Stability in p -th moment for multifactor uncertain differential equation

Abstract

Uncertain differential equation plays an important role in dealing with dynamical systems with uncertainty. The uncertain factor of influencing dynamical system is not alone in many situations. Multifactor uncertain differential equation is a type of differential equation driven by multiple Liu processes. Stability analysis on uncertain differential equation is to investigate the qualitative properties, which is significant both in theory and application for uncertain differential equations. This paper focuses on the stability in p-th moment for multifactor uncertain differential equation, including the concept of stability in p-th moment, and the sufficient condition for multifactor uncertain differential equation being stable in p-th moment. The relationship between stability in p-th moment and stability in measure is also discussed. Moreover, applications in finance and population dynamics are documented.

Keywords

Project scheduling uncertainty theory uncertain differential equation stability uncertain process

1 Introduction

Probability theory has been widely used to deal with indeterminacy phenomenon for a long time. For further describing the irregular movement of the pollen in the liquid, Wiener [37] designed Wiener process in the framework of probability theory. Then stochastic calculus was founded by Ito [7] to deal with the integral and differential of a stochastic process with respect to Wiener process. Following that, stochastic differential equation was proposed and applied to many areas such as filtration [8] and finance [1].

As we know, a premise of applying probability theory is that the available probability distribution is close enough to the real frequency. However, we are lack of data to estimate the distribution via statistics in many cases. As a result, we have no choice but to invite some domain experts to evaluate the belief degree that each event happens. But the human beings usually estimate a much wider range of values than the object really takes, which was shown by Liu [17] through a survey. Therefore, it cannot be treated via probability theory. For dealing with the belief degree mathematically, an uncertainty theory was presented by Liu [10] in 2007, and refined by Liu [12] in 2009 based on normality, duality, subadditivity and product axioms. Until now, uncertainty theory has been widely applied to many fields, such as uncertain programming [12], uncertain risk analysis [14], uncertain inference [15], and uncertain logic [16].

In order to describe the evolution of an uncertain phenomenon, an uncertain process as a sequence of uncertain variables indexed by space or time was founded by Liu [21] in 2008. Later in 2009, Liu [12] designed a Liu process, which is an uncertain process with stationary and independent normal uncertain increments. Meanwhile, a theory of uncertain calculus was established by Liu [11] to handle the integral and differential of an uncertain process. Then Liu [20] extended uncertain integral from single Liu process to multiple ones. Driven by Liu process, uncertain differential equation was firstly proposed by Liu [21] as a type of differential equation in 2008. Since uncertain differential equation is an important tool to deal with dynamical systems in uncertain environments, it was widely studied in recent years, and a lot of discoveries in both theory and practice have been received. Chen and Liu [2] solved a linear uncertain differential equation under linear growth condition and Lipschitz continuous condition, and obtained its analytic solution in 2010. Furthermore, Gao [6] extended the existence and uniqueness theorem under the conditions that the coefficients are local Lipschitz continuous. After that, analytic methods were proposed by Liu [23] and Yao [41] to solve a particular class of nonliear uncertain differential equations, which were generalized by Liu [24] and Wang [36] later. More importantly, Yao-Chen formula presented by Yao and Chen [42] is an important theoretical result which established a relationship between ordinary differential equations and uncertain differential equations. Following that, Yao [43] investigated a numerical method to obtain the uncertainty distributions of extreme values and time integral of the solution of an uncertain differential equation. Based on above theoretical development, uncertain differential equation has been successfully applied in many fields such as uncertain finance [13], uncertain game [38], heat conduction [40] and wave equation [4].

With many applications of uncertain differential equations, the properties of them were also developed. The concept of stability for an uncertain differential equation was firstly presented by Liu [11] to describe the perturbation of the noise to the solutions in 2009, which has been widely investigated by many scholars. Yao et al. [44] gave a sufficient condition for an uncertain differential equation being stable in measure. After that, a sufficient condition for uncertain differential equation being stable in p-th moment was came up with Sheng and Wang [31], where p is a positive number. Then Yao et al. [45] discussed stability in mean for uncertain differential equation in 2015. Following that, Sheng and Gao [32] studied exponential stability for uncertain differential equation in 2016. Then Liu et al. [22] investigated the almost sure stability for uncertain differential equation. A sufficient condition is proved by Yang et al. [39] for an uncertain differential equation being stable in inverse distribution. Then Wang and Ning [35] presented a sufficient condition for uncertain delay differential equation being stable in p-th moment. Following that, Ma et al. [26] studied the sufficient condition for uncertain differential equation with jumps being stable in p-th moment.

