Almost sure and p th moment exponential stability of backward uncertain differential equations

Abstract

Driven by a Liu process, backward uncertain differential equation is a type of differential equations with given final value. So far, the concepts of stability in measure, stability in mean and stability in pth moment for backward uncertain differential equations have been proposed. As a supplement, this paper is concerned with two other kinds of stability of backward uncertain differential equations, and proposes the concepts of almost sure stability and pth moment exponential stability. In addition, some sufficient conditions for backward uncertain differential equations being stable almost surely or pth moment exponentially are derived.

Keywords

Backward uncertain differential equation almost sure stability uncertain variable uncertain process

1 Introduction

When no adequate samples are available to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur. To rationally model belief degrees, Liu [8] founded uncertainty theory in 2007. Later, uncertainty theory was perfected by Liu [10] with the fundamental concept of uncertain measure. As another fundamental concept in uncertainty theory, uncertain variable was presented by Liu [8]. Nowadays, uncertainty theory has become a branch of mathematics for modeling human uncertainty.

In order to model the evolution of uncertain phenomena, the concept of uncertain process was started by Liu [9] as a sequence of uncertain variables indexed by time. Inaddition, Liu designed a Liu process [10] in contrast to Wiener process. Meanwhile, uncertain integral was also proposed by Liu [10] to integrate an uncertain process with respect to a Liu process.

Based on uncertain calculus, an uncertain differential equation driven by a Liu process was proposed by Liu [9] as a type of differential equation, which is different from fuzzy differential equations [1 , 7]. Following that, many researchers have done a lot of work in theory and application about uncertain differential equations. Note that the analytic methods do not always work, some numerical methods were proposed to solve uncertain differential equations, such as Adams-Simpson method [12], Milne method [5]. As to the stability of the solution, there are mainly five types of stability for an uncertain differential equation, namely stability in measure, stability in mean, stability in pth moment, almost sure stability and pth moment exponential stability. Up to now, uncertain differential equation has been applied to finance, and uncertain stock model [20], uncertain interest rate model [3] and uncertain currency model [11, 14] have sprang up. In addition, uncertain differential equation has also been introduced to differential game [16], heat conduction [17] and optimal control [21].

So far, uncertain differential equation has been successfully extended in many directions, including uncertain differential equation with jumps [19], uncertain delay differential equation [13] and backward uncertain differential equation [6], etc. The concept of backward uncertain differential equation was introduced by Ge and Zhu [6]. As to the solution of a backward uncertain differential equation, an existence and uniqueness theorem was given by Ge and Zhu [6]. For the stability analysis of the solutions, the concepts of stability in measure, stability in mean and stability in pth moment for a backward uncertain differential equation were proposed by Wang and Ning [15]. Meanwhile, they derived some sufficient conditions for a backward uncertain differential equation being stable in measure, in mean and in pth moment in [15].

In this paper, we continue to study another two types of stability of backward uncertain differential equations, namely almost sure stability and pth moment exponential stability. For them, we first give their definitions, and then provide some stability theorems. The structure of this paper is organized as follows. Section 2 introduces some basic concepts and theorems about uncertain variable, uncertain process and backward uncertain differential equations. In Section 3, we define almost sure stability for a backward uncertain differential equation, and derive two sufficient conditions for backward uncertain differential equations and linear backward uncertain differential equations, respectively. Then, the concept and a sufficient condition of pth moment exponential stability of a backward uncertain differential equation are presented in Section 4. Finally, Section 5 makes a brief conclusion.

2 Preliminaries

In this section, we introduce some basic concepts and theorems about uncertain variable, uncertain process and backward uncertain differential equation.

2.1 Uncertain variable

Definition 2.1. (Liu [8, 10]) Let ℒ be a σ-algebra on a nonempty set Γ. A set function ℳ : ℒ → [0, 1] is called an uncertain measure if it satisfies the following axioms:

Axiom 1. (Normality Axiom) ℳ {Γ} =1;

Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1;

Axiom 3. (Subadditivity Axiom) For every countable sequence of {Λ_i} ∈ ℒ, we have $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$

Axiom 4. (Product Axiom) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯ The product uncertain measure ℳ is an uncertain measure satisfying $ℳ {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}}$ where Λ_k ∈ ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [8]) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, ℒ, ℳ) to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

Definition 2.3. (Liu [8]) Let ξ be an uncertain variable. The expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} ℳ {ξ \geq r} d r - \int_{- \infty}^{0} ℳ {ξ \leq r} d r$ provided that at least one of the above two integrals is finite.

