Continuous dependence on solutions of uncertain differential equations via uncertain measure

Abstract

This paper presents some continuous dependence theorems on solutions of uncertain differential equations based on uncertain measure. We first introduce some properties on solution of uncertain differential equation. And then, we provide a continuous dependence theorem, and a continuity theorem to the initial value. In the proposed continuity theorem, the solution is regarded as a ternary function of initial values. Furthermore, we discuss how the solution continuously depends on initial value and parameter, and propose two theorems, namely, continuous dependence theorem on parameter, and continuity theorem on parameter to the initial value.

Keywords

Uncertain variable numerical series positive numerical series convergence in measure

1. Introduction

Nondeterministic phenomenon in dynamic system, such as perturbation and white noise, is usually described by random variable. In 1923, Wiener [1] developed Wiener process to handle dynamic systems with perturbation, which is a stochastic process with stationary and independent normal random increments. Based on Winer process, Ito [2] proposed stochastic calculus to deal with the integral and differential of stochastic processes with respect to Wiener process in mid-twentieth century. Following that, stochastic differential equation, a type of differential equation driven by Wiener process, was proposed and widely applied to many areas such as finance (Black and Scholes [3], Merton [4]) and filtration (Kalman and Bucy [5]). However, lots of surveys show that sometimes it is not suitable to regard the perturbation as random variable. As we know, a premise of applying probability theory is that we should obtain a probability distribution that is close enough to the real frequency via statistics. That’s to say, we should have enough samples. However, there are situations where people have none or no sufficient historical data. In this case, we have to invite some experts to evaluate their belief degree about the possible events. According to Kahneman and Tversky’s prospect theory [6], human beings tend to overweight unlikely events, so the belief degree generally has a much larger range than the real frequency. Additionally, Liu [7] showed that human beings usually estimate a much wider range of values that the object actually takes. Hence, it is inappropriate to treat the belief degree as a random variable and to model nondeterministic phenomenon in this case by probability theory.

To deal with belief degree mathematically, Liu [8] proposed uncertainty theory in 2007 and redefined in 2010 [9]. So far, uncertainty theory has been employed in many theoretical studies and applications. For a more detailed exposition of uncertainty theory, the readers may consult the recent book [7]. As an uncertain counterpart of wiener process, Liu [10] proposed a type of canonical Liu process that is a Lipschitz continuous uncertain process with stationary independent increments and increments are normal uncertain variables. Based on canonical Liu process, Liu [10] founded uncertain calculus to deal with the integral and differential of an uncertain process with respect to canonical process. Then Liu and Yao [11] extended uncertain integral from single canonical process to multiple ones. After that, Chen and Ralescu [12] proposed a type of Liu process via the uncertain integral of an uncertain process.

Uncertain differential equation, as a type of differential equations driven by the canonical Liu processes, was proposed by Liu [13]. In recent years, many researchers have done a lot of work about uncertain differential equation. Chen and Liu [14] proved existence and uniqueness theorem for the solutio of uncertain differential equation. Yao [15] presented some methods to solve two different types of nonlinear uncertain differential equation. Yao and Chen [16] proved that the solution of an uncertain differential equation can be represented by a spectrum of ordinary differential equations. Gao and Yao [17] provided a continuous dependence theorem on solution with respect to the parameters of an uncertain differential equation. Yao [18] studied uncertain differential equation with jumps. Li and Peng [19] proposed multifactor uncertain differential equation. Yao et al. [20] studied stability in mean for uncertain differential equation. Zhang et al. [21] proposed a hamming method for solving uncertain differential equations. Li et al. [22] gave an uncertain differential equation for SIS epidemic model. Chen and Li [23] proposed some uncertain differential mean value theorems and analysis the stability of the uncertain differential equation.

Continuous dependence theorem is a useful tool in the fundamental theory of mathematics, and it has been widely used in many fields. However, at present, there are few studies on employing uncertain measure to study the continuous dependence on solutions of the uncertain differential equation. Thus, in this paper, we will discuss some continuous dependence on solutions of uncertain differential equations via uncertain measure. The rest of the paper is organized as follows. In section 2, we will introduce some basic concepts and theorems about uncertainty theory and uncertain differential equation. In section 3, some continuous dependence theorem in uncertain measure are presented. At last, some conclusions are given in section 4.

2. Preliminaries

In order to model human’s belief degree, an uncertainty theory was founded by Liu [8] in 2007, and refined by Liu [9] in 2010. Uncertainty theory is developed based on the below four axioms.

Definition 1. [8] Let Γ be a nonempty set, and $L$ a σ-algebra on Γ. A set function $M$ 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 for any $Λ \in L$ ;

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

In this case, the triplet ( $Γ, L, M$ ) is called an uncertainty space.

Besides, a product axiom was given by [10] for the operation of uncertain variables in 2009.

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 $L_{kitsc}$ for k = 1, 2, …, respectively.

