Exponential stability of uncertain differential equation with jumps

1 Introduction

In order to model dynamic stochastic systems, Ito [8] proposed a stochastic differential equation which was driven by Wiener process. Based on stochastic differential equation, Black and Scholes [1] assumed the price of a stock and obtained the famous European option pricing formulas for the stock. The evolution of some undetermined phenomena behaves not like randomness but like uncertainty.

In order to address uncertainties, professor Liu [10] founded uncertainty theory and later perfected it in [12]. In the framework of uncertainty theory, a concept of uncertain variable was defined by Liu [10] as a measurable function on an uncertainty space. Later some scholars studied uncertain variables, such as [2, 24]. To describe an uncertain system evolving with time, Liu [11] proposed a concept of uncertain process as a sequence of uncertain variables driven by the time. Since then, Liu [11] designed a Liu process as counterpart of Wiener process. We can say that Liu process is a Lipschitz continuous uncertain process with stationary and independent normal uncertain increments. Subsequently, Liu [12] founded uncertain calculus to handle the integral and differential of an uncertain process with respect to Liu process. Later, Yang et al [30] proposed uncertain calculus with respect to Yao process. Uncertain differential equation is a type of differential equation driven by Liu process. Till now, uncertain differential equation has been applied in many fields such as population growth [34], uncertain finance [15], uncertain programming [19], uncertain environment [7], optimal control [35], differential game [29], string vibration [5], spring vibration [4], heat conduction [32], and epidemic spread [20].

Stability of uncertain differential equation was a hot spot in recent years. Liu [12] and Yao et al [28] have already studied stability in measure and stability in mean, respectively. Subsequently, Sheng and Wang [23] studied the stability in p-th moment. Liu et al. [18] discussed the almost sure stability. Sheng and Gao [25] investigated the exponential stability of uncertain differential equation. Later, Yang et al. [31] studied the stability in inverse distribution. Chen and Ning [3] investigated pth moment exponential stability of uncertain differential equation. As an extension of the previous work on stability, Zhang et al. [33] discussed stability in measure and stability in mean for the solution of multifactor uncertain differential equation. Ma et al. [22] introduced a concept of stability in distribution for multifactor uncertain differential equation.

As a supplement of uncertain differential equation, uncertain differential equation with jumps which essentially is a type of uncertain differential equation driven by both Liu process and uncertain renewal process, was first proposed by Yao [26]. Stability analysis of uncertain differential equation with jumps had a great practical and theoretical significance. Many researchers investigated the stability of uncertain differential equation with jumps. Ji and Ke [9] discussed almost sure stability for the uncertain differential equation with jumps. Ma et al. [21] proposed the stability in p-th moment and proved the sufficient condition for uncertain differential equation with jumps being stable in p-th moment. Recently, Gao [6] put forward the concept of stability in mean and testified the stability theorem for the uncertain differential equation with jumps being stable in mean under strong Lipschitz conditions.

In this paper, a new definition of exponential stability is introduced for uncertain differential equation with jumps. In Sections 2 and 3, brief basic concepts about uncertain calculus, uncertain process and uncertain differential equation with jumps are recalled as they are required. Section 4 gives a sufficient and necessary condition for exponential stability of a linear uncertain differential equation with jumps are derived. In Section 5, the relationships among exponential stability, stability in measure, and stability in mean for uncertain differential equation with jumps are discussed. The last section gives some conclusions and illustrates the future scope of the study.

2 Uncertain variable

In this part, we review some preliminary concepts and theorems in uncertainty theory. Let Γ be a nonempty set and L be a σ-algebra over Γ. Liu [10] proposed the uncertain measure and uncertain variable. The uncertainty distribution and the dependence of uncertain variables were defined.

Definition 2.1. (Liu [17]) An uncertain variable ξ is called normal if it has the normal uncertainty distribution Φ ( x ) = ( 1 + exp ( π ( e - x ) 3 σ ) ) - 1 , x ∈ R(1) denoted by N ( e , σ ) , where e and σ are real numbers satisfying σ > 0.

The operational law of uncertain variables was proposed by (Liu [13]) to calculate the inverse uncertainty distribution of strictly monotonous function as the following theorem.