Definition 2.4. (Liu [8]) Let ξ be an uncertain variable and p be a positive integer. Then E [ξ^p] is called the pth moment of ξ.

2.2 Uncertain process

To model the evolution of uncertain phenomena, Liu [9] proposed an uncertain process as a sequence of uncertain variables driven by time. Following that, Liu [10] designed a Liu process as one of the most important uncertain processes.

Definition 2.5. (Liu [10]) An uncertain process C_t (t ≥ 0) is called a Liu process if

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

C_t is a stationary independent increment process,

every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t².

Based on the Liu process, Liu integral is defined as an counterpart of Ito integral as follows.

Definition 2.6. (Liu [10]) Let X_t be an uncertain process and let C_t be a Liu process. For any partition of the closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = sup_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$

Then the uncertain integral of X_t with respect to C_t is defined by $\int_{a}^{b} X_{t} d C_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite. In this case, the uncertain process X_t is said to be uncertain integrable.

Theorem 2.1. (Chen and Liu [2]) Suppose that C_t is a Liu process, and X_t is an integrable uncertain process on interval [a, b] with respect to t. Then the inequality $| \int_{a}^{b} X_{t} (γ) d C_{t} (γ) | \leq K (γ) \int_{a}^{b} | X_{t} (γ) | d t$ holds, where K (γ) is the Lipschitz constant of C_t (γ).

Theorem 2.2. (Yao et al. [18]) Let C_t be a Liu process on uncertainty space (Γ, ℒ, ℳ). Then there exists an uncertain variable K such that K (γ) is a Lipschitz constant of the sample path C_t (γ) for each γ, $lim_{x \to + \infty} ℳ {γ \in Γ | K (γ) \leq x} = 1$ and $ℳ {γ \in Γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$

2.3 Backward uncertain differential equation

Definition 2.7. (Ge and Zhu [6]) Let C_t be a Liu process, and f and g are two real-valued functions. ${\begin{matrix} d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}, t \in [0, T) \\ X_{T} = ξ \end{matrix}$ (1) is called a backward uncertain differential equation with final value X_T = ξ.

Backward uncertain differential equation (1) is equivalent to the following form

$\begin{matrix} X_{t} & = & ξ - \int_{t}^{T} f (s, X_{s}) d s \\ - \int_{t}^{T} g (s, X_{s}) d C_{s}, t \in [0, T] . \end{matrix}$ (2)

Theorem 2.3. (Ge and Zhu [6]) Backward uncertain differential equation (1) has a unique solution X_t with given final value on [0, T] if the coefficients f (t, x)and g (t, x) satisfy the Lipschitz condition $\begin{matrix} | f (t, x_{1}) - f (t, x_{2}) | + | g (t, x_{1}) - g (t, x_{2}) | \\ \leq L | x_{1} - x_{2} |, \forall x_{1}, x_{2} \in R, t \in [0, T] \end{matrix}$ and the linear growth condition $| f (t, x) | + | g (t, x) | \leq L (1 + | x |), \forall x \in R, t \in [0, T] .$ where L is a positive constant.

Theorem 2.4. (Wang and Ning [15]) Let the real-valued functions g and u be non-negative and continuous on the interval [0, T] with T > 0. Assume that C is a positive constant. If u satisfies the integral inequality $u (t) \leq C + \int_{t}^{T} g (s) u (s) d s, \forall t \in [0, T]$ then $u (t) \leq C exp (\int_{t}^{T} g (s) d s), \forall t \in [0, T] .$

3 Almost sure stability

In this section, we give a definition of almost sure stability for backward uncertain differential equation (1), and two examples are presented to explain the definition. Finally, two sufficient conditions for backward uncertain differential equation (1) and linear backward uncertain differential equations being stable almost surely are provided, respectively.

Definition 3.1. Backward uncertain differential equation (1) is said to be stable almost surely if for any two solutions X_t and Y_t with different final values x_T and y_T, we have $ℳ {lim_{| x_{T} - y_{T} | \to 0} | X_{t} - Y_{t} | = 0} = 1, \forall t \in [0, T] .$

Example 3.1. Consider the backward uncertain differential equation ${\begin{matrix} d X_{t} = μ d t + σ d C_{t}, t \in [0, T) \\ X_{T} = ξ . \end{matrix}$ (3)

Clearly, its two solutions with final values x_T and y_T are $X_{t} = x_{T} - μ (T - t) + σ (C_{T} - C_{t})$ and $Y_{t} = y_{T} - μ (T - t) + σ (C_{T} - C_{t}),$ respectively.