Definition 2. [8] Let ( $Γ, L, M$ ) be an uncertainty space. An uncertain variable ξ is a measurable function from Γ 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 3. [10] The uncertain variables ξ₁, ξ₂, . . . , ξ_n are said to be independent if $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, . . . , B_n of real numbers.

In order to describe an uncertain variable in practice, a concept of uncertainty distribution is defined below.

Definition 4. [8] The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for any real number x.

Definition 5. [9] An uncertain variable ξ with uncertainty distribution Φ is said to be regular, if its inverse function Φ^-1 (α) exists and is unique for each α ∈ (0, 1).

Theorem 1. [9] Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f (x₁, x₂, ⋯ , x_n) is strictly increasing with respect to x₁, x₂, ⋯ , x_m and and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n then ξ = f (ξ₁, ξ₂, ⋯ , ξ_n) is an uncertain variable with an inverse uncertainty distribution $\begin{matrix} Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), \dots, \\ Φ_{m}^{- 1} (α), Φ_{m + 1}^{- 1} (1 - α), \dots, \\ Φ_{n}^{- 1} (1 - α)) . \end{matrix}$

Definition 6. [10] An uncertain process C_t is said to be a canonical 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 normal uncertain variable with expected value 0 and variance t², whose uncertainty distribution is $Φ_{t} (x) = (1 + exp (- \frac{π x}{\sqrt{3} t}))^{- 1}, x \in R$

Definition 7. [10] Let X_t be an uncertain process and C_t be a canonical process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $▵ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$ Then the uncertain integral of X_t is defined by $\int_{a}^{b} X_{t} {dC}_{t} = lim_{▵ \to 0} Σ_{i = 1}^{k} X_{ti} \cdot (C_{ti + 1} - C_{ti})$

provided that the limit exists almost surely and is finite. In this case, the uncertain process X_t is said to be integrable.

Definition 8. [10] Suppose that C_t is a liu Process, f and g are continuous functions. Given an initial value X₀, the uncertain differential equation ${dX}_{t} = f (t, X_{t}) dt + g (t, X_{t}) {dC}_{t}$ (1) is called an uncertain equation with an initial value X₀.

For the sake of simplicity, we get the equivalent integral form of the Eq (1), which expressed as follows $X_{t} = X_{0} + \int_{0}^{t} f (s, X_{s}) ds + \int_{0}^{t} g (s, X_{s}) {dC}_{t} .$ (2)

Lemma 1. [15] Let C_t be a Liu process. Then there exists an uncertain variable K (γ) such that K (γ) is a Lipschitz constant of the sample path C_t (γ) for each γ, and $lim_{x \to + \infty} M {γ | K (γ) \leq x} = 1 .$

Definition 9. [8] The uncertain sequence ξ_n is said to be converges in measure to ξ if $lim_{n \to + \infty} M {| ξ_{n} - ξ | \leq ɛ} = 0,$ for every ɛ > 0.

Remark 1. According to Definition 8, we know that function f (x) is converges in measure to A if $lim_{x \to + \infty} M {| f (x) - A | \leq ɛ} = 0,$ for every ɛ > 0.

3. Continuous dependence on solutions of uncertain differential equations

In this section, we first present a continuous dependence theorem, and a continuity theorem to the initial value. Furthermore, we discuss how the solution continuously depends on initial value and parameter, and propose two theorems respectively.

3.1 Continuity theorems in uncertain measure

Theorem 2. (Existence and uniqueness theorem) [14] The uncertain differential equation dX_t = f (t, X_t) dt + g (t, X_t) dC_t has a unique solution if the coefficients f (t, x) and g (t, x) satisfy linear growth condition $| f (t, x) + g (t, x) | \leq L (1 + | x |), \forall x \in R, t \geq 0$ and the lipschitz condition $\begin{array}{l} | f (t, x) - f (t, y) | + | g (t, x) - g (t, y) | \leq L | x - y |, \\ \forall x, y \in R, t \geq 0 \end{array}$ for some constant L.

Theorem 3. (Symmetry theorem) Assume the uncertain differential equation dX_t = f (t, X_t) dt + g (t, X_t) dC_t satisfies the existence and uniqueness theorem, and X_t is the unique solution with initial value X_{t
₀} = X₀, for all t ∈ [0, T]. We write $X_{t} = φ (t, t_{0}, X_{0}) .$ Then, change the relative position of (t, X_t) and (t₀, X₀), we can obtain that $X_{0} = φ (t_{0}, t, X_{t}) .$

Proof. For any t₁ ∈ [0, T], we have X_{t
₁} = φ (t₁, t₀, X₀). It follows form the existence and uniqueness theorem that X₀ and X_{t
₁} with the same uncertain process. Thus $X_{t} = φ (t, t_{1}, X_{t_{1}}),$ where X₀ = φ (t₀, t₁, X_{t
₁}). By the arbitrariness of t₁, we have $X_{0} = φ (t_{0}, t, X_{t}) .$

The Theorem is proved.□

Lemma 2. Assume coefficient f (t, x) and g (t, x) are continuous and satisfy local lipshitz condition within a finite time interval [0, T]. Then for any two solution X_t and $X_{t}^{*}$ of the uncertain differential equation, there exists a positive constant L such that $\begin{array}{l} | X_{t} (γ) - X_{t}^{*} (γ) | \leq | X_{0} (γ) - X_{0}^{*} (γ) | \\ \cdot \exp (L (1 + K_{γ}) \cdot t) . \end{array}$ (3) where K_γ is the Lipschitz constant to the sample path C_t (γ).