Considering the case of the uncertain factor influencing dynamical systems is usually not single, uncertain integral was extended with respect to multiple Liu processes. Then the analytic solutions of multifactor differential equation was discussed by Li et al. [9], and the existence and uniqueness theorem for its solution was proved by that paper. After that, Zhang et al. [46] investigated stability in mean and stability in measure for the solutions of multifactor uncertain differential equation. Sheng et al. [33] presented a sufficient condition for a multifactor uncertain differential equation being almost surely stable. Following that, Ma et al. [25] gave a sufficient condition for a multifactor uncertain differential equation being stable in distribution and discussed the relationships among stability in distribution, stability in measure, and stability in mean for multifactor uncertain differential equation. In addition, some researchers investigated the properties of fuzzy differential equation. Based on the discussion on the properties of symmetric fuzzy numbers [28, 29], Qiu et al. [30] gave an existence and uniqueness theorem for a solution to a fuzzy differential equation in the quotient space of fuzzy numbers.

Inspired by above results, this paper will propose a concept of stability in p-th moment for multifactor uncertain differential equation and give a sufficient condition for multifactor uncertain differential equation being stable in p-th moment. The relationship between stability in p-th moment and stability in measure is also discussed. The rest of this paper is organized as follows. In next section, this paper reviews some preliminary knowledge in uncertainty theory and the definition of multifactor uncertain differential equation. In Sections 3 and 4, the stability in p-th moment and a sufficient condition for multifactor uncertain differential equation being stable in p-th moment are proposed, and the relationship between the stability in p-th moment and stability in measure for multifactor uncertain differential equation is discussed. Finally, the application in reality is documented.

2 Preliminaries

In this section, some preliminary concepts from uncertainty theory as needed are reviewed for further understanding the paper.

Definition 1. (Liu [10]) Let Γ be a nonempty set, and $L$ be a σ-algebra over Γ. A set function $M : L \to [0, 1]$ is called an uncertain measure if it satisfies the following 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 $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$

Note that the triplet $(Γ, L, M)$ is called an uncertainty space.

Besides, the product uncertain measure on the product σ-algebra was defined by Liu [11] as follows.

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 are arbitrarily chosen events from $L_{k}$ for k = 1, 2, ⋯, respectively.

Definition 2. (Liu [10]) 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, the set{γ ∈ Γ|ξ (γ) ∈ B} is an event in $L$ .

Definition 3. (Liu [10]) The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for any real number x.

Peng and Iwamura [27] proved that a function $Φ : R \to [0, 1]$ is an uncertainty distribution if and only if it is a monotone increasing function except Φ (x) ≡0 and Φ (x) ≡1. An uncertainty distribution Φ (x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to + \infty} Φ (x) = 1 .$

Definition 4. (Liu [18]) Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ.

Example 1. An uncertain variable ξ is called linear if it has a linear uncertainty distribution $Φ (x) = {\begin{matrix} 0, & if x < a \\ (x - a) / (b - a), & if a \leq x < b \\ 1, & if x \geq b \end{matrix}$ denoted by Ł (a, b), where a and b are real numbers with a < b. The inverse uncertainty distribution of linear uncertain variable Ł (a, b) is $Φ^{- 1} (α) = (1 - α) a + α b .$

Example 2. An uncertain variable ξ is called 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. The inverse uncertainty distribution of normal uncertain variable $N (e, σ)$ is $Φ^{- 1} (α) = e + \frac{σ \sqrt{3}}{π} ln \frac{α}{1 - α} .$

Definition 5. (Liu [10]) Let ξ be an uncertain variable on an uncertainty space ( $Γ, Ł, M$ ). Then its expected value E [ξ] is $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} dx - \int_{- \infty}^{0} M {ξ \leq x} dx$ provided that at least one of the two integrals is finite.