Then $\begin{matrix} ℳ {lim_{| x_{T} - y_{T} | \to 0} | X_{t} - Y_{t} | = 0} \\ = ℳ {lim_{| x_{T} - y_{T} | \to 0} | x_{T} - y_{T} | = 0} = 1 \end{matrix}$ for any t ∈ [0, T], and backward uncertain differential equation (3) is stable almost surely.

Example 3.2. Consider the backward uncertain differential equation

{\begin{matrix} d X_{t} = - X_{t} d t + σ d C_{t}, t \in [0, T) \\ X_{T} = ξ . \end{matrix}

(4)

Since its solutions with different final values x_T and y_T are

X_{t} = (x_{T} - σ \int_{t}^{T} exp (s - T) d C_{s}) exp (T - t)

and

Y_{t} = (y_{T} - σ \int_{t}^{T} exp (s - T) d C_{s}) exp (T - t),

respectively, we obtain that

| X_{t} - Y_{t} | = | x_{T} - y_{T} | exp (T - t)

holds almost surely.

Then

\begin{matrix} ℳ {lim_{| x_{T} - y_{T} | \to 0} | X_{t} - Y_{t} | = 0} \\ = ℳ {lim_{| x_{T} - y_{T} | \to 0} | x_{T} - y_{T} | exp (T - t) = 0} = 1 \end{matrix}

for any t ∈ [0, T], and backward uncertain differential equation (4) is stable almost surely.

Theorem 3.1. Assume that backward uncertain differential equation (1) has a unique solution with each given final value. Then it is stable almost surely if the coefficients f (t, x) and g (t, x) satisfy the strong Lipschitz condition $\begin{matrix} | f (t, x_{1}) - f (t, x_{2}) | + | g (t, x_{1}) - g (t, x_{2}) | \\ \leq L (t) | x_{1} - x_{2} |, \forall x_{1}, x_{2} \in R, t \in [0, T] \end{matrix}$ where L (t) is a positive function satisfying $\int_{0}^{T} L (t) d t < + \infty .$

Proof. Assume that X_t and Y_t are two solutions of backward uncertain differential equation (1) with different final values x_T and y_T, respectively. That is, ${\begin{matrix} d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}, t \in [0, T) \\ X_{T} = x_{T} \end{matrix}$ and ${\begin{matrix} d Y_{t} = f (t, Y_{t}) d t + g (t, Y_{t}) d C_{t}, t \in [0, T) \\ Y_{T} = y_{T} . \end{matrix}$

Then for a Lipschitz continuous sample C_t (γ), we have $\begin{matrix} X_{t} (γ) & = & x_{T} - \int_{t}^{T} f (s, X_{s} (γ)) d s \\ - \int_{t}^{T} g (s, X_{s} (γ)) d C_{s} (γ) \end{matrix}$ and $\begin{matrix} Y_{t} (γ) & = & y_{T} - \int_{t}^{T} f (s, Y_{s} (γ)) d s \\ - \int_{t}^{T} g (s, Y_{s} (γ)) d C_{s} (γ) \end{matrix}$ for any t ∈ [0, T].

By using the strong Lipschitz condition and Theorem 2.1, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | x_{T} - y_{T} + \int_{t}^{T} [f (s, Y_{s} (γ)) - f (s, X_{s} (γ))] d s \\ + \int_{t}^{T} [g (s, Y_{s} (γ)) - g (s, X_{s} (γ))] d C_{s} (γ) | \\ \leq | x_{T} - y_{T} | + | \int_{t}^{T} [f (s, Y_{s} (γ)) - f (s, X_{s} (γ))] d s | \\ + | \int_{t}^{T} [g (s, Y_{s} (γ)) - g (s, X_{s} (γ))] d C_{s} (γ) | \\ \leq | x_{T} - y_{T} | + \int_{t}^{T} L (s) | X_{s} (γ) - Y_{s} (γ) | d s \\ + K (γ) \int_{t}^{T} L (s) | X_{s} (γ) - Y_{s} (γ) | d s \\ = | x_{T} - y_{T} | + \int_{t}^{T} (1 + K (γ)) L (s) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$

where K (γ) is the Lipschitz constant of C_t (γ).