Proof: Define $X_{t}^{v} (γ) = (X_{t} (γ) - X_{t}^{*} (γ))^{2}$ for each γ ∈ Γ. It follows from the linear growth and Lipschitz condition that $\begin{matrix} \frac{{dX}_{t}^{v} (γ)}{dt} & = 2 (X_{t} (γ) - X_{t}^{*} (γ)) \cdot (\frac{{dX}_{t} (γ)}{dt} - \frac{{dX}_{t}^{*} (γ)}{dt}) \\ = 2 (X_{t} (γ) - X_{t}^{*} (γ)) \cdot [(f (t, X_{t} (γ)) \\ - f (t, X_{t}^{*} (γ)) + K_{γ} (g (t, X_{t} (γ)) \\ - g (t, X_{t}^{*} (γ)))] \\ \leq 2 L (1 + K_{γ}) \cdot X_{t}^{v} (γ) . \end{matrix}$ Take $Φ_{t} = X_{t}^{v} (γ) \cdot exp (- 2 L (1 + K_{γ}) t)$ , next, we will consider monotonicity of Φ_t. Due to $\begin{matrix} \frac{d Φ_{t}}{dt} & = \frac{d (X_{t}^{v} (γ) \cdot exp (- 2 L (1 + K_{γ}) t))}{dt} \\ = exp (- 2 L (1 + K_{γ}) t) \cdot [\frac{{dX}_{t}^{v} (γ)}{dt} \\ - 2 L (1 + K_{γ}) \cdot X_{t}^{v} (γ)] \leq 0, \end{matrix}$ for each t ∈ [t₀, T]. So, we know that Φ_t is a monotonically decreasing function, and we can obtain Φ_t ≤ Φ₀. That means $X_{t}^{v} (γ) \leq X_{0}^{v} \cdot exp (L (1 + K_{γ}) \cdot t) .$

Therefore, for each t ∈ [t₀, T], $\begin{matrix} | X_{t} (γ) - X_{t}^{*} (γ) | \leq | X_{0} (γ) - X_{0}^{*} (γ) | \\ \cdot exp [L (1 + K_{γ}) \cdot t] . \end{matrix}$

The Theorem is proved.□

Example 1. Consider non-negative function φ (t) and f (t) satisfy $φ (t) \leq M + k \int_{0}^{t} φ (s) f (s) ds, t \in [0, T]$ where M and k are positive constant. Then there exists an event $M {φ (t) \geq M \cdot exp (k \int_{0}^{t} f (s) ds)} = 0 .$ Define $ν (t) = M + k \int_{0}^{t} φ (s) f (s) ds$ . Then, we have φ (t) ≤ ν (t). $\frac{d ν_{t}}{dt} = k φ (t) f (t),$ then, we can obtain $\frac{d ν_{t}}{dt} \leq k ν (t) f (t) .$ Furthermore, $\begin{matrix} \frac{d ν_{t}}{dt} \cdot exp (- k \int_{0}^{t} f (s) ds) \leq k ν (t) f (t) \\ \cdot exp (- k \int_{0}^{t} f (s) ds) \\ \frac{d ν_{t}}{dt} \cdot exp (- k \int_{0}^{t} f (s) ds) - k ν (t) f (t) \\ \cdot exp (- k \int_{0}^{t} f (s) ds) \leq 0 \\ = \frac{d}{dt} [ν (t) \cdot exp (- k \int_{0}^{t} f (s) ds)] \leq 0 . \end{matrix}$

Thus $ν (t) \cdot exp (- k \int_{0}^{t} f (s) ds) - ν_{0} \leq 0,$ That means $ν (t) \leq M \cdot exp (- k \int_{0}^{t} f (s) ds),$ Thus, we have $\begin{matrix} M {φ (t) \geq M \cdot exp (k \int_{0}^{t} f (s) ds)} \\ \leq M {ν (t) \geq M \cdot exp (k \int_{0}^{t} f (s) ds)} = 0 . \end{matrix}$

