The pth moment exponential stability of uncertain differential equation with jumps

1 Introduction

As far as we know, probability theory is used to study how likely an event is to happen. In real life, it is sometimes difficult for us to get data for a variety of reasons. The axiomatic system of probability theory is ineffective at this moment. In order to reasonably calculate the reliability of event occurrence, Liu [4] proposed the uncertainty theory and refined it in 2009 [6]. Liu [5] presented the uncertain process. Soon after, Liu [6] came up with Liu process. Meanwhile, Liu introduced uncertain calculus.

Liu [5] first proposed an uncertain differential equation through the Liu process in 2008. Yao [14] gave some calculation methods of the uncertain differential equation. There are many properties of uncertain differential equation, among which stability attracts many scholars to study it, such as Yao [15], Sheng [11], Yang [19], Zhang [20] and so on.

Uncertain renewal process which is describing the discontinuous uncertain system was put forward by Liu [5]. On this basis, Yao [13] put forward an uncertain differential equation with jumps. Then Yao [16] researched stability in measure for the solution. Different types of stability for uncertain differential equation with jumps have recently received more attentions, such as almost sure stability [3], the stability in pth moment [12], the stability in mean [2] and exponential stability [10].

The second part of this article introduces some basic concepts. Based on the above research, a new concept of the pth moment exponential stability for the uncertain differential equation with jumps is introduced in Section 3. Section 4 obtain a necessary and sufficient condition of stability in pth moment exponential of linear uncertain differential equation with jumps. And the relationships between pth moment exponential stability and stability in pth moment and exponential stability are given in Section 5. Finally, the article summarizes all results.

2 Basic knowledge

An uncertain differential equation with jump is formed by adding an uncertain renewal process on the basis of the uncertain differential equation. Considering the rapid change or jump at a certain moment, the uncertain differential equation with jump can be better characterized in economics, biology, phenomena in fields such as science and physics.

Definition 2.1. (Yao [13]) Let h, q and g be real functions, C _t be a Liu process and N _t be an uncertain renewal process. dY t = h ( t , Y t ) dt + q ( t , Y t ) dC t + g ( t , Y t ) dN t(1) is called an uncertain differential equation with jumps.

Theorem 2.1. (Yao [17]) C _t is a Liu process, N _t is an uncertain renewal process with iid interarrival times η₁, η₂, …. μ _t , σ _t and ν _t are real functions. Then the uncertain differential equation with jumps dY t = μ t Y t dt + σ t Y t dC t + ν t Y t dN t has a solution Y t = Y 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  = η₁ + η₂ + … + η _i for i ≥ 1.

Let Y _t and Z _t be two solutions of the uncertain differential equation with jumps (1) with different initial values Y₀ and Z₀, respectively.

Definition 2.2. (Yao [16]) The Equation (1) is stable in measure. Then lim | Y 0 - Z 0 | → 0 M { sup t ≥ 0 | Y t - Z t | ≤ ɛ } = 1 , for any given real number ɛ > 0.

Definition 2.3. (Gao [2]) The Equation (1) is stable in mean. Then lim | Y 0 - Z 0 | → 0 E [ sup t ≥ 0 | Y t - Z t | ] = 0 .

Definition 2.4. (Ma et al. [12]) The Equation (1) is stable in pth moment, if and only if lim | Y 0 - Z 0 | → 0 E [ sup t ≥ 0 | Y t - Z t | p ] = 0 .

Definition 2.5. (Liu [10]) The Equation (1) is called exponentially stable, if E [ | Y t - Z t | ] ≤ A exp ( - β t ) , ∀ t ≥ 0 ,(2) which A and β are two positive constants.

Definition 2.6. (Liu [4]) Holder’s Inequality. Set m and n be positive numbers and 1 m + 1 n = 1 . Set ξ, η be independent uncertain variables. Then E [ | ξ η | ] ≤ E [ | ξ | m ] m E [ | ξ | n ] n .(3)

Theorem 2.2. (Liu [7]) Set h (t) be an integrable function, the Liu integral ∫ 0 s h ( t ) dC t ∼ N ( 0 , ∫ 0 s | h ( t ) | dt ) .

is a normal uncertain variable at each time t.

Theorem 2.3. (Gao [2]) The Equation (1) is stable in mean, it must be stable in measure.

