Stability in credibility for fuzzy differential equation

1 Introduction

As a concept of describing a set with uncertain boundary, fuzzy set was initiated by Zadeh [22]. Then Zadeh [23] proposed the concept of possibility measure in 1978. Although possibility measure provides the theoretical basis for the measurement of fuzzy events, it doesn’t satisfy self duality. In order to define a self-dual measure, Liu and Liu [5] presented the concept of credibility measure in 2002, and a sufficient and necessary condition for credibility measure was given by Li and Liu [6] in 2006.

Credibility theory was founded by Liu [7] and was refined by Liu [8] in 2007, it is a branch of mathematics for studying the behavior of fuzzy phenomena. Based on credibility theory, a function from credibility space to the set of real numbers which called fuzzy variable was defined by Liu [7]. Later, in order to describe dynamic fuzzy phenomena, fuzzy process was proposed by Liu [8]. Especially, Liu process was first designed by Liu [9], which is a fuzzy process with stationary and independent normal fuzzy increments. Meanwhile, Liu integral and Liu formula were introduced by Liu [9]. Building on those foundations, many research works were made. For example, Liu [9] investigated geometric Liu process and proposed an alternative assumption that stock price follows geometric Liu process, then Liu’s stock model was constructed. A new stock model incorporating the mean reversion was initiated by Gao and Gao [4]. Qin and Li [14] gave the European options pricing formula for Liu’s stock model. The continuity of Liu process was proved by Dai [2]. Qin and Wen [12] considered some properties of complex Liu process. Dai [3] gave a reflection principle related to Liu process. You, Huo and Wang [17] extended Liu formula and Liu integral to the case of multi-dimensional and the existence and properties of Liu integral were also studied by You and Wang [18]. You, Ma and Huo [19] presented a definition of generalized Liu integral, and the properties of this kind of generalized fuzzy integral were proved.

Fuzzy differential equation driven by Liu process was first proposed by Liu [9] in 2008. Later, a great deal of results about fuzzy differential equation driven by Liu process appeared in theory. Zhu [25] extended the concept of fuzzy differential equation driven by Liu process to the case of multi-dimensional fuzzy differential equation driven by Liu process. You, Wang and Huo [20] studied the existence and uniqueness theorem for homogeneous fuzzy differential equation driven by Liu process. In addition, You, Wang and Huo [20] also studied the analytic solution for some special types of fuzzy differential equation driven by Liu process. Based on You’s existence and uniqueness theorem, a new existence and uniqueness theorem was proved by Chen and Qin [1]. Considering the complexity of solving fuzzy differential equation driven by Liu process, You and Hao [21] presented the fuzzy Euler approximation and designed corresponding numerical method. Except for the theory of fuzzy differential equation driven by Liu process, applications have also been widely studied. Zhu [24] applied fuzzy differential equation to fuzzy optimal control problems, furthermore, fuzzy optimal control was applied to fuzzy portfolio problem by Zhu [26]. A fuzzy control system with application to production planning problem was introduced by Qin, Bai and Dan [13].