It follows from Theorem 2.4 that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | x_{T} - y_{T} | exp (\int_{t}^{T} (1 + K (γ)) L (s) d s) \\ \leq | x_{T} - y_{T} | exp ((1 + K (γ)) \int_{0}^{T} L (s) d s) \end{matrix}$ for any t ∈ [0, T].

Since L (s) is integrable on [0, T] and K (γ) is finite, we obtain $lim_{| x_{T} - y_{T} | \to 0} | X_{t} (γ) - Y_{t} (γ) | = 0 .$

That is, $ℳ {lim_{| x_{T} - y_{T} | \to 0} | X_{t} - Y_{t} | = 0} = 1, \forall t \in [0, T] .$

So backward uncertain differential equation (1) is stable almost surely, and the theorem is proved. □

Example 3.3. Consider the backward uncertain differential equation ${\begin{matrix} d X_{t} = X_{t} d t + σ d C_{t}, t \in [0, T) \\ X_{T} = ξ . \end{matrix}$ (5)

Its two solutions with final values x_T and y_T are $X_{t} = (x_{T} - σ \int_{t}^{T} exp (T - s) d C_{s}) exp (t - T)$ and $Y_{t} = (y_{T} - σ \int_{t}^{T} exp (T - s) d C_{s}) exp (t - T),$ respectively. That is, backward uncertain differential equation (5) has a unique solution with each given final value.

Since f (t, x) = x and g (t, x) = σ satisfy $\begin{matrix} | f (t, x_{1}) - f (t, x_{2}) | + | g (t, x_{1}) - g (t, x_{2}) | \\ = | x_{1} - x_{2} | \leq | x_{1} - x_{2} | \end{matrix}$ with L (t) =1 for all $x_{1}, x_{2} \in R$ , backward uncertain differential equation (5) is stable almost surely by using Theorem 3.1.

Corollary 3.1. The linear backward uncertain differential equation ${\begin{matrix} d X_{t} = (u_{t} X_{t} + v_{t}) d t + (w_{t} X_{t} + z_{t}) d C_{t}, t \in [0, T) \\ X_{T} = ξ . \end{matrix}$ (6) is stable almost surely if u_t, v_t, w_t and z_t are real-valued bounded functions, and $\int_{0}^{T} | u_{t} | d t < + \infty, \int_{0}^{T} | w_{t} | d t < + \infty .$

Proof. Take f (t, x) = u_tx + v_t and g (t, x) = w_tx + z_t, and let M denote a common upper bound of |u_t|, |v_t|, |w_t| and |z_t|. Since the expression $\begin{matrix} | f (t, x) | + | g (t, x) | \\ = | u_{t} x + v_{t} | + | w_{t} x + z_{t} | \leq 2 M (1 + | x |) \end{matrix}$ holds, the linear growth condition is satisfied. Besides, we have $\begin{matrix} | f (t, x_{1}) - f (t, x_{2}) | + | g (t, x_{1}) - g (t, x_{2}) | \\ = | u_{t} | | x_{1} - x_{2} | + | w_{t} | | x_{1} - x_{2} | \\ = (| u_{t} | + | w_{t} |) | x_{1} - x_{2} | . \end{matrix}$

Letting L (t) = |u_t| + |w_t|, we have $\int_{0}^{T} L (t) d t = \int_{0}^{T} | u_{t} | d t + \int_{0}^{T} | w_{t} | d t < + \infty .$

The strong Lipstchitz condition is satisfied. By using Theorem 3.1, linear backward uncertaindifferential equation (6) is stable almost surely. □

4 pth moment exponential stability

This section is concerned with the pth moment exponential stability for backward uncertain differential equation (1). At first, pth moment exponential stability for backward uncertain differential equation (1) is defined as follows.

Definition 4.1. Backward uncertain differential equation (1) is said to be pth moment exponentially stable (0 < p < + ∞) if for any two solutions X_t and Y_t with different final values x_T and y_T, there exists a pair of positive constants C and λ such that $\begin{matrix} E | X_{t} - Y_{t} |^{p} \\ \leq C | x_{T} - y_{T} |^{p} exp (λ (T - t)), \forall t \in [0, T] . \end{matrix}$

In the case of p = 1, backward uncertain differential equation (1) is usually said to be exponentially stable.

Example 4.1. Consider backward uncertain differential equation (4).

