Almost sure stability for multifactor uncertain differential equation

Abstract

In many cases, the uncertain factor influencing dynamic systems is not single. Multifactor uncertain differential equation is a type of differential equation driven by the multiple Liu processes. Some concepts of stability in measure and stability in mean for multifactor uncertain differential equations have been proposed. This paper focuses on presenting a concept of the almost sure stability of multifactor uncertain differential equation. A sufficient condition for a multifactor uncertain differential equation being almost surely stable are provided. In addition, this paper discusses some examples to illustrate the theoretical considerations.

Keywords

Uncertainty theory multifactor uncertain differential equation almost sure stability

1 Introduction

We know that a premise of applying probability theory is that the obtained distribution is close enough to the real frequency. However, sometimes we can not obtain data to estimate a probability distribution. In this case, we have to invite some experts to evaluate their belief degree about the possible events. According to Kahneman and Tversky [1] that human beings usually overweight unlikely events. This fact makes the belief degree have much larger variance than the frequency. As a result, probability theory is not applicable to model the expert’s belief degree.

In order to deal with the belief degrees, Liu [2] introduced an uncertainty theory in 2007 and perfected in 2009 [3]. For describe the evolution of an uncertain phenomenon, Liu [4] proposed an uncertain process as a sequence of uncertain variables indexed by time or space in 2008. One year later, Liu [3] presented a Liu process which is also a type of stationary independent increment process. Meanwhile, Liu [3] founded uncertain calculus to deal with the integral and differential of an uncertain process with respect to Liu process, Chen and Ralescu [5] proposed an uncertain integral with respect to general Liu process. Then Liu and Yao [6] extended uncertain integral from single Liu process to multiple ones.

In 2008, Liu [4] first presented a type of uncertain differential equation driven by Liu process. The existence and uniqueness theorem of solution of uncertain differential equation was first proved by Chen and Liu [7] under linear growth condition and Lipschitz continuous condition. Furthermore, this theorem was verified by Gao [8] under local linear growth condition and local Lipschitz continuous condition. Stability of uncertain differential equation has recently received a lot of attentions. The concept of stability in measure of uncertain differential equation was presented by Liu [3], and Yao et al. [9] proved some stability theorems of uncertain differential equation. Following that, Yao et al. [10] discussed stability in mean, Sheng and Wang [11] considered stability in p-th moment, Sheng and Gao [12] considered an exponential stability of uncertain differential equations, and Liu et al. [13] investigated almost sure stability.

In many real cases, uncertain factor influencing dynamic systems are usually not alone and maybe more one. In 2012, Liu and Yao [6] discussed the uncertain integral with respect to multiple Liu processes. Li et al. [14] proposed the multifactor uncertain differential equation that driven by multiple Liu processes, and they proved the existence and uniqueness theorem for its solution. After that, Zhang et al. [15] investigated the stability in measure and in mean for the solutions of multifactor uncertain differential equation. As an extension of the previous work on stability of multifactor uncertain differential equation the objective of this paper is to develop the almost sure stability of multifactor uncertain differential equation. The rest of this paper is organized as follows. In Section 2, this paper reviews some basic concepts about uncertain theory and uncertain calculus. Section 3 presents the concept of almost sure stability for a multifactor uncertain differential equation. A sufficient condition will be given in Section 4. Finally, some conclusions are given in Section 5.

2 Preliminaries

In this section, we review some preliminary concepts in uncertainty theory and uncertain calculus.

2.1 Uncertainty theory

Definition 1. (Liu [2]) 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 three axioms:

Axiom 1: (Normality Axiom) ℳ {Γ} =1 for the universal set Γ .

Axiom 2: (Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ.

Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯ , we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$

A set Λ ∈ ℒ is called an event. The uncertain measure ℳ {Λ} indicates the degree of belief that Λ will occur. The triplet (Γ, ℒ, ℳ) is called an uncertainty space.

Besides, in order to obtain a product uncertain measure, Liu [3] defined the product uncertain measure on the product σ-algebra ℒ as follows:

Axiom 4: (Product Axiom) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯ Then 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 are arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively.

An uncertain variable is essentially a measurable function from an uncertainty space to the set of real numbers. In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows.