In order to get the average value of ξ in terms of uncertain measure, Liu [10] proposed the concept of expected value which represents the size of ξ as follows.

Definition 2.2. (Liu [10]) Let ξ be an uncertain variable. Then the expected value of ξ is defined as E [ ξ ] = ∫ 0 + ∞ M { ξ ≥ x } d x - ∫ - ∞ 0 M { ξ ≤ x } d x(2) provided that at least one of the two integrals is finite.

The expected value of a lognormal uncertain variable exp(ξ) is E [ exp ( ξ ) ] = { 3 σ exp ( e ) csc ( 3 σ ) , if σ ≤ π 3 + ∞ , otherwise .

In this section, we will introduce some fundamental concepts and properties concerning uncertain processes, Liu processes and uncertain differential equations with jumps.

Definition 3.1. (Liu [16]) Let T be a totally ordered set (e.g. time) and let ( Γ , L , M ) be an uncertainty space. An uncertain process is a function X _t  (γ) from to the set of real numbers such that X _t  ∈ B is an event for any Borel set B of real numbers at each time t.

Definition 3.2. (Liu [12]) An uncertain process C _t is said to be a Liu process if

C₀ = 0 and almost all sample paths are Lipschitz continuous,

Definition 3.3. (Liu [12]) Let X _t be an uncertain process and C _t be a 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 Liu integral of X _t with respect to C _t is defined as ∫ a b X t dC 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 finite.

Theorem 3.1. (Liu [13]) Let f (t) be an integrable function with respect to t. Then the Liu integral ∫ 0 s f ( t ) dC t is a normal uncertain variable at each time s, and ∫ 0 t f ( t ) dC t ∼ N ( 0 , ∫ 0 s | f ( t ) | ds ) .

Definition 3.4. (Liu [11]) Let ξ₁, ξ₂, … be iid positive uncertain variables. Define S₀ = 0 and S _n  = ξ₁ + ξ₂ + ⋯ + ξ _n for n ≥ 1. Then the uncertain process N t = max n ≥ 0 { n | S n ≤ t } is called an uncertain renewal process.

Definition 3.5. (Yao [26]) Let X _t be uncertain process and let N _t be an uncertain renewal process. Then the Yao integral of X _t on the interval [a, b] is defined by ∫ a b X t dN t = ∑ a < t ≤ b X t - ( N t - N t - ) provided that the sum exists almost surely and is finite.

Definition 3.6. (Yao [26]) Let C _t be a Liu process, N _t be an uncertain renewal process. Suppose that f, p and h are some given real functions. Then dX t = f ( t , X t ) dt + g ( t , X t ) dC t + h ( t , X t ) dN t(3) is called an uncertain differential equation with jumps.

Example 3.1. Let C _t be a Liu process, N _t be an uncertain renewal process with iid interarrival times ξ₁, ξ₂, …, and μ _t , σ _t and ν _t be some real functions. Then the uncertain differential equation with jumps dX t = μ t X t dt + σ t X t dC t + ν t X t dN t has a solution X t = X 0 exp ( ∫ 0 t μ s ds + ∫ 0 t σ s dC s ) · ∏ i = 1 N t ( 1 + ν S i ) , where S₀ = 0 and S _i  = ξ₁ + ξ₂ + … + ξ _i for i ≥ 1.

Definition 3.7. (Yao [27]) An uncertain differential equation with jumps (1) is said to be stable in measure if for any two solutions X _t and Y _t with different initial values X₀ and Y₀, we have lim | X 0 - Y 0 | → 0 M { sup t ≥ 0 | X t - Y t | ≤ ɛ } = 1 , for any given real number ɛ > 0.

Definition 3.8. (Gao [6]) Let X _t and Y _t be two solutions of the uncertain differential equation with jumps (1) with different initial values X₀ and Y₀, respectively. If we have lim | X 0 - Y 0 | → 0 E [ sup t ≥ 0 | X t - Y t | ] = 0 , the uncertain differential equation with jumps (1) is said to be stable in mean.