As shown in Example 3.2, its solutions with different final values x_T and y_T are $X_{t} = (x_{T} - σ \int_{t}^{T} exp (s - T) d C_{s}) exp (T - t)$ and $Y_{t} = (y_{T} - σ \int_{t}^{T} exp (s - T) d C_{s}) exp (T - t),$ respectively, and $| X_{t} - Y_{t} | = | x_{T} - y_{T} | exp (T - t)$ holds almost surely.

For any p > 0, we have $\begin{matrix} E | X_{t} - Y_{t} |^{p} \\ = | x_{T} - y_{T} |^{p} exp (p (T - t)) \\ \leq C | x_{T} - y_{T} |^{p} exp (λ (T - t)), \end{matrix}$ where C = 1 and λ = p. Hence, we conclude that backward uncertain differential equation (4) is pth moment exponentially stable.

In what follows, we study the relationship between p₁th moment exponential stability and p₂th moment exponential stability, where p₁ and p₂ are positive numbers.

Theorem 4.1. For any two real numbers p₁ and p₂ (0 < p₁ < p₂ < + ∞), if backward uncertain differential equation (1) is p₂th moment exponentially stable, then it is p₁th moment exponentially stable.

Proof. Assume that X_t and Y_t are two solutions of backward uncertain differential equation (1) with different final values x_T and y_T, respectively. According to the definition of p₂th moment exponential stability, there exists a pair of positive constants C₂ and λ₂ such that $\begin{matrix} E | X_{t} - Y_{t} |^{p_{2}} \\ \leq C_{2} | x_{T} - y_{T} |^{p_{2}} exp (λ_{2} (T - t)), \forall t \in [0, T] . \end{matrix}$

By using Hölder’s inequality, we have $\begin{matrix} E | X_{t} - Y_{t} |^{p_{1}} \\ = E [| X_{t} - Y_{t} |^{p_{1}} \cdot 1] \\ \leq \sqrt[\frac{p_{2}}{p_{1}}]{E | X_{t} - Y_{t} |^{p_{1} \cdot \frac{p_{2}}{p_{1}}}} \cdot \sqrt[\frac{p_{2}}{p_{2} - p_{1}}]{E [1^{\frac{p_{2}}{p_{2} - p_{1}}}]} \\ = \sqrt[\frac{p_{2}}{p_{1}}]{E | X_{t} - Y_{t} |^{p_{2}}} \\ \leq \sqrt[\frac{p_{2}}{p_{1}}]{C_{2} | x_{T} - y_{T} |^{p_{2}} exp (λ_{2} (T - t))} \\ = C_{2}^{\frac{p_{1}}{p_{2}}} | x_{T} - y_{T} |^{p_{1}} exp (λ_{2} \frac{p_{1}}{p_{2}} (T - t)) \\ = C_{1} | x_{T} - y_{T} |^{p_{1}} exp (λ_{1} (T - t)) \end{matrix}$ where $C_{1} = C_{2}^{\frac{p_{1}}{p_{2}}}$ and $λ_{1} = λ_{2} \frac{p_{1}}{p_{2}}$ .

According to Definition 4.1, backward uncertain differential equation (1) is p₁th moment exponentially stable. Thus, p₂th moment exponential stability implies p₁th moment exponential stability for backward uncertain differential equation (1) with p₁ < p₂, and the theorem is proved. □

Example 4.2. Consider backward uncertain differential equation (4).

Since $\begin{matrix} E | X_{t} - Y_{t} |^{2} \\ = | x_{T} - y_{T} |^{2} exp (2 (T - t)) \\ \leq | x_{T} - y_{T} |^{2} exp (2 (T - t)) \end{matrix}$ holds, backward uncertain differential equation (4) is 2th moment exponentially stable.

For any 0 < p < 2, we have $\begin{matrix} E | X_{t} - Y_{t} |^{p} \\ = E [| X_{t} - Y_{t} |^{p} \cdot 1] \\ \leq \sqrt[2 / p]{E | X_{t} - Y_{t} |^{p \cdot 2 / p}} \cdot \sqrt[2 / (2 - p)]{E [1^{2 / (2 - p)}]} \\ = \sqrt[2 / p]{E | X_{t} - Y_{t} |^{2}} \\ \leq \sqrt[2 / p]{| x_{T} - y_{T} |^{2} exp (2 (T - t))} \\ = | x_{T} - y_{T} |^{p} exp (p (T - t)) \\ = C_{1} | x_{T} - y_{T} |^{p} exp (λ_{1} (T - t)), \end{matrix}$ where C₁ = 1 and λ₁ = p.