Theorem 1. (Liu [10]) Let ξ be an uncertain variable with uncertainty distribution Φ. Then $E [ξ] = \int_{0}^{+ \infty} (1 - Φ (x)) dx - \int_{- \infty}^{0} Φ (x) dx .$

An uncertain process is a sequence of uncertain variables indexed by time. A formal definition of uncertain process is stated as follows.

Definition 6. (Liu [21]) 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 $T \times (Γ, L, M)$ 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 7. (Liu [19]) Uncertain processes X_1t, X_2t, ⋯ , X_nt are said to be independent if for any positive integer k and any times t₁, t₂, ⋯ , t_k, the uncertain vectors $ξ_{i} = (X_{{it}_{1}}, X_{{it}_{2}}, \dots, X_{{it}_{k}}), i = 1, 2, \dots, n$ are independent, i.e., for any Borel sets B₁, B₂, ⋯ , B_n of k-dimensional real vectors, we have $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}} .$

An uncertain process X_t is said to have independent increments if $X_{t_{0}}, X_{t_{1}} - X_{t_{0}}, X_{t_{2}} - X_{t_{1}}, \dots, X_{t_{k}} - X_{t_{k - 1}}$ are independent uncertain variables where t₀ is the initial time and t₁, t₂, ⋯ , t_k are any times with t₀ < t₁ < ⋯ < t_k . An uncertain process X_t is said to have stationary increments if, for any given t > 0, the increments X_s+t - X_s are identically distributed uncertain variables for all s > 0.

Definition 8. (Liu [11]) 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_t+s - C_s is a normal uncertain variable with an uncertainty distribution $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

The expected value of a normal uncertain variable ξ is E [ξ] = e, and the expected value of a lognormal uncertain variable exp(ξ) is $E [exp (ξ)] = {\begin{matrix} \sqrt{3} σ exp (e) csc (\sqrt{3} σ), & if σ \leq π / \sqrt{3} \\ + \infty, & otherwise . \end{matrix}$ (1)

Definition 9. (Liu [11]) Let X_t be an uncertain process and let 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 $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$

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

Definition 10. (Liu [18]) Let C_t be a Liu process and let Z_t be an uncertain process. If there exist uncertain processes μ_t and σ_t 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 differentiable and has an uncertain differential $d Z_{t} = μ_{t} d t + σ_{t} d C_{t} .$

Theorem 2. (Yao et al. [44]) 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 $lim_{x \to + \infty} M {γ \in Γ | K (γ) \leq x} = 1$ and $\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq 2 {(1 + exp (\frac{- π y}{\sqrt{3}}))}^{- 1} - 1 . \end{matrix}$

Definition 11. (Li et al. [32]) Suppose C_1t, C_2t, ⋯ , C_nt are independent Liu processes, and f, g₁, g₂, ⋯ , g_n are some given functions. Then $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ (2) is called a multifactor uncertain differential equation with respect to C_1t, C_2t, ⋯ , C_nt. A solution is an uncertain process X_t that satisfies (2) identically in t.

The multifactor uncertain differential equation (2) is equivalent to the uncertain integral equation $X_{t} = X_{0} + \int_{0}^{t} f (t, X_{s}) ds + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (t, X_{s}) {dC}_{is}$ (3)

Definition 12. (Zhang et al. [46]) A multifactor uncertain differential equation (2) 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 $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \geq ɛ} = 0,$ for any given real number ɛ > 0 .

3 Stability in p-th moment

In this section, we discuss stability in p-th moment for multifactor uncertain differential equation and investigate a sufficient condition for multifactor uncertain differential equation being stable in p-th moment.