As a branch of differential equation theory, stability plays an important role in both theory and application. Thus, the study of stability is a major field for scholars and research works concerning stability for fuzzy differential equation were made in the past two decades. For example, sufficient conditions for stability and asymptotic stability of solutions of fuzzy differential equations were obtained by Le [10]. Mizukoshi [11] discussed the fuzzy initial problem with parameters and initial conditions given by fuzzy sets, the concept of fuzzy equilibrium stability was also introduced. Wang, Qiu, Gao and Wang [16] studied a network-based fuzzy control for nonlinear industrial processes by applying fuzzy differential equation and distributed fuzzy H_∞ filtering problems was researched by Wang, Qiu and Fu [15] via fuzzy modelling technique. Different from these fuzzy differential equation based on fuzzy set theory, this paper will discuss fuzzy differential equations driven by Liu process. Until now, some concepts of stability for fuzzy differential equations driven by Liu process has been studied, for example, by finding the equilibrium point X e of n-dimensional fuzzy differential equation that satisfies f  (t, X _e ) =0 and g  (t, X _e ) =0 for all t, Zhu [25] called it Lyapunov stability in credibility when Cr {∥  X _t  -  X _e  ∥ < ɛ} =1 with any crisp initial vector X ₀ is close enough to the equilibrium point X _e , where X _e is an equilibrium point of n-dimensional fuzzy differential equation driven by Liu process and X _t is a solution of n-dimensional fuzzy differential equation driven by Liu process. Zhu [25] also obtained some sufficient and necessary conditions of stability for fuzzy linear differential equations driven by Liu process. Different from the concepts of stability for fuzzy differential equation driven by Liu process in Zhu [25], we will call it stability in credibility when Cr {∥  X _t  -  Y _t  ∥ < ɛ} =1 with any initial vector X ₀ is close enough to another initial vector Y ₀ , where X ₀ and Y ₀ are two different initial vectors, X _t and Y _t are two different solution of fuzzy differential equation driven by Liu process corresponding to different initial vectors X ₀ and Y ₀ .

In order to deal with a fuzzy dynamic system with fuzzy disturbance reflected by a fuzzy process, based on credibility theory, this paper aims to discuss the stability in credibility for fuzzy differential equation driven by Liu process. The rest of the paper was organized as follows. In Section 2, some basic concepts in credibility theory and fuzzy process were reviewed. In Section 3, the concept of stability in credibility for fuzzy differential equation driven by Liu process was proposed. Finally, some theorems for fuzzy differential equation driven by Liu process being stability in credibility were proved in Section 4.

2 Preliminaries

In this section, we will introduce some useful definitions and theorems about credibility theory and fuzzy differential equation driven by Liu process.

Definition 2.1. (Liu [8]) Let Θ be a nonempty set, and P is the power of Θ. Each element in P is called an event. Then credibility measure Cr is introduced as a set function if it satisfies the following condition:

(Normality) Cr {Θ} =1 ;

Let T be an index set and let ( Θ , P , Cr ) be a credibility space. As a function from T × ( Θ , P , Cr ) to the set of real numbers, the fuzzy process X (t, θ) is a fuzzy variable for each t = t^*. X (t, θ^*) is called a sample path of fuzzy process if θ^* is fixed. Instead of longer notation X (t, θ), in the following sections we will use the symbol X _t . And we use ∥ x ∥ to denote the Euclidean norm of a vector x in the sequel.

Definition 2.2. (Liu [9]) A fuzzy process C _t is said to be a Liu process if

C₀ = 0,

Theorem 2.1. (Dai [2]) Let C _t be a Liu process. For any given θ ∈ Θ with Cr {θ} >0, the path C _t  (θ) is Lipschitz continuous, i.e. there exists a Lipschitz constant K(θ) satisfying | C t - C s | ≤ K ( θ ) | t - s | . Theorem 2.2. (Liu Formula, Liu [9]) Let C _t be a standard Liu process, and let h (t, c) be a continuously differentiable function. Define X _t  = h (t, C _t ). Then we have the following chain rule d X t = ∂ h ∂ t ( t , C t ) d t + ∂ h ∂ c ( t , C t ) d C t .

Definition 2.3. (Liu Integral, Liu [9]) Let X _t be a fuzzy process and let C _t be a standard Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as Δ = max 1 ≤ i ≤ k ∣ t i + 1 - t i ∣ .

Then the fuzzy integral of fuzzy process X _t with respect to C _t is ∫ a b X t d C t = lim Δ → 0 ∑ i = 1 k X t i ( C t i + 1 - C t i ) , provided that the limit exists almost surely and is a fuzzy variable.