Definition 2. (Liu [2]) The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for any $x \in R .$

The inverse function Φ^-1 is called the inverse uncertainty distribution of ξ.

Theorem 1. (Liu [16]) Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f is a strictly increasing function for ξ₁, ξ₂, ⋯ , ξ_m and f is a strictly decreasing function for ξ_m+1, ξ_m+2, ⋯ , ξ_n, then $ξ = f (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ is an uncertain variable with inverse uncertainty distribution $\begin{matrix} Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), Φ_{m + 2}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$

An uncertain variable ξ is said to be normal if it has a normal uncertainty distribution $Φ (x) = {(1 + exp (\frac{π (e - x)}{\sqrt{3} σ}))}^{- 1}, x \in R$ denoted by 𝒩 (e, σ) where e and σ are real numbers with σ > 0 . In this case, the uncertain variable exp(ξ) has a lognormal uncertainty distribution $Ψ (x) = {(1 + exp (\frac{π (e - ln x)}{\sqrt{3} σ}))}^{- 1}, x \in R^{+}$ denoted by ℒOGN (e, σ) .

2.2 Uncertain calculus

In order to model the evolution of uncertain phenomena, an uncertain process was proposed by Liu [4] as a sequence of uncertain variables driven by time or space. After that, Liu [3] designed a process which is one of the most important uncertain process, it is named as Liu process thereafter.

Definition 3. (Liu [3]) An uncertain process C_t is said to be a Liu process if

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

C_t has stationary and independent increments,

every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t², whose uncertainty distribution is $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Based on Liu process, Liu integral was defined as follows.

Definition 4. (Liu [3]) Let X_t be an uncertain process and C_t be Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$

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 integrable.

For example, an integrable function f (t) is an integrable uncertain process, and the uncertain integral $\int_{0}^{s} f (t) d C_{t} \sim N (0, \int_{0}^{s} | f (t) | d t)$ (1) is a normal uncertain variable.

Definition 5. (Liu [16]) Let C_t be Liu process and Z_t be an uncertain process. If there exist uncertain processes μ_s and σ_s such that $Z_{t} = Z_{0} + \int_{0}^{t} μ_{s} d s + \int_{0}^{t} σ_{s} d C_{s}$ for any t ≥ 0, then Z_t is said to be a general Liu process with drift μ_t and diffusion σ_t and has an uncertain differential $d Z_{t} = μ_{t} d t + σ_{t} d C_{t} .$

Definition 6. (Liu [4]) Suppose C_t is Liu process, and f and g are two given functions. Then $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (2) is called an uncertain differential equation. A solution is an uncertain process that satisfies (2) identically in t .

Theorem 2. (Chen and Liu [5]) Let C_t be a Liu process, and X_t an integrable uncertain process on [a, b] with respect to t. Then the inequality $| \int_{a}^{b} X_{t} (γ) d C_{t} (γ) | \leq K (γ) \int_{a}^{b} | X_{t} (γ) | d t$ (3) holds, where K (γ) is the Lipschitz constant of the sample path C_t (γ).

Theorem 3. (Yao etal. [9]) Let C_t be a Liu process. 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 $lim_{x \to + \infty} M {K \leq x} = 1$ (4) and

$\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3} σ}))}^{- 1} - 1 . \end{matrix}$ (5)

Definition 7. (Liu and Yao [6]) Suppose that C_1t, C_2t, ⋯ , C_nt are independent Liu processes, and Z_t is an uncetain process. If there are uncertain propesses μ_t and σ_it, i = 1, 2, ⋯ , n such that $Z_{t} = Z_{0} + \int_{0}^{t} μ_{s} d s + \sum_{i = 1}^{n} \int_{0}^{t} σ_{is} d C_{is}$ (6) for any t ≥ 0, then Z_t has an uncertain differential $d Z_{t} = μ_{t} d t + \sum_{i = 1}^{n} σ_{it} d C_{it}$ and Z_t is called a differentiable uncertain process with drift μ_t and diffusions σ_it, i = 1, 2, ⋯ , n.