Hence, backward uncertain differential equation (4) is pth moment exponentially stable with 0 < p < 2.

Wang and Ning [15] defined pth moment stability of backward uncertain differential equation (1) as follows.

Definition 4.2. (Wang and Ning [15]) Backward uncertain differential equation (1) is said to be stable in pth moment (0 < p < + ∞) if for any two solutions X_t and Y_t with different final values x_T and y_T, we have $lim_{| x_{T} - y_{T} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] = 0 .$

Now, we analyze the relationship between pth moment exponential stability and pth moment stability for backward uncertain differential equation (1), and the analysis result is stored in the following theorem.

Theorem 4.2. If backward uncertain differential equation (1) is pth moment exponentially stable, then it is pth moment stable.

Proof. Assume that X_t and Y_t are two solutions of backward uncertain differential equation (1) with different final values x_T and y_T, respectively. According to the definition of pth moment exponential stability, there exist a pair of positive constants C and λ such that $\begin{matrix} E | X_{t} - Y_{t} |^{p} \\ \leq C | x_{T} - y_{T} |^{p} exp (λ (T - t)), \forall t \in [0, T] . \end{matrix}$

Hence, the inequality $\begin{matrix} 0 & \leq & lim_{| x_{T} - y_{T} | \to 0} E | X_{t} - Y_{t} |^{p} \\ \leq & lim_{| x_{T} - y_{T} | \to 0} C | x_{T} - y_{T} |^{p} exp (λ (T - t)) = 0, \\ \forall t \in [0, T] \end{matrix}$ holds. And we immediately obtain $lim_{| x_{T} - y_{T} | \to 0} E | X_{t} - Y_{t} |^{p} = 0, \forall t \in [0, T]$ which means that pth moment exponential stability implies stability in pth moment, and the theorem is proved. □

Example 4.3. Consider backward uncertain differential equation (4).

According to Example 4.1, backward uncertain differential equation (4) is pth moment exponentially stable with any p > 0. That is, $E | X_{t} - Y_{t} |^{p} \leq C | x_{T} - y_{T} |^{p} exp (λ (T - t)),$ where X_t and Y_t represent the solutions with different final values x_T and y_T, respectively, C = 1 and λ = p.

Hence, we have $\begin{matrix} 0 & \leq & lim_{| x_{T} - y_{T} | \to 0} E | X_{t} - Y_{t} |^{p} \\ \leq & lim_{| x_{T} - y_{T} | \to 0} C | x_{T} - y_{T} |^{p} exp (λ (T - t)) = 0 \end{matrix}$ for any t ∈ [0, T], which means $lim_{| x_{T} - y_{T} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] = 0 .$

Hence, we conclude that backward uncertain differential equation (4) is pth moment stable.

Finally, we give a sufficient condition for backward uncertain differential equation (1) being pth moment exponentially stable.

Theorem 4.3. Assume that backward uncertain differential equation (1) has a unique solution with each given final value. Then it is pth moment exponentially stable if the coefficients f (t, x) and g (t, x) satisfy the condition

$\begin{matrix} | f (t, x_{1}) - f (t, x_{2}) | \leq λ / p | x_{1} - x_{2} |, \\ | g (t, x_{1}) - g (t, x_{2}) | \leq G (t) | x_{1} - x_{2} |, \\ \forall x_{1}, x_{2} \in R, t \in [0, T) \end{matrix}$ (7) for some positive constant λ, where G (t) is a positive function satisfying $\int_{0}^{T} G (t) d t < \frac{π}{\sqrt{3} p} .$

By the condition (7) and Theorem 2.1, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | x_{T} - y_{T} + \int_{t}^{T} [f (s, Y_{s} (γ)) - f (s, X_{s} (γ))] d s \\ + \int_{t}^{T} [g (s, Y_{s} (γ)) - g (s, X_{s} (γ))] d C_{s} (γ) | \\ \leq | x_{T} - y_{T} | + | \int_{t}^{T} [f (s, Y_{s} (γ)) - f (s, X_{s} (γ))] d s | \\ + | \int_{t}^{T} [g (s, Y_{s} (γ)) - g (s, X_{s} (γ))] d C_{s} (γ) | \\ \leq | x_{T} - y_{T} | + \int_{t}^{T} λ / p | X_{s} (γ) - Y_{s} (γ) | d s \\ + K (γ) \int_{t}^{T} G (s) | X_{s} (γ) - Y_{s} (γ) | d s \\ = | x_{T} - y_{T} | + \int_{t}^{T} (λ / p + K (γ) G (s)) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$

where K (γ) is the Lipschitz constant of C_t (γ).