Definition 3.9. (Ma et al [21]) Let X _t and Y _t be two solutions of the uncertain differential equation with jumps (1) with different initial values X₀ and Y₀, respectively. If we have lim | X 0 - Y 0 | → 0 E [ sup t ≥ 0 | X t - Y t | p ] = 0 , p > 0 the uncertain differential equation with jumps (1) is said to be stable in p-th moment.

Definition 3.10. (Ji et al [9]) Let X _t and Y _t be two solutions of the uncertain differential equation with jumps (1) with different initial values X₀ and Y₀, respectively. If we have M { γ ∈ Γ | lim | X 0 - Y 0 | → 0 sup t ≥ 0 | X t ( γ ) - Y t ( γ ) | = 0 } = 1 , the uncertain differential equation with jumps (1) is said to be stable almost surely.

Theorem 3.2. (Gao [6]) If the uncertain differential equation with jumps (1) is stable in mean, then it is stable in measure.

In this section, we propose a definition of exponential stability for an uncertain differential equation with jumps.

Definition 4.1. The uncertain differential equation with jumps dX t = f ( t , X t ) dt + g ( t , X t ) dC t + h ( t , X t ) dN t(4) is said to be exponentially stable, if for any solutions X _t and Y _t , there exist positive constants A and α such that E [ | X t - Y t | ] ≤ A exp ( - α t ) , ∀ t ≥ 0 .(5)

Example 4.1. Consider the uncertain differential equation dX t = ( 1 - X t ) dt + σ dC t + ν dN t .(6) Since the two solutions with different initial values X₀ and Y₀ are X t = exp ( - t ) ( X 0 + ∫ 0 t exp ( s ) ds + ∫ 0 t σ exp ( s ) dC s + ∫ 0 t ν exp ( s ) dN s ) and Y t = exp ( - t ) ( Y 0 + ∫ 0 t exp ( s ) ds + ∫ 0 t σ exp ( s ) dC s + ∫ 0 t ν exp ( s ) dN s ) , we have E [ | X t - Y t | ] = exp ( - t ) | X 0 - Y 0 | ≤ 2 | X 0 - Y 0 | exp ( - t ) = A exp ( - t ) , where the constants A = 2|X₀ - Y₀| and α = 1. Hence the uncertain differential equation (4) is exponentially stable.

Example 4.2. Consider the uncertain differential equation dX t = ( 1 + X t ) dt + σ dC t + ν dN t .(7) Since the two solutions with different initial values X₀ and Y₀ are X t = exp ( t ) ( X 0 + ∫ 0 t exp ( - s ) ds + ∫ 0 t σ exp ( - s ) dC s + ∫ 0 t ν exp ( - s ) dN s ) , and Y t = exp ( t ) ( Y 0 + ∫ 0 t exp ( - s ) ds + ∫ 0 t σ exp ( - s ) dC s + ∫ 0 t ν exp ( - s ) dN s ) , we have E [ | X t - Y t | ] = exp ( t ) | X 0 - Y 0 | → + ∞ , as t→ + ∞. Thus there do not exist the positive numbers A and α such that E [|X _t  - Y _t |] ≤ A exp(- αt). Hence uncertain differential equation (5) is not exponentially stable.

In this section, we give a sufficient and necessary conditions for the linear uncertain differential equation with jumps being exponentially stable.

Theorem 5.1. Let u _it , v _it , w _it , i = 1, 2 be real functions. Then the linear uncertain differential equation with jumps

dX t = ( u 1 t X t + u 2 t ) dt + ( v 1 t X t + v 2 t ) dC t + ( w 1 t X t + w 2 t ) dN t(8) is exponentially stable if and only if w_1t is a monotone and integrable function on [0, + ∞) and

∫ 0 t u 1 s ds < - α t , ∫ 0 + ∞ | v 1 s | ds < π 3 , ∫ 0 + ∞ w 1 s ds < + ∞ .(9)

