for (S2), (S4).

Corollary 3.2. Assume that for SSDE Equation (2.2) exists the Lyapunov like function which satisfies the conditions of Theorem 3.1, and exist the positive numbers such that

.

If, then (S7) holds.

Proof. The proof for (S7) is proved analogous to the proof of the affirmations for (S4).

4. Some Applications of Stochastic Set Differential Equations

For example, in a finance market we consider some stock price at time denoted by 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. 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 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 is certainly in this interval which admissible prices.

Example 4.1. (Stock prices) Let denote the price of a stock at time, where (i.e. interval-valued). We can model the evolution of and the relative change of price, evolves according to the SSDE under the form

(4.1)

for all, for certain constants, called the drift and 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 SSDE (4.1) satisfies the following interval-valued stochastic differential equation

(4.2)

for all. That is,

Since, and are the solutions of the following stochastic differential system

(4.3)

(4.4)

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

and

.

Its graphical representation can be seen in Figure 1.

From here it is easily verifiable stability criteria of solution to Equation (4.1).

5. 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

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

(4.5)

where

are continuous multifunctions, state set and is different controls, inclusion: admissible control, feedback control and contraction control. The problems of the existence and properties of solutions to SSCDEs Equation (4.5) is still open.

6. Acknowledgements

The authors gratefully acknowledge the referees for their careful reading and many valuable remarks which improved the presentation of the paper. The authors thank the partial financial support of AM editorial office.

REFERENCES