Definition 8. (Li et al. [14]) Let C_1t, C_2t, ⋯ , C_nt be independent Liu processes, and f and g_i are given functions, i = 1, 2, ⋯ , n. Then $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ (7) is called a multifactor uncertain differential equation with respect to C_it, i = 1, 2, ⋯ , n. A solution is an uncertain process X_t that satisfies (7) identically at each time t .

A multifactor uncertain differential equation is essentially a type of differential equation driven by multiple Liu processes.

The uncertain differential Equation (7) and the uncertain integral Equation (8) $X_{t} = X_{0} + \int_{0}^{t} f (t, X_{s}) d s + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (t, X_{s}) d C_{is}$ (8) is equivalent.

Definition 9. (Zhang et al. [15]) Suppose $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it} .$ (9) is a multifactor uncertain differential equation. Then the Equation (9) is called stable in measure if $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \leq ε} = 1$ (10) for any given real number ε > 0; the Equation (9) is called stable in mean if $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |] = 0,$ (11) where X_t and Y_t are any two solutions with different initial values X₀ and Y₀.

3 Almost sure stability

In this section, we investigate a definition of almost sure stability for a multifactor uncertain differential equation.

Definition 10. A multifactor uncertain differential equation $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ (12) is said to be almost surely stable if for any solutions X_t and Y_t with different initial values X₀ and Y₀, we have $M {lim_{| X_{0} - Y_{0} | \to 0} | X_{t} - Y_{t} | = 0} = 1, \forall t > 0 .$ (13)

Example 1. Consider the multifactor uncertain differential equation $d X_{t} = μ d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} + σ_{3} C_{3 t} .$ (14)

Since the two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} = X_{0} + μ t + σ_{1} C_{1 t} + σ_{2} C_{2 t} + σ_{3} C_{3 t}, \\ Y_{t} = Y_{0} + μ t + σ_{1} C_{1 t} + σ_{2} C_{2 t} + σ_{3} C_{3 t}, \end{matrix}$ we have $| X_{t} - Y_{t} | = | X_{0} - Y_{0} |, \forall t > 0,$ so $M {lim_{| X_{0} - Y_{0} | \to 0} | X_{t} - Y_{t} | = 0} = 1, \forall t > 0 .$

Hence the multifactor uncertain differential Equation (15) is almost surely stable.

Example 2. Consider the multifactor uncertain differential equation $d X_{t} = - X_{t} d t + σ_{1} d C_{1 t} + σ_{2} C_{2 t} + σ_{3} C_{3 t} .$ (15)

Since the two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & exp (- t) X_{0} + σ_{1} exp (- t) \int_{0}^{t} exp (s) d C_{1 s} \\ + σ_{2} exp (- t) \int_{0}^{t} exp (s) d C_{2 s} \\ + σ_{3} exp (- t) \int_{0}^{t} exp (s) d C_{3 s}, \\ Y_{t} & = & exp (- t) Y_{0} + σ_{1} exp (- t) \int_{0}^{t} exp (s) d C_{1 s} \\ + σ_{2} exp (- t) \int_{0}^{t} exp (s) d C_{2 s} \\ + σ_{3} exp (- t) \int_{0}^{t} exp (s) d C_{3 s}, \end{matrix}$ we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = exp (- t) | X_{0} - Y_{0} | \\ \leq | X_{0} - Y_{0} |, \forall t > 0, γ \in Γ . \end{matrix}$

Then $M {lim_{| X_{0} - Y_{0} | \to 0} | X_{t} - Y_{t} | = 0} = 1 .$

Hence the multifactor uncertain differential Equation (15) is almost surely stable.

Example 3. Consider the multifactor uncertain differential equation $d X_{t} = X_{t} d t + σ_{1} d C_{1 t} + σ_{2} C_{2 t} + σ_{3} C_{3 t} .$ (16)

Since the two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & exp (t) X_{0} + σ_{1} exp (t) \int_{0}^{t} exp (- s) d C_{1 s} \\ + σ_{2} exp (t) \int_{0}^{t} exp (- s) d C_{2 s} \\ + σ_{3} exp (t) \int_{0}^{t} exp (- s) d C_{3 s}, \\ Y_{t} & = & exp (t) Y_{0} + σ_{1} exp (t) \int_{0}^{t} exp (- s) d C_{1 s} \\ + σ_{2} exp (t) \int_{0}^{t} exp (- s) d C_{2 s} \\ + σ_{3} exp (t) \int_{0}^{t} exp (- s) d C_{3 s}, \end{matrix}$ we have $| X_{t} (γ) - Y_{t} (γ) | = exp (t) | X_{0} - Y_{0} | \to + \infty, \forall γ \in Γ$ almost surely as t → ∞ . Hence the multifactor uncertain differential Equation (16) is not almost surely stable.