Theorem 2.4. (Ma et al. [12]) The equation dY t = ( u 1 t Y t + u 2 t ) dt + ( v 1 t Y t + v 2 t ) dC t + ( w 1 t Y t + w 2 t ) dN t is stable in pth moment, when ∫ 0 + ∞ u 1 s ds < + ∞ , ∫ 0 + ∞ | v 1 s | ds < π 3 , ∫ 0 + ∞ w 1 s ds < + ∞ .(4)

Definition 3.1. Let 0< p < + ∞. dY t = h ( t , Y t ) dt + q ( t , Y t ) dC t + g ( t , Y t ) dN t(5) is called pth moment exponentially stable, if there are positive constants G and β, such that E [ | Y t - Z t | p ] ≤ G | Y 0 - Z 0 | p exp ( - β t ) , ∀ t ≥ 0 .(6)

Example 3.1. dY t = ( 1 - Y t ) dt + σ dC t + ν dN t .(7) The two solutions with different initial values Y₀ and Z₀ are

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 ) .

and

Z t = exp ( - t ) ( Z 0 + ∫ 0 t exp ( s ) ds + ∫ 0 t σ exp ( s ) dC s + ∫ 0 t ν exp ( s ) dN s )

We have E [ | Y t - Z t | p ] = exp ( - pt ) | Y 0 - Z 0 | p ≤ 2 | Y 0 - Z 0 | p exp ( - pt ) = G | Y 0 - Z 0 | p exp ( - pt ) . It is pth moment exponentially stable, when G = 2 and β = p.

Example 3.2. dY t = ( 1 + Y t ) dt + σ dC t + ν dN t .(8) Suppose the two solutions with different initial values Y₀ and Z₀ are

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 ) .

and

Z t = exp ( t ) ( Z 0 + ∫ 0 t exp ( - s ) ds + ∫ 0 t σ exp ( - s ) dC s + ∫ 0 t ν exp ( - s ) dN s ) ,

We have E [ | Y t - Z t | p ] = exp ( pt ) | Y 0 - Z 0 | p → + ∞ , as t→ + ∞. Thus it is not satisfied the in Equation (6). The Equation (8) is not pth moment exponentially stable.

Theorem 4.1. Let u _it , v _it , w _it , i = 1, 2 be real functions. The linear uncertain differential equation with jumps dY t = ( u 1 t Y t + u 2 t ) dt + ( v 1 t Y t + v 2 t ) dC t + ( w 1 t Y t + w 2 t ) dN t is pth moment exponentially stable if and only if w_1t is a monotone and integrable function on [0, + ∞) and ∫ 0 t u 1 s ds ≤ - β t p , ∫ 0 + ∞ | v 1 s | ds < π 3 p , ∫ 0 + ∞ w 1 s ds < + ∞ .(9)

Proof. Let η₁, η₂, … denote the iid positive interrarrival times of N _t and inf j ≥ 1 η j ( γ ) , S₀ = 0 and S _i  = η₁ + η₂ + … + η _i for i ≥ 1. Assuming there is a positive number H such that M { γ ∈ Γ | inf i ≥ 1 η i ( γ ) ≥ H } = 1 , we get d ( Y t - Z t ) = u 1 t ( Y t - Z t ) dt + v 1 t ( Y t - Z t ) dC t + w 1 t ( Y t - Z t ) dN t ,(10) According to Theorem 2.1, the Equation (10) has a solution

Y t ( γ ) - Z t ( γ ) = ( Y 0 - Z 0 ) exp ( ∫ 0 t u 1 s ds + ∫ 0 t v 1 s dC s ( γ ) ) · ∏ i = 1 N t ( γ ) ( 1 + w 1 S i ( γ ) ) .