Definition 2.4. (Fuzzy Differential Equation Driven by Liu Process, Liu [9]) Suppose C _t is a standard Liu process, and f, g are some given functions. Then

d X t = f ( t , X t ) d t + g ( t , X t ) d C t(1) is called a fuzzy differential equation driven by Liu process. A solution is a fuzzy process X _t that satisfies above equality identically in t.

Definition 2.5. (n-dimensional Fuzzy Differential Equation Driven by Liu Process, Zhu [25]) Let C _t be a standard n-dimensional Liu process. Then

d X t = f ( t , X t ) d t + g ( t , X t ) d C t , X t 0 = x 0(2) is called an n-dimensional fuzzy differential equation driven by Liu process, where X _t is n-dimensional state vector, x ₀ is the crisp n-dimensional initial state vector, f  (t, X _t ) is some given vector function of time t and state X _t , and g  (t, X _t ) is some given matrix-valued function of time t and state X _t in C^n×n.

Theorem 2.3. (Existence and Uniqueness Theorem, Chen and Qin [1]) The fuzzy differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t has a unique solution if the coefficients f (t, x) and g (t, x) satisfy the linear growth condition | f ( t , x ) | + | g ( t , x ) | ≤ L ( 1 + | x | ) , ∀ x ∈ R , 0 ≤ t ≤ T and the Lipschitz condition | f ( t , x ) − f ( t , y ) | + | g ( t , x ) + g ( t , y ) | ≤ L | x − y | , ∀ x , y ∈ R , 0 ≤ t ≤ T , for some constants L, T.

In this section, we will present a concept of stability in credibility for fuzzy differential equation driven by Liu process.

Definition 3.1. The fuzzy differential equation (1) is said to be stability in credibility if for any two solutions X _t and Y _t corresponding to different initial values X₀ and Y₀, we have 0 Cr { | X t - Y t | < ɛ } = 1 , for all t ≥ 0 , where ɛ is any given number and ɛ > 0.Example 3.1. In order to understand the concept of stability in credibility preferably, let us consider the fuzzy differential equation d X t = a 2 d t + b d C t . where a and b are two constants. Its two solutions corresponding to different initial values X₀ and Y₀ are X t = X 0 + a 2 t + bC t , and Y t = Y 0 + a 2 t + bC t , respectively. Then we have | X t - Y t | = | X 0 - Y 0 | . As a result, for any given ɛ > 0, we always have lim | X 0 − Y 0 | → 0 Cr { | X t − Y t | < ε } = lim | X 0 − Y 0 | → 0 Cr { | X 0 − Y 0 | < ε } = 1 , ∀ t ≥ 0. Hence the fuzzy differential equation is stability in credibility. Definition 3.2. The n-dimensional fuzzy differential equation (2) is said to be stability in credibility, if for any two solutions X _t and Y _t corresponding to different initial values X ₀ and Y ₀ , we have lim ‖ X 0 − Y 0 ‖ → 0 Cr { ‖ X t − Y t ‖ < ε } = 1 , f o r a l l t ≥ 0 , Example 3.2. Consider an m-dimensional fuzzy differential equation d X t = a d t + b 2 V t d C t , where V _t is an m × n integrable fuzzy matrix process, a is a vector and b is a constant, C _t is an m-dimensional Liu process. Its two solutions corresponding to different initial values italicX ₀ and Y ₀ are X t = X 0 + ∫ 0 t a ds + ∫ 0 t b 2 V s d C s and Y t = Y 0 + ∫ 0 t a ds + ∫ 0 t b 2 V s d C s , respectively. Then we have ∥ X t - Y t ∥ = ∥ X 0 - Y 0 ∥ . As a result, we always have lim ‖ X 0 − Y 0 ‖ → 0 Cr { ‖ X t − Y t ‖ < ε } = lim ‖ X 0 − Y 0 ‖ → 0 Cr { ‖ X 0 − Y 0 ‖ < ε } = 1 , ∀ t ≥ 0. Thus the m-dimensional fuzzy differential equation is stability in credibility. Note that some fuzzy differential equations driven by Liu process are not stability in credibility, the next example will show it. Example 3.3. Consider fuzzy differential equation d X t = aX t d t + d C t , where a is a constant. Its two solutions corresponding to different initial values X₀ and Y₀ are X t = exp ( at ) X 0 + exp ( at ) ∫ 0 t exp ( - as ) ds and Y t = exp ( at ) Y 0 + exp ( at ) ∫ 0 t exp ( - as ) ds , respectively. Then we have | X t - Y t | = | exp ( at ) ( X 0 - Y 0 ) | . ∥As a result, for any given number ɛ > 0, we have lim | X 0 - Y 0 | → 0 Cr { | X t - Y t | < ɛ } = lim | X 0 - Y 0 | → 0 Cr { exp ( at ) | X 0 - Y 0 | < ɛ } ≠ 1 , ∀ t ≥ 0 . Hence the fuzzy differential equation is not stability in credibility.