4 Stability theorem

In this section, we give a sufficient condition for a multifactor uncertain differential equation being almost surely stable.

Theorem 4. The multifactor uncertain differential equation $d X_{t} = f (t, X_{t}) + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ (17) is almost surely stable if the coefficient functions f (t, x) and g_i (t, x), i = 1, 2, ⋯ , n satisfy the linear growth condition $| f (t, x) | + \sum_{i = 1}^{n} | g_{i} (t, x) | \leq L (1 + | x |), \forall x \in R, t \geq 0$ (18) for some constant L, and the strong Lipschitz conditions $| f (t, x) - f (t, y) | \leq L_{1 t} | x - y |, \forall x, y \in R, t \geq 0$ (19) and

$\begin{matrix} \sum_{i = 1}^{n} | g_{i} (t, x) - g_{i} (t, y) | \\ \leq L_{2 t} | x - y |, \forall x, y \in R, t \geq 0 . \end{matrix}$ (20) where L_1t and L_2t are two functions satisfying $\int_{0}^{+ \infty} L_{1 s} d s < + \infty$ (21) and $\int_{0}^{+ \infty} L_{2 s} d s < + \infty .$ (22)

Proof. It follows from Li et al. [14] that for any given initial value, the multifactor uncertain differential Equation (17) has a unique solution. Assume that X_t and Y_t are two solutions of the multifactor uncertain differential Equation (17) with two different initial values X₀ and Y₀, respectively, i.e., $\begin{matrix} X_{t} & = & X_{0} + \int_{0}^{t} f (s, X_{s}) d s \\ + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (s, X_{s}) d C_{is}, \\ Y_{t} & = & Y_{0} + \int_{0}^{t} f (s, Y_{s}) d s \\ + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (s, Y_{s}) d C_{is} . \end{matrix}$

Then we have $\begin{matrix} X_{t} - Y_{t} \\ = (X_{0} - Y_{0}) + \int_{0}^{t} (f (s, X_{s}) - f (s, Y_{s})) d s \\ + \sum_{i = 1}^{n} \int_{0}^{t} (g_{i} (s, X_{s}) - g_{i} (s, Y_{s})) d C_{is} \\ = (X_{0} - Y_{0}) + \int_{0}^{t} (f (s, X_{s}) - f (s, Y_{s})) d s \\ + \int_{0}^{t} \sum_{i = 1}^{n} (g_{i} (s, X_{s}) - g_{i} (s, Y_{s})) d C_{is} . \end{matrix}$

According to the strong Lipschitz condition, we can obtain the in equality as follows, $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | \sum_{i = 1}^{n} (g_{i} (s, X_{s} (γ)) - g_{i} (s, Y_{s} (γ))) | d | C_{is} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | \sum_{i = 1}^{n} (g_{i} (s, X_{s} (γ)) - g_{i} (s, Y_{s} (γ))) | K_{i} (γ) d s \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | \sum_{i = 1}^{n} (g_{i} (s, X_{s} (γ)) - g_{i} (s, Y_{s} (γ))) | K (γ) d s \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} L_{1 s} | X_{s} (γ) - Y_{(} γ) | d s \\ + \int_{0}^{t} L_{2 s} K (γ) | X_{s} (γ) - Y_{s} (γ) | d s \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} (L_{1 s} + L_{2 s} K (γ)) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$

where K_i (γ) are the Lipschitz constants of C_it (γ), and K_i are independent uncertain variables, i = 1, 2, ⋯ , n, respectively. In here, we denoted $K (γ) = ⋁_{i = 1}^{n} K_{i} (γ)$ .