Theorem 4. (Continuous dependence theorem) Suppose coefficient f (t, x) and g (t, x) are continuous and satisfy local lipshitz condition within a finite time interval [0, T]. X_t is a solution of the uncertain differential equation, write as $\begin{matrix} X_{t} & = X_{0} + \int_{0}^{t} f (s, X_{s}) ds + \int_{0}^{t} g (s, X_{s}) {dC}_{s} \\ = φ (t, t_{0}, X_{0}), t \in [0, T] \end{matrix}$ with the initial value X₀. Then for any ɛ > 0, there exists a positive number σ = σ (ɛ, 0, T), when $M {({\bar{t}}_{0} - t_{0})^{2} + ({\bar{X}}_{0} - X_{0})^{2} > σ^{2}} = 0,$ such that $M {| φ (t, t_{0}, X_{0}) - φ (t, {\bar{t}}_{0}, {\bar{X}}_{0}) | \geq ɛ} = 0$

Proof. Due to coefficient f (t, x) and g (t, x) satisfy local lipshitz condition during the time interval [0, T], thus, when $M {({\bar{t}}_{0} - t_{0})^{2} + ({\bar{X}}_{0} - X_{0})^{2} > σ^{2}} = 0$ , by using Lemma 1, we have $\begin{array}{l} ℳ {| φ (t, t_{0}, X_{0}) - φ (t, {\bar{t}}_{0}, {\bar{X}}_{0}) | \geq ε} \\ = ℳ {| (X_{0} + \int_{0}^{t} f (s, X_{s}) d s + \int_{0}^{t} g (s, X_{s}) d C_{s}) \\ - ({\bar{X}}_{0} + \int_{0}^{t} f (s, {\bar{X}}_{s}) d t + \int_{0}^{t} g (s, {\bar{X}}_{s}) d C_{s}) | \geq ε} \\ \leq ℳ {| ({\bar{X}}_{0} - X_{0}) + \int_{0}^{t} f (s, X_{s}) - f (s, {\bar{X}}_{s}) d s \\ + \int_{0}^{t} g (s, X_{s}) - g (s, {\bar{X}}_{s}) d C_{s} | \geq ε} \\ \leq ℳ {| ({\bar{X}}_{0} - X_{0}) + L (1 + K_{γ}) \\ \int_{0}^{t} (X_{v} - {\bar{X}}_{v}) d v | \geq ε} \\ \leq ℳ {| ({\bar{X}}_{0} - X_{0}) + ({\bar{X}}_{0} - X_{0}) \\ \cdot \exp (L (1 + K_{γ}) t) | \geq ε} = 0. \end{array}$ (5)

Furthermore, we will consider the solution on the boundary. Suppose $X_{t}^{'} = ψ (t, t_{0}, X_{0})$ is the solution on the boundary of the uncertain differential equation. It follows from Theorem 3 that $\begin{array}{l} | φ_{t} (γ) - ψ_{t} (γ) | \leq | φ_{0} (γ) - ψ_{0} (γ) | \\ \cdot \exp (L (1 + K_{γ}) \cdot (t - t_{0})) . \end{array}$ (6)

Define $σ_{1} = ɛ \cdot \frac{exp (- L (1 + K_{γ}) \cdot (t - t_{0}))}{2}$ , there exists a positive number σ₂, when $M {| t - t_{0} | \geq σ_{2}} = 0$ , such that $M {| φ_{t} (γ) - φ_{0} (γ) | \geq σ_{1}} = 0 .$

Take σ = min {σ₁, σ₂}, then, when $M {({\bar{t}}_{0} - t_{0})^{2} + ({\bar{X}}_{0} - X_{0})^{2} > σ^{2}} = 0,$ we have $\begin{matrix} M {| φ (t, t_{0}, X_{0}) - ψ (t, t_{0}, X_{0}) |^{2} > ɛ^{2}} \\ \leq M {| (φ ({\bar{t}}_{0}, t_{0}, X_{0}) - ψ ({\bar{t}}_{0}, t_{0}, X_{0})) \\ \cdot exp (L (1 + K_{γ}) \cdot (t - t_{0})) |^{2} > ɛ^{2}} \\ \leq M {| [(φ ({\bar{t}}_{0}, t_{0}, X_{0}) - φ (t_{0}, t_{0}, X_{0})) \\ + (φ (t_{0}, t_{0}, X_{0}) - ψ ({\bar{t}}_{0}, t_{0}, X_{0}))] \\ \cdot exp (L (1 + K_{γ}) \cdot (t - t_{0})) |^{2} > ɛ^{2}} \\ \leq M {| 2 [(φ ({\bar{t}}_{0}, t_{0}, X_{0}) - φ (t_{0}, t_{0}, X_{0}))^{2} \\ + (φ (t_{0}, t_{0}, X_{0}) - ψ ({\bar{t}}_{0}, t_{0}, X_{0}))^{2}] \cdot \\ exp (2 L (1 + K_{γ}) \cdot (t - t_{0})) | > ɛ^{2}} \\ \leq M {| 2 [σ_{1}^{2} + ({\bar{X}}_{0} (γ) - X_{0} (γ))^{2}] \\ \cdot exp (2 L (1 + K_{γ}) \cdot (t - t_{0})) | > ɛ^{2}} \\ \leq M {| 4 σ_{1}^{2} \cdot exp (2 L (1 + K_{γ}) \cdot (t - t_{0})) | > ɛ^{2}} \\ = M {| 2 σ_{1} \cdot exp (L (1 + K_{γ}) \cdot (t - t_{0})) | > ɛ} = 0 \end{matrix}$ Thus, we obtain that $M {| φ (t, t_{0}, X_{0}) - ψ (t, t_{0}, X_{0}) | > ɛ} = 0 .$