Proof: Let X _t and Y _t be two solutions of the uncertain differential equation with jumps (6) with different initial values X₀ and Y₀, respectively. We have dX t = ( u 1 t X t + u 2 t ) dt + ( v 1 t X t + v 2 t ) dC t + ( w 1 t X t + w 2 t ) dN t and dY t = ( u 1 t Y t + u 2 t ) dt + ( v 1 t Y t + v 2 t ) dC t + ( w 1 t X t + w 2 t ) dN t . Let ξ₁, ξ₂, … denote the iid interrarrival times of N _t . Define S₀ = 0 and S _i  = ξ₁ + ξ₂ + … + ξ _i for i ≥ 1. Assuming that there is a positive number H such that M { γ ∈ Γ | inf i ≥ 1 ξ i ( γ ) ≥ H } = 1 . The linear uncertain differential equation with jumps d ( X t - Y t ) = u 1 t ( X t - Y t ) dt + v 1 t ( X t - Y t ) dC t + w 1 t ( X t - Y t ) dN t has a solution X t - Y t = ( X 0 - Y 0 ) exp ( ∫ 0 t u 1 s ds + ∫ 0 t v 1 s dC s ) · ∏ i = 1 N t ( 1 + w 1 s i ) . For almost all of γ, | X t ( γ ) - Y t ( γ ) | ≤ | X 0 - Y 0 | exp ( ∫ 0 t u 1 s ds ) · exp ( ∫ 0 t v 1 s dC s ( γ ) ) · ∏ i = 1 ∞ ( 1 + w 1 s i ( γ ) ) ≤ | X 0 - Y 0 | exp ( ∫ 0 t u 1 s ds ) · exp ( ∫ 0 t v 1 s dC s ( γ ) ) · exp ( ∑ i = 1 ∞ w 1 s i ( γ ) ) . Since w_1t is a monotone and integrable function on [0, + ∞) and ∫ 0 + ∞ w 1 s ds < + ∞ , we get

∑ i = 1 ∞ w 1 s i ( γ ) · inf ξ i + 1 ( γ ) ≤ ∑ i = 1 ∞ w 1 s i ( γ ) · ξ i + 1 ( γ ) = ∫ 0 + ∞ w 1 s ds < + ∞ ,(10) for almost all γ. The inequation (8) is equivalent to ∑ i = 1 ∞ w 1 s i ( γ ) ≤ 1 H ∫ 0 + ∞ w 1 s ds < + ∞ . That is to say exp ( ∑ i = 1 ∞ w 1 s i ) ≤ exp ( 1 H ∫ 0 + ∞ w 1 s ds ) < + ∞ . There is a real number 0< G < + ∞ such that exp ( 1 H ∫ 0 + ∞ w 1 s ds ) = G . We immediately have

| X t - Y t | ≤ G | X 0 - Y 0 | exp ( ∫ 0 t u 1 s ds ) · exp ( ∫ 0 t v 1 s dC s ) .(11) Taking expected value on both sides for inequation (9), we get

E [ | X t - Y t | ] ≤ G | X 0 - Y 0 | exp ( ∫ 0 t u 1 s ds ) · E [ exp ( ∫ 0 t v 1 s dC s ) ] .(12) Hence, the linear uncertain differential equation (9) is exponentially stable if and only if ∫ 0 t u 1 s ds < - α t , and E [ exp ( ∫ 0 t v 1 s dC s ) ] < + ∞ .(13) Since ∫ 0 t v 1 s dC s ∼ N ( 0 , ∫ 0 t | v 1 s | ds ) , the inquality (11) is equivalent to ∫ 0 + ∞ | v 1 s | ds < π 3 , according to the expected value of the lognormal uncertain variables. It can be deduced from the inequality (9) that E [ | X t - Y t | ] ≤ G | X 0 - Y 0 | · E [ exp ( ∫ 0 t v 1 s dC s ) ] · exp ( - α t ) . Taking A = G | X 0 - Y 0 | · E [ exp ( ∫ 0 t v 1 s dC s ) ] , we get E [ | X t - Y t | ] ≤ A exp ( - α t ) . The theorem is proved.□

Example 5.1. Consider the uncertain differential equation dX t = - X t dt + exp ( - t ) X t dC t + σ dN t .

Since ∫ 0 t u 1 s ds = - t , ∫ 0 + ∞ | v 1 s | ds = ∫ 0 + ∞ exp ( - t ) ds = 1 < π 3 , and ∫ 0 + ∞ w 1 s ds = 0 , the uncertain differential equation with jumps satisfies the condition (7). Therefore, it is exponentially stable.