Then accordling to the Gronwall’s in equality, for any t ≥ 0, we can also obtain $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{t} L_{1 s} d s) \\ exp (K (γ) \int_{0}^{t} L_{2 s} d s) \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} d s) \\ exp (K (γ) \int_{0}^{+ \infty} L_{2 s} d s) . \end{matrix}$

Since $\int_{0}^{+ \infty} L_{1 s} d s < + \infty,$ and $\int_{0}^{+ \infty} L_{2 s} d s < + \infty,$ and K (γ) is finite, we can obtain that $| X_{t} (γ) - Y_{t} (γ) | \to 0$ as long as |X₀ - Y₀|→0. In other words, we have $M {lim_{| X_{0} - Y_{0} | \to 0} | X_{t} - Y_{t} | = 0} = 1 .$

Hence, it follows from the definition of almost sure stability that the multifactor uncertain differential equation is almost surely stable under the strong Lipschitz condition. The theorem is proved.

Example 4. Consider the multifactor uncertain differential equation

$\begin{matrix} d X_{t} & = & exp (- t) X_{t} d t + exp (- t^{2} - X_{t}^{2}) d C_{1 t} \\ + exp (- t^{2} - X_{t}^{2}) C_{2 t} . \end{matrix}$ (23)

Since the three functions f (t, x) = x exp(t) and g₁ (t, x) = exp(t² - x²) and g₂ (t, x) = exp(t² - x²) satisfy the linear growth condition $| f (t, x) | + | g_{1} (t, x) | + | g_{2} (t, x) | \leq 1 + | x |$ and the strong Lipschitz condition $\begin{matrix} | f (t, x) - f (t, y) | + | g_{1} (t, x) - g_{1} (t, y) | \\ + | g_{2} (t, x) - g_{2} (t, y) | \\ \leq (exp (t) + exp (t^{2}) + exp (t^{2})) | x - y | \end{matrix}$ for all $x, y \in R$ and t ≥ 0 . Hence the multifactor uncertain differential Equation (23) is almost surely stable.

Remark 1. Theorem 4 gives the sufficient condition but not the necessary condition for multifactor uncertain differential equation being almost surely stable.

Theorem 5. Suppose that u_1t, u_2t, v_it and w_it, i = 1, 2, ⋯ , n are real functions. Then the linear multifactor uncertain differential equation $d X_{t} = (u_{1 t} X_{t} + u_{2 t}) d t + \sum_{i = 1}^{n} (v_{it} X_{t} + w_{it}) d C_{it}$ (24) is almost surely stable if $\int_{0}^{+ \infty} | u_{1 t} | d t < + \infty$ (25) and $\int_{0}^{+ \infty} | v_{it} | d t < + \infty, i = 1.2 . \dots, n .$ (26)

Proof. Assume that X_t and Y_t are two solutions of the linear multifactor uncertain differential Equation (24) with two different initial values X₀ and Y₀, respectively, i.e., $\begin{matrix} d X_{t} & = & (u_{1 t} X_{t} + u_{2 t}) d t + \sum_{i = 1}^{n} (v_{it} X_{t} + w_{it}) d C_{it}, \\ d Y_{t} & = & (u_{1 t} Y_{t} + u_{2 t}) d t + \sum_{i = 1}^{n} (v_{it} Y_{t} + w_{it}) d C_{it} . \end{matrix}$

Then we have $\begin{matrix} d (X_{t} - Y_{t}) \\ = u_{1 t} (X_{t} - Y_{t}) d t + \sum_{i = 1}^{n} v_{it} (X_{t} - Y_{t}) d C_{it}, \end{matrix}$ and $\begin{matrix} X_{t} - Y_{t} \\ = (X_{0} - Y_{0}) exp (\int_{0}^{+ \infty} u_{1 t} d s) \\ exp (\sum_{i = 1}^{n} \int_{0}^{+ \infty} v_{it} d C_{it}) . \end{matrix}$

Thus, we have $\begin{matrix} | X_{t} - Y_{t} | \\ = | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} u_{1 t} d t) \\ \prod_{i = 1}^{n} exp (\int_{0}^{+ \infty} v_{it} d C_{it}) . \end{matrix}$