The Theorem is proved.□

Example 2. Let X_t = φ (t, X₀, Y₀) and Y_t = ψ (t, X₀, Y₀) are two solutions of the equation ${\begin{matrix} \frac{{dX}_{t}}{dt} = X_{t} Y_{t} + t^{2}, \\ 2 \frac{{dY}_{t}}{dt} = - Y_{t}^{2} . \end{matrix}$ with the initial value φ (1, X₀, Y₀) = X₀ and ψ (1, X₀, Y₀) = Y₀. Consider the continuous dependence of solution on initial value X₀, Y₀. According to the equation $2 \frac{{dY}_{t}}{dt} = - Y_{t}^{2}$ , we obtain a specific solution Y_t ≡ 0. Then, by equation $\frac{{dX}_{t}}{dt} = X_{t} Y_{t} + t^{2}$ , we have $X_{t} = \frac{t^{2}}{3} + c$ . Clearly, the solution φ and ψ are both continuous function about (X₀, Y₀).

If y ≠ 0, by $2 \frac{{dY}_{t}}{dt} = - Y_{t}^{2}$ , we have $Y_{t} = \frac{2}{t + a}$ . Then, $\frac{{dX}_{t}}{dt} = X_{t} \frac{2}{t + a} + t^{2}$ . So $\begin{matrix} X_{t} & = exp (\int \frac{2}{t + c_{1}} dt) [c_{2} + \int t^{2} \\ exp (- \int \frac{2}{t + c_{1}} dt)] \\ = (t + c_{1})^{2} [c_{2} + \int (\frac{t}{t + c_{1}})^{2} dt] \\ = (t + c_{1})^{2} [c_{2} + \int (1 + \frac{c_{1}^{2}}{(t + c_{1})^{2}} - \frac{2 c_{1}}{t + c_{1}}) dt] \\ = (t + c_{1})^{2} [c_{2} + t - \frac{c_{1}^{2}}{t + c_{1}} - 2 c_{1} ln (t + c_{1})] \end{matrix}$ By the initial value X (1) = X₀ and Y (1) = Y₀, we obtain $c_{1} = \frac{2}{Y_{0}} - 1$ . Then, $\begin{matrix} Y & = ψ (t, X_{0}, Y_{0}) \\ = \frac{2}{t + \frac{2}{Y_{0}} - 1} \\ = \frac{2 Y_{0}}{2 + Y_{0} (t - 1)} \end{matrix}$ And $\begin{matrix} X_{0} & = (1 + c_{1})^{2} [c_{2} + 1 - \frac{c_{1}^{2}}{1 + c_{1}} - 2 c_{1} ln (1 + c_{1})] \\ = \frac{4}{Y_{0}^{2}} [c_{2} - 1 - \frac{2 - Y_{0}}{2} + 2 (\frac{2}{Y_{0}} - 1) ln \frac{2}{Y_{0}}] \end{matrix}$ where $c_{2} = \frac{X_{0} - Y_{0}^{2}}{4} + 1 + \frac{Y_{0} (\frac{2}{Y_{0}} - 1)^{2}}{2} + 2 (\frac{2}{Y_{0}} - 1)$ $ln \frac{2}{Y_{0}}$ . It is clear that, φ (t, X₀, Y₀) and Y_t = ψ (t, X₀, Y₀) are both continuous dependence on solution (X₀, Y₀).

Furthermore, when the solution X_t = φ (t, t₀, X₀) is regarded as a ternary function of variables t and initial values (t₀, X₀), according to the existence and uniqueness theorem and continuous dependence theorem on initial values, we can get that the solution X_t = φ (t, t₀, X₀) is continuous with respect to the ternary function. In conclusion, we can further extend the continuous dependence theorem of the solution to the continuity theorem in measure of the solution to the initial valueąč.