Definition 13. Let X_t and Y_t be two solutions of multifactor uncertain differential equation $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ with different initial values X₀ and Y₀, respectively. Then the multifactor uncertain differential equation (2) is said to be stable in p-th moment if $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] = 0, \forall p > 0 .$

Example 3. Consider the multifactor uncertain differential equation $d X_{t} = μ d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} + σ_{3} d C_{3 t} .$ (4)

The two solutions with different initial values X₀ and Y₀ are $X_{t} = X_{0} + μ t + σ_{1} C_{1 t} + σ_{2} C_{2 t} + σ_{3} C_{3 t}$ and $Y_{t} = Y_{0} + μ t + σ_{1} C_{1 t} + σ_{2} C_{2 t} + σ_{3} C_{3 t},$ respectively. Since $| X_{t} - Y_{t} | = | X_{0} - Y_{0} |, \forall t \geq 0,$ we have $sup_{t \geq 0} | X_{t} - Y_{t} |^{p} = | X_{0} - Y_{0} |^{p} .$

So we can get $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] = 0, \forall p > 0 .$

Thus, the multifactor uncertain differential equation (4) is stable in in p-th moment.

Example 4. Consider the multifactor uncertain differential equation $d X_{t} = X_{t} dt + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} + σ_{3} d C_{3 t} .$ (5)

Since the two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & exp (t) X_{0} + σ_{1} exp (t) \int_{0}^{t} exp (- s) {dC}_{1 t} \\ + σ_{2} exp (t) \int_{0}^{t} exp (- s) {dC}_{2 t} \\ + σ_{3} exp (t) \int_{0}^{t} exp (- s) {dC}_{3 t} \end{matrix}$ and $\begin{matrix} Y_{t} & = & exp (t) Y_{0} + σ_{1} exp (t) \int_{0}^{t} exp (- s) {dC}_{1 t} \\ + σ_{2} exp (t) \int_{0}^{t} exp (- s) {dC}_{2 t} \\ + σ_{3} exp (t) \int_{0}^{t} exp (- s) {dC}_{3 t}, \end{matrix}$ respectively. Since $| X_{t} - Y_{t} | = exp (t) | X_{0} - Y_{0} |, \forall t \geq 0,$ we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ = lim_{| X_{0} - Y_{0} | \to 0} E [{exp}^{p} (t) | X_{0} - Y_{0} |^{p}] = + \infty . \end{matrix}$

Therefore, the multifactor uncertain differential equation (5) is not stable in p-th moment.

Theorem 3. The multifactor uncertain differential equation $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ is stable in p-th moment if the coefficients f (t, x) and g_i (t, x) , i = 1, 2, ⋯ , n satisfy the linear growth condition $| f (t, x) | + \sum_{i = 1}^{n} | g (t, x) | \leq L (1 + | x |), \forall x \in R, t \geq 0$ for some constant L, and the strong Lipschitz condition $| f (t, x) - f (t, y) | \leq L_{1} (t) | x - y |, \forall x, y \in R, t \geq 0$ and $\begin{matrix} \sum_{i = 1}^{n} | g_{i} (t, x) - g_{i} (t, y) | \\ \leq L_{2} (t) | x - y |, \forall x, y \in R, t \geq 0, \end{matrix}$ where L₁ (t) and L₂ (t) are two positive functions satisfying $\int_{0}^{+ \infty} L_{1} (t) d t < + \infty$ and $\int_{0}^{+ \infty} L_{2} (t) d t < π / \sqrt{3} p .$

Proof. It follows from Li et al. [9] that for any given initial value, the multifactor uncertain differential equation (2) has a unique solution. Assume that X_t and Y_t are the solutions of the multifactor uncertain differential equation with different initial values X₀ and Y₀, respectively. Then for any Lipschitz continuous sample path C_it (γ) , i = 1, 2, …, n, we have $\begin{matrix} X_{t} (γ) & = & X_{0} + \int_{0}^{t} f (s, X_{s} (γ)) ds \\ + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (s, X_{s} (γ) {dC}_{is} (γ) \end{matrix}$ and $\begin{matrix} Y_{t} (γ) & = & Y_{0} + \int_{0}^{t} f (s, Y_{s} (γ)) ds \\ + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (s, Y_{s} (γ) {dC}_{is} (γ) . \end{matrix}$