Hence, the linear multifactor uncertain differential Equation (24) is almost surely stable if $exp (\int_{0}^{+ \infty} u_{1 s} d s) < + \infty$ (27) and $\prod_{i = 1}^{n} exp (\int_{0}^{+ \infty} v_{it} d C_{it}) < + \infty .$ (28)

Since the in equality (27) is equivalent to the following in equality $\int_{0}^{+ \infty} | u_{1 t} | d t < + \infty .$

Note that $\int_{0}^{+ \infty} v_{it} d C_{it} \sim N (0, \int_{0}^{+ \infty} | v_{it} | d t)$ by the Equation (2). Thus the in equality (28) is equivalent to $\int_{0}^{+ \infty} | v_{1 t} | d t < + \infty, i = 1, 2, \dots, n .$ So we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} | X_{t} - Y_{t} | \\ = lim_{| X_{0} - Y_{0} | \to 0} | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} u_{1 t} d s) \\ \prod_{i = 1}^{n} exp (\int_{0}^{+ \infty} v_{it} d C_{it}) \\ = 0, \end{matrix}$ that is, $M {lim_{| X_{0} - Y_{0} | \to 0} | X_{t} - Y_{t} | = 0} = 1 . \forall t \geq 0$

Therefore it follows from the definition of almost sure stability, the linear multifactor uncertain differential Equation (24) is almost surely stable. The theorem is proved.

Example 5. Consider the uncertain differential equation $d X_{t} = (exp (- t) X_{t} + μ) d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} .$ (29)

Since u_1t = exp(- t), v_1t = 0 and v_2t = 0 satisfy the condition (25), the linear multifactor uncertain differential equation (29) is almost surely stable.

Example 6. Consider the uncertain differential equation $d X_{t} = X_{t} d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} .$ (30)

Since u_1t = 1 does not satisfy the condition (25), the multifactor uncertain differential Equation (30) is not almost surely stable.

5 Conclusions

This paper proposed a concept of almost surely stable for a multifactor uncertain differential equation, and some theorems on almost surely stable were proved, in which the sufficient conditions the generally multifactor uncertain differential equation being almost surely stable were provided. Meanwhile, this paper gave also a sufficient condition for a linear multifactor uncertain differential equation being almost surely stable.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grants Nos. 61262023, 61462086, 61563050) and Doctoral Fund of Xinjiang University (No. BS150206).

References

Kahneman

and Tversky

, Prospect Theory: An analysis of decision under risk, Econometrica 47(2) (1979), 263–292.

Liu

, Uncertainty Theory, 2nd ed, Springer-Verlag, Berlin, 2007.

Liu

, Some research problems in uncertainty theory, Journal of Uncertain Systems 3(1) (2009), 3–10.

Liu

, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems 2(1) (2008), 3–16.

Chen

and Ralescu

D.A.

, Liu process and uncertain calculus, Journal of Uncertainty Analysis Applications 1 (2013), Article 2.

Liu

and Yao

, Uncertain integral with respect to multiple canonical processes, Journal of Uncertain Systems 6(4) (2012), 250–255.

Chen

and Liu

, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making 9(1) (2010), 69–81.

Gao

, Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, Journal of Uncertain Systems 6(3) (2012), 223–232.

Yao

, Gao

and Gao

, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making 12(1) (2013), 3–13.

10.

Yao

, Ke

and Sheng

, Stability in mean for uncertain differential equation, Fuzzy Optimization and Decision Making 14(3) (2015), 365–379.

11.

Sheng

and Wang

, Stability in p-th moment for uncertain differential equation, Journal of Intelligent Fuzzy Systems 26(3) (2014), 1263–1271.

12.

Sheng

and Gao

, Exponential stability for uncertain differential equation, Soft Computing 20(9) (2016), 3673–3678.

13.

Liu

, Ke

and Fei

, Almost sure stability for uncertain differential equation, Fuzzy Optimization and Decision Making 13(4) (2014), 463–473.

14.

, Peng

and Zhang

, Multifactor uncertain differential equation, Journal of Uncertainty Analysis Applications 3 (2015), Article 7.

15.

Zhang

, Gao

and Yang

, The stability of multifactor uncertain differential equation, Journal of Intelligent Fuzzy Systems 30(6) (2016), 3281–3290.

16.

Liu

, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.