Theorem 5. (Continuity theorem to the initial value) Suppose coefficient f and g are continuous and satisfy local lipshitz condition on G_T. Define solution X_t = φ (t, t₀, X₀) as a ternary function on parameters t, t₀ and X₀. Then φ (t, t₀, X₀) is continuous in measure to parameters t, t₀ and X₀ during the finite interval G_T. That means, for any ɛ > 0, $(t^{'}, t_{0}^{'}, X_{0}^{'}) \in G_{T}$ , there exists a positive number σ, when $M {| (t^{'} - t)^{2} + (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} | > σ^{2}} = 0,$ such that $M {| φ (t, t_{0}, X_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'}) | > ɛ} = 0 .$

Proof: Clearly, the solution X_t = φ (t, t₀, X₀) is continuous at time t, thus, for any ɛ > 0, there exists a positive number σ₁, when $M {| t^{'} - t | \geq σ_{1}} = 0,$ we have $M {| φ (t, t_{0}, X_{0}) - φ (t^{'}, t_{0}, X_{0}) | > \frac{ɛ}{2}} = 0 .$ Furthermore, by Theorem 3 we can obtain that, for the ɛ, there exists a positive σ₂, when $M {| (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} | > σ_{2}^{2}} = 0,$ we have $M {| φ (t, t_{0}, X_{0}) - φ (t, t_{0}^{'}, X_{0}^{'}) | > \frac{ɛ}{2}} = 0 .$ Take σ = min {σ₁, σ₂}, $(t^{'}, t_{0}^{'}, X_{0}^{'}) \in G_{T}$ , when $M {| (t^{'} - t)^{2} + (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} | > σ^{2}} = 0,$ we have $\begin{matrix} M {| φ (t, t_{0}, X_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'}) | > ɛ} \\ = M {| (φ (t, t_{0}, X_{0}) - φ (t^{'}, t_{0}, X_{0})) \\ + (φ (t^{'}, t_{0}, X_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'})) | > ɛ} \\ \leq M {| φ (t, t_{0}, X_{0}) - φ (t^{'}, t_{0}, X_{0}) | > \frac{ɛ}{2}} \\ + M {| φ (t^{'}, t_{0}, X_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'}) | > \frac{ɛ}{2}} = 0 \end{matrix}$ Thus, φ (t, t₀, X₀) is continuous in measure to parameters t, t₀ and X₀ during the finite interval G_T.

The Theorem is proved.□

Example 3. Let X_t = φ_n (t, t₀, X₀) is the solutions of the equation $\frac{{dX}_{t}}{dt} = X_{t} \cdot exp (X_{t} + t^{2}) .$ with $φ_{n} (\frac{1}{n}) = \frac{1}{n^{2}}$ . Consider the function f (t, X_t) = X_t · exp(X_t + t²) is continuous and satisfy local Lipshitz condition on G_T, where the largest interval G_T is (- ∞ , + ∞). According to $(\frac{1}{n}, \frac{1}{n^{2}}) \to (0, 0)$ , when n→ ∞. We can obtain that, for any σ > 0, A, B ∈ R, $(t^{'}, t_{0}^{'}, X_{0}^{'}) \in [A, B]$ , there exists N > 0, when n > N, we have $M {| (t^{'} - t)^{2} + (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} | > σ^{2}} = 0 .$ It follows from Theorem 4 that, for any ɛ > 0, for the above σ, when n > N, φ_n (t) exist during the interval [A, B], and $M {| φ (t, t_{0}, X_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'}) | > ɛ} = 0 .$

3.2 Continuity dependence theorem on parameter

Next,we will further discuss the solution continuously depends on initial value and parameter in measure. And investigates continuous dependence theorems in uncertain differential equation. The uncertain differential equation depends on parameter is expressed as ${dX}_{t} = f (t, X_{t}, p) dt + g (t, X_{t}, p) {dC}_{t},$ (7) where p ∈ [-1, 1].

Lemma 3. The uncertain differential equation dX_t = f (t, X_t, p) dt + g (t, X_t, p) dC_t with parameter p has a unique solution if the coefficients f (t, X_t, p) and g (t, X_t, p) satisfy the linear growth condition $\begin{matrix} | f (t, X_{t}, p) | \leq L (1 + | X |), \\ | g (t, X_{t}, p) | \leq L (1 + | X |), \end{matrix}$ and Lipschitz condition $\begin{matrix} | f (t, X_{t}, p) - f (t, Y_{t}, p) | \leq L | X - Y |, \\ | g (t, X_{t}, p) - g (t, Y_{t}, p) | \leq L | X - Y |, \end{matrix}$ where L is Lipschitz constant.