Therefore, | Y t ( γ ) - Z t ( γ ) | ≤ | Y 0 - Z 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 ) ) . That is | Y t ( γ ) - Z t ( γ ) | ≤ | Y 0 - Z 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 ) ) . So | Y t - Z t | ≤ | Y 0 - Z 0 | exp ( ∫ 0 t u 1 s ds ) · exp ( ∫ 0 t v 1 s dC s ) · exp ( ∑ i = 1 ∞ w 1 S i ) .(11) Let’s take the expectation of both sides of this inequality(11), E [ | Y t - Z t | p ] ≤ | Y 0 - Z 0 | p exp ( p ∫ 0 t u 1 s ds ) · E [ exp ( p ∫ 0 t v 1 s dC s ) · exp ( p ∑ i = 1 ∞ w ( 1 S i ) ) ] . 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 j ≥ 1 ξ j ( γ ) ≤ ∑ i = 1 ∞ w 1 S i ( γ ) · ξ j + 1 ( γ ) = ∫ 0 + ∞ w 1 s ds < + ∞ . And also ∑ i = 1 ∞ w 1 S i ( γ ) ≤ 1 H ∫ 0 + ∞ w 1 s ds < + ∞ . So ∑ i = 1 ∞ w 1 S i ≤ 1 H ∫ 0 + ∞ w 1 s ds < + ∞ , and that is exp ( p ∑ i = 1 ∞ w 1 S i ) ≤ exp ( p H ∫ 0 + ∞ w 1 s ds ) < + ∞ . Therefore, E [ | Y t - Z t | p ] ≤ | Y 0 - Z 0 | p · exp ( p ∫ 0 t u 1 s ds ) · exp ( p H ∫ 0 + ∞ w 1 s ds ) · E [ exp ( p ∫ 0 t v 1 s dC s ) ] . We already know ∫ 0 t u 1 s ds < - α t p , exp ( p ∫ 0 t u 1 s ds ) < exp ( - α t ) . Consider the inequality E [ exp ( p ∫ 0 t v 1 s dC s ) ] < + ∞ .(12) Since p ∫ 0 t v 1 s dC s ∼ N ( 0 , p ∫ 0 t | v 1 s | ds ) , inquality (12) is true if and only if p ∫ 0 + ∞ | v 1 s | ds < π 3 . It implies that ∫ 0 + ∞ | v 1 s | ds < π 3 p . Taking

G = exp ( p H ∫ 0 + ∞ w 1 s ds ) · E [ exp ( p ∫ 0 t v 1 s dC s ) ] ,

we get E [ | Y t - Z t | p ] ≤ G | Y 0 - Z 0 | p · exp ( - β t ) . The theorem is true.

Example 4.1. dY t = - Y t dt + ( sint t Y t + 5 ) dC t + 1 ( 1 + t ) 3 Y t dN t .(13) Note u 1 t = - 1 , v 1 t = sint t , w 1 t = 1 ( 1 + t ) 3 . ∫ 0 t u 1 t ds = ∫ 0 t ( - 1 ) ds = - t ≤ - β t p , if and only if β ≤ p. ∫ 0 + ∞ v 1 t ds = ∫ 0 + ∞ sint t ds = π 2 < π 3 p ,

if and only if 0 < p < 2 3 , w_1t is also a monotone and integrable function on [0, + ∞) ∫ 0 + ∞ w 1 t ds = ∫ 0 + ∞ 1 ( 1 + t ) 3 ds = 1 2 < + ∞ , if and only if β ≤ p < 2 3 the uncertain differential Equation (13) is pth moment exponentially stable by the Theorem 4.1.

Example 4.2. dY t = - Y t dt + exp ( - t 2 ) Y t dC t + ( exp ( t ) Y t + 2 ) dN t .(14) Note ∫ 0 + ∞ v 1 t ds = ∫ 0 + ∞ exp ( - t 2 ) ds = π 2 < π 3 p , if and only if 0 < p < 2 3 π 3 , when β ≤ p < 2 3 π 3 , u_1t, v_1t satisfy the conditions (9). w_1t = exp (t) is a monotone and integrable function on [0, + ∞), but ∫ 0 + ∞ w 1 t dt = ∫ 0 + ∞ exp ( t ) dt → + ∞ , it does not satisfy the conditions (9). The Equation (14) is not pth moment exponentially stable.

In order to distinguish the pth moment exponential stability by various means, this section compares pth moment exponential stability with other stability.

Theorem 5.1. The Equation (1) is pth moment exponentially stable, it is must be exponentially stable.

Proof. The proof of Theorem 5 is obtained by Definition 3.1 and Definition 2.5, when p = 1, A = G|X₀ - Y₀| ^p .

Remark 5.1. The following example proves that the converse of Theorem 5 is not true.

Example 5.1.

dY t = - Y t dt + exp ( - t ) Y t dC t + σ dN t .(15) We suppose that the solutions Y _t and Z _t have different initial values Y₀ and Z₀, respectively. d ( Y t - Z t ) = - ( Y t - Z t ) dt + exp ( - t ) ( Y t - Z t ) dC t . Thus Y t - Z t = ( Y 0 - Z 0 ) exp ( - t + ∫ 0 t exp ( - s ) dC s ) . Taking the expected value of |Y _t  - Z _t |, we get E [ | Y t - Z t | ] = | Y 0 - Z 0 | exp ( - t ) · E [ exp ( ∫ 0 t exp ( - s ) dC s ) ] . Also, we take the pth moment and the expected value on both sides of this equation,

E [ | Y t - Z t | p ] = | Y 0 - Z 0 | p exp ( - pt ) · E [ exp ( p ∫ 0 t exp ( - s ) dC s ) ] .