Example 5.2. Consider the uncertain differential equation dX t = X t dt + exp ( - t ) X t dC t + σ dN t . Since u_1t = 1 does not satisfy the condition (7), it is not exponentially stable.

Theorem 6.1. If the uncertain differential equation with jumps

dX t = ( u 1 t X t + u 2 t ) dt + ( v 1 t X t + v 2 t ) dC t + ( w 1 t X t + w 2 t ) dN t(14) is exponentially stable, then it is stable in mean.

Proof: Let X _t and Y _t be two solutions of the uncertain differential equation with jumps (11) with initial values X₀ and Y₀, respectively. According to Definition 4.1, we have E [ | X t - Y t | ] ≤ A exp ( - α t ) . Noting that A → 0 as |X₀ - Y₀|→0, we obtain lim | X 0 - Y 0 | → 0 E [ sup t ≥ 0 | X t - Y t | ] = 0 . Therefore it is stable in mean from the Definition 3.8.□

Remark 6.1. Generally, stability in mean does not imply exponential stability. The following example can explain it.

Example 6.1. Consider the uncertain differential equation dX t = 2 X t ( 1 + t ) 2 dt + σ dC t + ν dN t .(15) Let X _t and Y _t be two solutions of the uncertain differential equation with jumps (15) with initial values X₀ and Y₀, respectively. Then dY t = 2 Y t ( 1 + t ) 2 dt + σ dC t + ν dN t . We have d ( X t - Y t ) = 2 ( X t - Y t ) ( 1 + t ) 2 dt(16) with an initial value X₀ - Y₀. The differential equation (14) has the solution

X t - Y t = exp ( 2 t 1 + t ) ( X 0 - Y 0 ) . Therefore, we obtain lim | X 0 - Y 0 | → 0 E [ sup t ≥ 0 | X t - Y t | ] = 0 . The uncertain differential equation (15) is stable in mean. However, there do not exist positive constants A and α such that E [ | X t - Y t | ] ≤ A exp ( - α t ) . Thus the uncertain differential equation (15) is not exponentially stable.

Theorem 6.2. If the uncertain differential equation with jumps is exponentially stable, it is stable in measure.

Proof: It is easily proved from Theorems 3.2 and 6.1.□

Remark 6.2. Generally, stability in measure does not imply exponential stability. Example 6.1 can explain it. It has been proved that the uncertain differential equation (15) is stable in mean, it is stable in measure. However, it is not exponentially stable.

Example 6.2. Consider the uncertain differential equation

dX t = exp ( - t 2 ) dt + exp ( - t - X t ) dC t + exp ( - t 2 - X t ) dN t .(17) that mentioned in [21] has been proved that it is stable in measure. However, ∫ 0 t u 1 s ds = 0 . Therefore, it is not exponentially stable.

Remark 6.3. Generally, stability in p-th moment does not imply exponential stability.

Example 6.3. The following linear uncertain differential equation with jumps dX t = exp ( - t ) X t dt + exp ( - t 2 ) tX t dC t + ( exp ( - t 2 ) X t + t ) dN t that mentioned in [21] has been proved that it is stable in p-th moment. However, ∫ 0 t u 1 s ds = 1 - exp ( - t ) . Therefore, it is not exponentially stable.

Remark 6.4. Generally, almost sure stability does not imply exponential stability.

Example 6.4. The following linear uncertain differential equation with jumps dX t = t 2 dt + exp ( - t 2 - X t 2 ) dC t + exp ( - t ) X t dN t . that mentioned in [9] has been proved that it is stable almost surely. However, ∫ 0 t u 1 s ds = 0 . Therefore, it is not exponentially stable.

In this paper, we proposed the exponential stability for the uncertain differential equation with jumps, and gave a sufficient and necessary condition for the linear uncertain differential equation with jumps. Then the relationships among the stability in measure, the stability in mean and exponential stability are discussed. Further researches may research the numerical solutions of the uncertain differential equation with jumps and cover the applications in the finance.