By the strong Lipschitz condition, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | ds \\ + \sum_{i = 1}^{n} \int_{0}^{t} | g_{i} (s, X_{s} (γ) - g_{i} (s, Y_{s} (γ) | | {dC}_{is} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | ds \\ + \int_{0}^{t} K_{i} (γ) | \sum_{i = 1}^{n} (g_{i} (s, X_{s} (γ) - g_{i} (s, Y_{s} (γ)) | ds \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | ds \\ + \int_{0}^{t} K (γ) | \sum_{i = 1}^{n} (g_{i} (s, X_{s} (γ) - g_{i} (s, Y_{s} (γ)) | ds \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} L_{1} | X_{s} (γ) - Y_{s} (γ) | ds \\ + \int_{0}^{t} L_{2 s} K (γ) | X_{s} (γ) - Y_{s} (γ) | ds \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} (L_{1} + L_{2 s} K (γ)) | \\ X_{s} (γ) - Y_{s} (γ) | ds, \end{matrix}$ where K_i (γ) is the Lipschitz constant of the Lipschitz constant of C_it (γ), and K_i are independent uncertain variables, i = 1, 2, ⋯ , n, respectively. We denote $K (γ) = ⋁_{i = 1}^{n} K_{i} (γ)$ here. It follows from the Gr $\ddot{o}$ nwall’s inequality that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{t} L_{1 s} ds) \\ exp (K (γ) \int_{0}^{t} L_{2 s} ds) \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} ds) \\ exp (K (γ) \int_{0}^{+ \infty} L_{2 s} ds) . \end{matrix}$

Thus we have $\begin{matrix} | X_{t} - Y_{t} | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} ds) \\ exp (K (γ) \int_{0}^{+ \infty} L_{2 s} ds) \end{matrix}$ and $\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq 2 (1 + exp (- π x / \sqrt{3})^{- 1}) - 1 \end{matrix}$ by Theorem 3. Taking p-th moment on both sides, we have $\begin{matrix} E [| X_{t} - Y_{t} |^{p}] \\ \leq | X_{0} - Y_{0} |^{p} exp (p \int_{0}^{+ \infty} L_{1 s} ds) \\ E [exp (pK \int_{0}^{+ \infty} L_{2 s} ds)], \\ \forall t \geq 0 . \end{matrix}$

Since $\int_{0}^{+ \infty} L_{1 s} ds < + \infty,$ we immediately have $exp (p \int_{0}^{+ \infty} L_{1 s} ds) < + \infty .$

As for $E [exp (pK \int_{0}^{+ \infty} L_{2 s} ds)],$ we write $r = \int_{0}^{+ \infty} L_{2 s} ds < π / \sqrt{3} p$ for simplicity. It follows from the definition of expected value that $\begin{matrix} E [exp (rpK)] = \int_{0}^{+ \infty} M {exp (rpK) \geq x} dx \\ = \int_{0}^{1} M {exp (rpK) \geq x} dx \\ + \int_{1}^{+ \infty} M {exp (rpK) \geq x} dx \\ = 1 + \int_{1}^{+ \infty} M {exp (rpK) \geq x} dx \\ = 1 + r \int_{0}^{+ \infty} exp (rpy) M {K \geq y} dy \\ \leq 1 + r \\ \int_{0}^{+ \infty} exp (rpy) (1 - (2 (1 + exp (- π x / \sqrt{3}))^{- 1})) dy \\ = 1 + 2 r \int_{0}^{+ \infty} exp (rpy) (1 + exp (- π x / \sqrt{3}))^{- 1} dy \\ = 1 + 2 \int_{1}^{+ \infty} 1 / (1 + x^{π / (\sqrt{3} rp)}) dx < + \infty . \end{matrix}$

That is, $\begin{matrix} E [exp (pK \int_{0}^{+ \infty} L_{2 s} ds)] \\ \leq 1 + 2 \int_{1}^{+ \infty} 1 / (1 + x^{π / (p \sqrt{3} \int_{0}^{+ \infty} L_{2 s} ds)}) dx \\ < + \infty . \end{matrix}$