It follows from Theorem 2.2 that ∫ 0 t exp ( - s ) dC s ∼ N ( 0 , ∫ 0 t exp ( - s ) ds ) .

For all t > 0, ∫ 0 t exp ( - s ) ds = 1 - exp ( - t ) < π 3 . We have E [ exp ( ∫ 0 t exp ( - s ) dC s ) ] < + ∞ . Taking A = | Y 0 - Z 0 | E [ exp ( ∫ 0 t exp ( - s ) dC s ) ] and β = 1, the inequality (2) is satisfied. The Equation (15) is exponentially stable.

However, when ∫ 0 + ∞ p exp ( - s ) ds = p > π 3 , E [ exp ( p ∫ 0 + ∞ exp ( - s ) dC s ) ] → + ∞ . The Equation (15) is not pth moment exponentially stable.

Theorem 5.2. The Equation (1) is pth moment exponentially stable, it is stable in pth moment.

Proof. Equation (1) is pth moment exponentially stable. There are positive constants G and β such that E [ | Y t - Z t | p ] ≤ G | Y 0 - Z 0 | p exp ( - β t ) , ∀ t ≥ 0 . By taking limit, we obtain 0 ≤ lim | Y 0 - Z 0 | → 0 E [ sup | Y t - Z t | p ] ≤ lim | Y 0 - Z 0 | → 0 G | Y 0 - Z 0 | p = 0 . Thus lim | Y 0 - Z 0 | → 0 E [ sup | Y t - Z t | p ] = 0 . The Equation (1) is stable in pth moment.

Remark 5.2. The following example implies that the converse of Theorem 5.2 is not true.

Example 5.2. dY t = u t dt + exp ( - t ) Y t dC t + v t dN t .(16) The solutions Y _t and Z _t have different initial values Y₀ and Z₀. d ( Y t - Z t ) = ( Y t - Z t ) exp ( - t ) dC t . Thus Y t - Z t = ( Y 0 - Z 0 ) exp ( ∫ 0 t exp ( - s ) dC s ) . Taking pth moment of |Y _t  - Z _t |, we get

E | Y t - Z t | p = | Y 0 - Z 0 | p E [ exp ( p ∫ 0 t exp ( - s ) dC s ) ] .

It follows from Theorem 2.2 that ∫ 0 t p exp ( - s ) dC s ∼ N ( 0 , ∫ 0 t p exp ( - s ) ds ) . We have E [ exp ( p ∫ 0 t exp ( - s ) dC s ) ] < + ∞ . The above inequality is true if and only if ∫ 0 ∞ p exp ( - s ) ds = p < π 3 is true. But there do not exist positive constants G and β such that inequality (6) holds. Equation (16) is not pth moment exponentially stable. According to Theorem 2.4, it is obviously pth moment stable.

Theorem 5.3. For any two real numbers 0< p₁ < p₂ < + ∞, the Equation (1) is p₂th moment exponentially stable, then it is p₁th moment exponentially stable.

Proof. We get E [ | Y t - Z t | p 2 ] ≤ G 1 | Y 0 - Z 0 | p 2 exp ( - β 1 t ) , ∀ t ≥ 0 . Set p = p 2 p 1 and q = p 2 p 2 - p 1 . By Holder’s inequality, E [ | Y t - Z t | p 1 ] = E [ | Y t - Z t | p 1 · 1 ] ≤ E | Y t - Z t | p 1 p p E | 1 | q q ≤ E | Y t - Z t | p 1 p p = ( E | Y t - Z t | p 1 p ) p 1 p 2 = ( E | Y t - Z t | p 2 ) p 1 p 2 ≤ ( G 1 | Y 0 - Z 0 | p 2 exp ( - α 1 t ) ) p 1 p 2 = G 1 p 1 p 2 | Y 0 - Z 0 | p 1 exp ( - p 1 p 2 α 1 t ) = G 1 1 p | Y 0 - Z 0 | p 1 exp ( - 1 p α 1 t ) . Set G = G 1 1 p , β = 1 p α 1 . The equation is p₁th moment exponentially stable.

For uncertain differential equations with jump, we not only need to know its solutions, but also know the stability of the solutions. It is of great significance to study the stability of solutions of differential equations. The paper mainly presented a new concept of the pth moment exponential stability for uncertain differential equation. And a sufficient condition of the the pth moment stable is given. At last, two counterexamples were given to illustrate the feasibility of the conditions.