Theorem 6. (Continuous dependence theorem on parameter) Suppose that the coefficient f (t, X_t, p) and g (t, X_t, p) are continuous and satisfy local Lipchitz condition in T_p, (t, X_t, p) ∈ T_p. X_t = φ (t, t₀, X₀, p₀) is the solution of the uncertain differential equation (6). Then, for any ɛ > 0, there exists a positive number σ > 0, when $M {({\bar{t}}_{0} - t_{0})^{2} + ({\bar{X}}_{0} - X_{0})^{2} + (p - p_{0})^{2} > σ^{2}} = 0,$ where $({\bar{X}}_{0} = {\bar{X}}_{t_{0}})$ , such that $M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t, {\bar{t}}_{0}, {\bar{X}}_{0}, p) | \geq ɛ} = 0 .$ (8)

Proof: Due to X_t = φ (t, t₀, X₀, p₀) is the solution of uncertain integral equation $\begin{matrix} φ (t, t_{0}, X_{0}, p_{0}) = X_{0} + \int_{0}^{t} f (s, X_{s}, p_{0}) ds \\ + \int_{0}^{t} g (s, X_{s}, p_{0}) {dC}_{s} . \end{matrix}$ So $\begin{matrix} M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t, {\bar{t}}_{0}, {\bar{X}}_{0}, p) | \geq ɛ} \\ = M {| X_{0} + \int_{0}^{t} f (s, X_{s}, p_{0}) ds \\ + \int_{0}^{t} g (s, X_{s}, p_{0}) {dC}_{s} - ({\bar{X}}_{0} + \int_{0}^{t} f (s, {\bar{X}}_{s}, p) ds \\ + \int_{0}^{t} g (s, {\bar{X}}_{s}, p) {dC}_{s}) | \geq ɛ} \\ = M {| (X_{0} - {\bar{X}}_{0}) + \int_{0}^{t} f (s, X_{s}, p_{0}) \\ - f (s, X_{s}, p) ds + \int_{0}^{t} g (s, X_{s}, p_{0}) \\ - g (s, X_{s}, p) {dC}_{s} \\ + \int_{0}^{t} f (s, X_{s}, p) - f (s, {\bar{X}}_{s}, p) ds \\ + \int_{0}^{t} g (s, X_{s}, p) - g (s, {\bar{X}}_{s}, p) {dC}_{s} | \geq ɛ} \\ \leq M {| (X_{0} - {\bar{X}}_{0}) + \int_{0}^{t} f (s, X_{s}, p_{0}) \\ - f (s, X_{s}, p) ds + \int_{0}^{t} g (s, X_{s}, p_{0}) \\ - g (s, X_{s}, p) {dC}_{s} | \geq \frac{ɛ}{2}} \\ + M {| \int_{0}^{t} f (s, X_{s}, p) - f (s, {\bar{X}}_{s}, p) ds \\ + \int_{0}^{t} g (s, X_{s}, p) - g (s, {\bar{X}}_{s}, p) {dC}_{s} | \geq \frac{ɛ}{2}} \\ = A_{1} + A_{2} . \end{matrix}$ (9) where $A_{1} = M {| (X_{0} - {\bar{X}}_{0}) + \int_{0}^{t} f (s, X_{s}, p_{0}) - f (s,$ $X_{s}, p) ds + \int_{0}^{t} g (s, X_{s}, p_{0}) - g . (s, X_{s}, p) . 5 {ptdC}_{s} | \geq \frac{ɛ}{2}}$ , $A_{2} = M {| \int_{0}^{t} f (s, X_{s}, p) - f . (s, {\bar{X}}_{s}, p) . 6 ptds + \int_{0}^{t} g (s,$ $X_{s}, p) - g (s, {\bar{X}}_{s}, p) {dC}_{s} | \geq \frac{ɛ}{2}}$ .

Due to X_t is continuous on t, t ∈ [0, T]. Then, for any ɛ > 0, define $σ_{2} = min {exp (- L (1 + K_{r}) t) \cdot \frac{ɛ}{2 LT}, \frac{ɛ}{6}}$ , there exists a positive number σ₁, when $M {| t - 0 | > σ_{1}} = 0$ such that $M {| X_{t} - X_{0} | > σ_{2}} = 0 .$ Moreover, due to f and g are continuous, we obtain that, there exists a positive number σ₃, when $M {| p - p_{0} | > \frac{ɛ}{6}} = 0$ , we have $M {| \int_{0}^{t} f (s, X_{s}, p_{0}) - f (s, X_{s}, p) ds | > \frac{ɛ}{6}} = 0,$ and $M {| \int_{0}^{t} g (s, X_{s}, p_{0}) - g (s, X_{s}, p) ds | > \frac{ɛ}{6 K_{r}}} = 0 .$ Take σ = min {σ₁, σ₂, σ₃}, when $M {({\bar{t}}_{0} - t_{0})^{2} + ({\bar{X}}_{0} - X_{0})^{2} + (p - p_{0})^{2} > σ^{2}} = 0,$ we have $\begin{matrix} A_{1} & = M {| (X_{0} - {\bar{X}}_{0}) + \int_{0}^{t} f (s, X_{s}, p_{0}) \\ - f (s, X_{s}, p) ds + \int_{0}^{t} g (s, X_{s}, p_{0}) \\ - g (s, X_{s}, p) {dC}_{s} | \geq \frac{ɛ}{2}} \\ \leq M {| (X_{0} - {\bar{X}}_{0}) | \geq \frac{ɛ}{6}} \\ + M {| \int_{0}^{t} f (s, X_{s}, p_{0}) \\ - f (s, X_{s}, p) ds | \geq \frac{ɛ}{6}} \\ + M {| K_{r} \int_{0}^{t} g (s, X_{s}, p_{0}) \\ - g (s, X_{s}, p) ds | \geq \frac{ɛ}{6}} = 0 . \end{matrix}$ Furthermore, by using Lemma 1, we have $\begin{matrix} A_{2} & = M {| \int_{0}^{t} f (s, X_{s}, p) - f (s, {\bar{X}}_{s}, p) ds \\ + \int_{0}^{t} g (s, X_{s}, p) - g (s, {\bar{X}}_{s}, p) {dC}_{s} | \geq \frac{ɛ}{2}} \\ \leq M {| L \cdot \int_{0}^{t} (X_{s} - {\bar{X}}_{s}) ds \\ + L \cdot \int_{0}^{t} (X_{s} - {\bar{X}}_{s}) {dC}_{s} | \geq \frac{ɛ}{2}} \\ \leq M {| (X_{0} - {\bar{X}}_{0}) \cdot exp (L (1 + K_{γ}) t) | \geq \frac{ɛ}{2}} \\ = 0 . \end{matrix}$ Therefore, $M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t, {\bar{t}}_{0}, {\bar{X}}_{0}, p) | \geq ɛ} = 0 .$