Hence, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] = 0, \forall p > 0$ and the multifactor uncertain differential equation (2) is stable in p-th moment. The theorem is proved. □

Example 5. Consider the multifactor uncertain differential equation

$\begin{matrix} d X_{t} & = & exp (- t) X_{t} d t + exp (- t - X_{t}^{2}) d C_{1 t} \\ + exp (- t^{2} - X_{t}^{2}) d C_{2 t} \\ + exp (- t^{2} - X_{t}^{2}) d C_{3 t} . \end{matrix}$ (6)

Since the four functions f (t, x) = x exp(- t), g₁ (t, x) = exp(- t - x²), g₂ (t, x) = exp(- t² - x²) and g₃ (t, x) = exp(- t² - x²) satisfy the linear growth condition $\begin{matrix} | f (t, x) | + | g_{1} (t, x) | + | g_{2} (t, x) | + | g_{3} (t, x) | \\ \leq 1 + | x | \end{matrix}$ and the strong Lipschitz condition $\begin{matrix} | f (t, x) - f (t, y) | + | g_{1} (t, x) - g_{1} (t, y) | \\ + | g_{2} (t, x) - g_{2} (t, y) | + | g_{3} (t, x) - g_{2} (t, y) | \\ \leq [exp (- t) + exp (- t^{2}) + exp (- t^{2}) \\ + exp (- t^{2})] | x - y | \\ \leq [exp (- t) + 3 exp (- t^{2})] | x - y |, \\ \forall x, y \in R, t \geq 0, \end{matrix}$ the multifactor uncertain differential equation (6) is stable in p-th moment.

Remark 1. The above theorem gives a sufficient (but not necessary) condition for multifactor uncertain differential equation being stable in p-th moment.

Here, we give an example to illustrate it.

Example 6. Consider the multifactor uncertain differential equation $d X_{t} = - X_{t} d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} + σ_{3} d C_{3 t} .$ (7)

Obviously, the solutions X_t and Y_t with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & exp (- t) X_{0} + σ_{1} exp (- t) \int_{0}^{t} exp (s) d C_{1 s} \\ + σ_{2} exp (- t) \int_{0}^{t} exp (s) d C_{2 s} \\ + σ_{3} exp (- t) \int_{0}^{t} exp (s) d C_{3 s}, \\ Y_{t} & = & exp (- t) Y_{0} + σ_{1} exp (- t) \int_{0}^{t} exp (s) d C_{1 s} \\ + σ_{2} exp (- t) \int_{0}^{t} exp (s) d C_{2 s} \\ + σ_{3} exp (- t) \int_{0}^{t} exp (s) d C_{3 s} . \end{matrix}$

Since $\begin{matrix} E | X_{t} - Y_{t} |^{p} & = & exp (- pt) | X_{0} - Y_{0} |^{p} \\ \leq & | X_{0} - Y_{0} |^{p}, \forall t \geq 0, \end{matrix}$ the multifactor uncertain differential equation (7) is stable in p-th moment. However, the coefficient f (x, t) = - x does not satisfy the strong Lipschitz condition.

Theorem 4. If the multifactor uncertain differential equation $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ is stable in p-th moment, then it is stable in measure.

Proof. Assume that X_t and Y_t are two solutions of the multifactor uncertain differential equation with different initial values X₀ and Y₀, respectively. Then it follows from Definition 3.1 that $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p}] = 0, \forall t > 0 .$

Then for any given real number ɛ > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M {| X_{t} - Y_{t} | \geq ɛ} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} \frac{E [| X_{t} - Y_{t} |^{p}]}{ɛ^{p}} = 0, \forall t > 0 \end{matrix}$ by Markov inequality. Thus stability in p-th moment implies stability in measure. □

Theorem 5. For any two real numbers 0< p₁ < p₂ < + ∞, if a multifactor uncertain differential equation is stable in p₂-th moment, then it is stable in p₁-th moment.