4 Theorems of stability in credibility

In this section, we will give some sufficient conditions for fuzzy differential equation driven by Liu process being stability in credibility.

Theorem 4.1. Suppose that the fuzzy differential equation (1) has a unique solution for each initial value. Then it is stability in credibility, if coefficients f (t, x) and g (t, x) satisfy the strong Lipschitz condition | f ( t , x ) - f ( t , y ) | + | g ( t , x ) + g ( t , y ) | ≤ L ( t ) | x - y | , ∀ x , y ∈ R , t ≥ 0 for some integrable function L (t) on [0, + ∞).

Proof. Let X _t and Y _t be two solutions corresponding to different initial values X₀ and Y₀, respectively. Then for each θ ∈ Θ, we have d | X t - Y t | = | f ( t , X t ) d t - f ( t , Y t ) d t + g ( t , X t ) d C t - g ( t , Y t ) d C t | = | ( f ( t , X t ) - f ( t , Y t ) ) d t + ( g ( t , X t ) - g ( t , Y t ) ) d C t | ≤ | ( f ( t , X t ) - f ( t , Y t ) ) | d t + | ( g ( t , X t ) - g ( t , Y t ) ) | | d C t | ≤ L ( t ) | X t - Y t | d t + L ( t ) | X t - Y t | | d C t | ≤ L ( t ) | X t - Y t | d t + L ( t ) | K ( θ ) | | X t - Y t | d t = L ( t ) ( 1 + | K ( θ ) | ) | X t - Y t | d t ,(3) where K (θ) is Lipschitz constant of Liu process. Taking integral on both sides of (3), we have | X t - Y t | ≤ | X 0 - Y 0 | exp ( 1 + | K ( θ ) | ∫ 0 t L ( s ) d s ) .

For any given ɛ > 0, we always have Cr { | X t - Y t | < ɛ } ≥ Cr { | X 0 - Y 0 | exp ( 1 + | K ( θ ) | ∫ 0 t L ( s ) d s ) < ɛ } .

Since Cr { | X 0 - Y 0 | exp ( 1 + | K ( θ ) | ∫ 0 t L ( s ) d s ) < ɛ } → 1 as |X₀ - Y₀|→0, we obtain lim | X 0 - Y 0 | → 0 Cr { | X t - Y t | < ɛ } = 1 .

Hence the fuzzy differential equation is stability in credibility.

If it is not easy to determine whether or not f (t, x) and g (t, x) satisfy strong Lipschitz condition, we can judge whether the fuzzy differential equation is stability in credibility according to the following corollary.

Corollary 4.1. Let f (t, x) and g (t, x) be bounded real value functions on [0, + ∞). If f (t, x) and g (t, x) have derivatives with respect to x, and satisfy | f x ′ ( t , x ) | + | g x ′ ( t , x ) | ≤ L ( t ) , ∀ t ≥ 0 for some integrable function L (t) on [0, + ∞), then the fuzzy differential equation (1) is stability in credibility.