It follows from Theorem 2.4 that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | x_{T} - y_{T} | exp (\int_{t}^{T} (λ / p + K (γ) G (s)) d s) \\ = | x_{T} - y_{T} | exp (\int_{t}^{T} λ / p d s) \\ exp (\int_{t}^{T} K (γ) G (s) d s) \\ \leq | x_{T} - y_{T} | exp (\int_{t}^{T} λ / p d s) \\ exp (\int_{0}^{T} K (γ) G (s) d s) \end{matrix}$ holds almost surely for any t ∈ [0, T], where K is a nonnegative uncertain variable such that $\begin{matrix} ℳ {γ \in Γ | K (γ) \geq x} \\ = 1 - ℳ {γ \in Γ | K (γ) < x} \\ \leq 2 {(1 + exp (\frac{π x}{\sqrt{3}}))}^{- 1} \end{matrix}$ by Theorem 2.2. Taking pth moment on both sides of the inequality $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | x_{T} - y_{T} | exp (\int_{t}^{T} λ / p d s) \\ exp (\int_{0}^{T} K (γ) G (s) d s), \end{matrix}$ we have $\begin{matrix} E {| X_{t} - Y_{t} |}^{p} \\ \leq {| x_{T} - y_{T} |}^{p} exp (p \int_{t}^{T} λ / p d s) \\ E [exp (pK \int_{0}^{T} G_{s} d s)] \\ = E [exp (pK \int_{0}^{T} G_{s} d s)] {| x_{T} - y_{T} |}^{p} \\ exp (λ (T - t)) . \end{matrix}$

Since $\int_{0}^{T} G_{s} d s < \frac{π}{\sqrt{3} p},$ it follows from definition of expected value that $\begin{matrix} E [exp (pK \int_{0}^{T} G_{s} d s)] \\ = \int_{0}^{+ \infty} ℳ {γ | exp (K (γ) p \int_{0}^{T} G_{s} d s) \geq x} d x \\ = \int_{0}^{+ \infty} ℳ {γ | K (γ) \geq \frac{ln x}{p \int_{0}^{T} G_{s} d s}} d x \\ \leq 2 \int_{0}^{+ \infty} \frac{1}{1 + x^{π / (p \sqrt{3} \int_{0}^{T} G_{s} d s)}} d x < + \infty . \end{matrix}$

Taking $C = E [exp (pK \int_{0}^{T} G_{s} d s)]$ , we have $\begin{matrix} E | X_{t} - Y_{t} |^{p} \\ \leq C | x_{T} - y_{T} |^{p} exp (λ (T - t)), \forall t \in [0, T] \end{matrix}$ and backward uncertain differential equation (1) is pth moment exponentially stable. The theorem is thus verified. □

Example 4.4. Consider backward uncertain differential equation (4).

As shown in Example 3.2, its solutions with different final values x_T and y_T are $X_{t} = (x_{T} - σ \int_{t}^{T} exp (s - T) d C_{s}) exp (T - t)$ and $Y_{t} = (y_{T} - σ \int_{t}^{T} exp (s - T) d C_{s}) exp (T - t),$ respectively. That is, backward uncertain differential equation (4) has a unique solution with each given final value.

Since f (t, x) = - x and g (t, x) = σ satisfy condition (7), uncertain differential equation (4) is pth moment exponentially stable by using Theorem 4.3.

5 Conclusion

In this paper, we first proposed the concepts of almost sure stability and pth moment exponential stability for backward uncertain differential equations. Furthermore, some sufficient conditions for backward uncertain differential equations being stable almost surely and pth moment exponentially were derived. Meanwhile, this paper gave a sufficient condition for a linear backward uncertain differential equation being almost surely stable. In the future, we will focus on the numerical methods for solving backward uncertain differential equations.

Footnotes

Acknowledgments

This work is supported by Natural Science Foundation of Shandong Province (ZR2016AP12), and a Project of Shandong Province Higher Educational Science and Technology Program (J17KB124).

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