Proof. Assume that X_t and Y_t are two solutions of the multifactor uncertain differential equation with different initial values X₀ and Y₀. Then it follows from the definition of stability in p₂-th moment that $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p_{2}}] = 0, \forall t > 0 .$

By H $\ddot{o}$ lder’s inequality, we have $\begin{matrix} E [| X_{t} - Y_{t} |^{p_{1}}] \\ = E [| X_{t} - Y_{t} |^{p_{1}} \cdot 1] \\ \leq \sqrt[(\frac{p_{2}}{p_{1}})]{E [| X_{t} - Y_{t} |^{p_{1} \cdot \frac{p_{2}}{p_{1}}}]} \cdot \\ \sqrt[(\frac{p_{2}}{p_{2} - p_{1}})]{E [| X_{t} - Y_{t} |^{\frac{p_{2}}{p_{2} - p_{1}}}]} \\ = \sqrt[(\frac{p_{2}}{p_{1}})]{E [| X_{t} - Y_{t} |^{p_{2}}]}, \forall t > 0 . \end{matrix}$

Thus stability in p₂-th moment implies stability in p₁-th moment when p₁ < p₂. □

4 Linear multifactor uncertain differential equation

In this section, we will give a sufficient and necessary condition for a linear multifactor uncertain differential equation being stable in p-th moment.

Theorem 6. 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

$d X_{t} = (u_{1 t} X_{t} + u_{2 t}) d t + \sum_{i = 1}^{n} (v_{it} X_{t} + w_{it}) d C_{it}$ (8) is stable in p-th moment if

$\int_{0}^{+ \infty} | u_{1 t} | d t < + \infty$ (9) and

$\int_{0}^{+ \infty} | v_{it} | d t < \frac{π}{\sqrt{3} p}, i = 1, 2, \dots, n .$ (10)

Proof. Assume that X_t and Y_t are two solutions of the linear multifactor uncertain differential equation with two different initial values X₀ and Y₀, respectively, i.e.,

$\begin{matrix} d X_{t} & = & (u_{1 t} X_{t} + u_{2 t}) d t + \sum_{i = 1}^{n} (v_{it} X_{t} + w_{it}) d C_{it}, \\ d Y_{t} & = & (u_{1 t} Y_{t} + u_{2 t}) d t + \sum_{i = 1}^{n} (v_{it} X_{t} + w_{it}) d C_{it} . \end{matrix}$

Then we have

$\begin{matrix} d (X_{t} - Y_{t}) & = & [u_{1 t} (X_{t} - Y_{t})] d t \\ + \sum_{i = 1}^{n} [v_{it} (X_{t} - Y_{t})] d C_{it} . \end{matrix}$

Thus

$\begin{matrix} X_{t} - Y_{t} & = & (X_{0} - Y_{0}) exp (\int_{0}^{+ \infty} u_{1 s} ds) \\ exp (\sum_{i = 1}^{n} \int_{0}^{+ \infty} v_{is} {dC}_{is}) . \end{matrix}$

Taking p-th moment on both sides, we have $\begin{matrix} | X_{t} - Y_{t} |^{p} \\ = | X_{0} - Y_{0} |^{p} \cdot exp (p \int_{0}^{+ \infty} u_{1 s} ds) \\ \cdot \prod_{i = 1}^{n} exp (p \int_{0}^{+ \infty} v_{is} {dC}_{is}) . \end{matrix}$

Thus $\begin{matrix} E [| X_{t} - Y_{t} |^{p}] \\ = | X_{0} - Y_{0} |^{p} \cdot exp (p \int_{0}^{+ \infty} u_{1 s} ds) \\ \cdot E [\prod_{i = 1}^{n} exp (p \int_{0}^{+ \infty} v_{is} {dC}_{is})] . \end{matrix}$

Hence, the linear multifactor uncertain differential equation (8) is stable in p-th moment if and only if $exp (p \int_{0}^{+ \infty} u_{1 s} ds) < + \infty$ (11) and $E [\prod_{i = 1}^{n} exp (p \int_{0}^{+ \infty} v_{is} {dC}_{is})] < + \infty, \forall t \geq 0 .$ (12)

Clearly, the inequality (11) is equivalent to $\int_{0}^{+ \infty} | u_{1 s} | ds < + \infty .$