Proof. For the bounded real value functions f (t, x) and g (t, x) we have | f ( t , x ) | + | g ( t , x ) | < K ( 1 + | x | ) , where K is a constant which satisfy |f (t, x) | + |g (t, x) | < K. From mean value theorem, we deduce that | f ( t , x ′ ) - f ( t , x ″ ) | + | g ( t , x ′ ) - g ( t , x ″ ) | = f x ′ ( t , ξ ) | x ′ - x ″ | + g x ′ ( t , η ) | x ′ - x ″ | ≤ L ( t ) | x ′ - x ″ | + L ( t ) | x ′ - x ″ | = 2 L ( t ) | x ′ - x ″ | , where ξ, η ∈ (x′, x″). Because L (t) is bounded on [0, + ∞), the functions f (t, x) and g (t, x) satisfy the Lipschitz condition. It follows from the existence and uniqueness theorem that the fuzzy differential equation has a unique solution. From Theorem 4.1, we can obtain that the fuzzy differential equation is stability in credibility.

Different from Theorem 4.1 and Corollary 4.1, we have the following corollary when fuzzy differential equation is general linear fuzzy differential equation driven by Liu process.

Corollary 4.2. Suppose u_1t, u_2t, v_1t, v_2t are bounded functions with respect to t on [0, + ∞). If u_1t and v_1t are integrable on [0, + ∞), then the linear fuzzy differential equation driven by Liu process d X t = ( u 1 t X t + u 2 t ) d t + ( v 1 t X t + v 2 t ) d C t(4) is stability in credibility.

Proof. For the linear fuzzy differential equation (4), we have f (t, x) = u_1tx + u_2t and g (t, x) = v_1tx + v_2t. Since | u 1 t X t + u 2 t | + | v 1 t X t + v 2 t | ≤ | u 1 t | | X t | + | u 2 t | + | v 1 t | | X t | + | v 2 t | < K | X t | + K + K | X t | + K = 2 K ( | X t | + 1 ) and | ( u 1 t X t + u 2 t ) - ( u 1 t Y t + u 2 t ) | + | ( v 1 t X t + v 2 t ) - ( v 1 t Y t + v 2 t ) | = | u 1 t ( X t - Y t ) | + | v 1 t ( X t - Y t ) | ≤ | u 1 t | | X t - Y t | + | v 1 t | | X t - Y t | = ( | u 1 t | + | v 1 t | ) | X t - Y t | ≤ 2 K ( X t - Y t ) , where K is a constant which make u_1t < K, u_2t < K, v_1t < K, v_2t < K hold. It follows from the existence and uniqueness theorem that fuzzy differential equation (4) has a unique solution. Since L (t) = |u_1t| + |v_1t| is integrable function on [0, + ∞), from Theorem 4.1, we can obtain that the fuzzy differential equation is stability in credibility.

According to Definition 3.2, we can extend Theorem 4.1 to the case of n-dimensional fuzzy differential equation driven by Liu process.

Theorem 4.2. Suppose that n-dimensional fuzzy differential equation (2) has a unique solution for each initial value. Then it is stability in credibility, if coefficients f  (t, x ) and g  (t, x ) satisfy strong Lipschitz condition ∥ f ( t , x ) - f ( t , y ) ∥ + ∥ g ( t , x ) - g ( t , y ) ∥ ≤ L ( t ) ∥ x - y ∥ , for ∀ x , y ∈ R m , t ≥ 0 for some integrable function L (t) on [0, + ∞).

Proof. The proof of this theorem is similar to Theorem 4.1.

As a tool to deal with dynamic system in fuzzy environment, fuzzy differential equation driven by Liu process holds an important place in both theory and application. Until now, some concepts of stability for fuzzy differential equation driven by Liu process have been proposed. The main results of the paper lies in the following aspects:

a concept of stability in credibility for fuzzy differential equation driven by Liu process was proposed;