Since $p \int_{0}^{+ \infty} v_{is} {dC}_{is} \sim N (0, p \int_{0}^{+ \infty} v_{is} {dC}_{is}),$ the inequality (12) is equivalent to $\int_{0}^{+ \infty} v_{is} {dC}_{is} < \frac{π}{\sqrt{3} p}$ from the expected value of lognormal uncertain variable (1). The theorem is proved. □

Example 7. Consider the multifactor uncertain differential equation $d X_{t} = μ d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} + σ_{3} d C_{3 t} .$ (13)

Since u_1t = 0, v_1t = 0, v_2t = 0 and v_3t = 0 satisfy the conditions (9) and (10), the multifactor uncertain differential equation (13) is stable in p-th moment.

Example 8. Consider the multifactor uncertain differential equation $d X_{t} = X_{t} d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} + σ_{3} d C_{3 t} .$ (14) $σ_{1} > 0, σ_{2} > 0, σ_{3} > 0$

Since $\int_{0}^{t} u_{1 s} ds = + \infty$ does not satisfy the condition, the uncertain differential equation (14) is not stable in p-th moment.

5 Application in finance and population dynamics

In this section, we present examples of application of multifactor uncertain differential equation in finance and population dynamics.

5.1 Application in finance

Stock price model problem was widely investigated by scholars recently. Multifactor uncertain differential equations were firstly applied into an uncertain stock model by Liu [11] in 2009. Let m, a, σ and ω be real numbers and let C_1t, C_2t be independent Liu processes. Assume a stock price X_t follows a multifactor uncertain differential equation $d X_{t} = (m - {aX}_{t}) d t + σ d C_{1 t} + ω d C_{2 t} .$ (15)

By Theorem 5, the stock price is stable in p-th moment obviously.

5.2 Application in population dynamics

Uncertain differential equation was firstly applied in population dynamics by Sheng [34] in 2017. Let d be the mortality rate, b be the reproduction rate, P_t be the population of the species at time t. Then the rate of change can be expressed by the equation $\frac{d P_{t}}{d t} = {bP}_{t} - {dP}_{t} .$ (16)

Clearly, the solution of the equation (16) is $P_{t} = P_{0} exp ((b - d) t),$ (17) where P₀ is the initial population. However, it is unrealistic that the mortality rate and the reproduction rate are regarded as constants. Since resources, ecology environment change etc. influence the population at the same time, multifactor uncertain differential equation was presented for population dynamics by Zhang et al. [46] as follows $\frac{d P_{t}}{d t} = (b + σ_{1} \frac{d C_{1 t}}{d t}) P_{t} - (d + σ_{2} \frac{d C_{1 t}}{d t}) P_{t}$ (18) where b is the expected reproduction rate, σ₁ is the volatility for the reproduction rate, d is the expected mortality rate and σ₂ is the volatility for the mortality rate, C_1t and C_2t are independent Liu processes.

The equation (18) is equivalent to the following multifactor uncertain differential equation $d P_{t} = (b - d) P_{t} d t + σ_{1} P_{t} d C_{1 t} - σ_{2} P_{t} d C_{2 t} .$ (19)

If b = d, then the equation (19) is stable in p-th moment. Compared with ordinary differential equation, multifactor uncertain differential equation is more appropriate to model the population dynamics.

6 Conclusion

The concept of stability in p-th moment for multifactor uncertain differential equation was proposed in this paper, then the sufficient condition of stability in p-th moment for the multifactor uncertain differential equation was provided. In addition, the relationship between stability in p-th moment and stability in measure was discussed. Finally, applications in finance and population dynamics were documented.

In order to investigate the property of multifactor uncertain differential equation, the following problem is worth further extended. As mentioned in Introduction, we know that the stability of equation contains stability in mean, in measure, in p-th moment, in distribution, almost sure stability and so on. Scholars have obtained these stability of multifator uncertain differential equation, but the exponential stability of multifactor uncertain differential equation is also an interesting question that can be analyzed.

Footnotes

Acknowledgments

This work was supported by the Foundation of National Natural Science Foundation of China (Nos. 11471152 and 11601210).

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