The Theorem is proved.□

Similarly, we regard the solution Xt = φ (t, t₀, X₀, p₀) as a quaternary function of variables t and initial values (t₀, X₀, p₀). Then, we can obtain the Continuity theorem on parameter in measure to the initial value.

Theorem 7. (Continuity theorem on parameter to the initial value) Suppose coefficient f and g are continuous and satisfy local lipshitz condition on T_p. Define solution X_t = φ (t, t₀, X₀, p₀) as a quaternary function on parameters t, t₀, X₀ and p₀. Then φ (t, t₀, X₀, p₀) is continuous in measure to parameters t, t₀, X₀ and p₀ during the finite interval T_p. That means, for any ɛ > 0, $(t^{'}, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) \in T_{p}$ , there exists a positive number σ, when $\begin{matrix} M {| (t^{'} - t)^{2} + (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} \\ + (p_{0}^{'} - p_{0})^{2} | > σ^{2}} = 0, \end{matrix}$ such that $M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) | > ɛ} = 0 .$

Proof: According to Theorem 5, for any ɛ > 0, $(t, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) \in T_{p}$ , there exists a positive number σ₂, when $\begin{matrix} M {| (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} \\ + (p_{0}^{'} - p_{0})^{2} | > σ_{2}^{2}} = 0, \end{matrix}$ such that $M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) | > \frac{ɛ}{2}} = 0 .$ Furthermore, it is clear that X_t = φ (t, t₀, X₀, p₀) is continuous at time t, then, there exists a positive number σ₁ that, when $M {| t - t^{'} | > σ_{1}} = 0,$ we have $M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t^{'}, t_{0}, X_{0}, p_{0}) | > \frac{ɛ}{2}} = 0 .$ Take σ = min {σ₁, σ₂}, $(t^{'}, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) \in T_{p}$ , when $\begin{matrix} M {| (t^{'} - t)^{2} + (t_{0}^{'} - t_{0})^{2} + (X_{0}^{'} - X_{0})^{2} \\ + (p_{0}^{'} - p_{0})^{2} | > σ^{2}} = 0, \end{matrix}$ we have $\begin{matrix} M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t^{'}, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) | > ɛ} \\ \leq M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t^{'}, t_{0}, X_{0}, p_{0}) | > \frac{ɛ}{2}} \\ + M {| φ (t, t_{0}, X_{0}, p_{0}) - φ (t, t_{0}^{'}, X_{0}^{'}, p_{0}^{'}) | > \frac{ɛ}{2}} \\ = 0 . \end{matrix}$ Therefore, the solution φ (t, t₀, X₀, p₀) is continuous in measure to parameters t, t₀, X₀ and p₀ during the limited interval T_p.

The Theorem is proved.□

Example 4. Let f (t, X_t), g (t, X_t), $\frac{\partial f}{\partial X}$ and $\frac{\partial g}{\partial X}$ are continuous function and f, g satisfy Lipschitz condition on T. X_t = φ (t, t₀, X₀) is solution of function $\frac{dX}{dt} = f (t, X_{t}) dt + g (t, X_{t}) {dC}_{t}$ with initial value φ (t₀, t₀, X₀) = X₀. Then, consider function $\frac{\partial φ}{\partial X}$ existence and continuous. Due to f, g, $\frac{\partial f}{\partial x}$ and $\frac{\partial g}{\partial x}$ continuous and f, g satisfy Lipschitz condition on T. It follows from Theorem 4 that, the solution X_t = φ (t, t₀, X₀) is continuous in measure on t, t₀ and X₀.

Suppose X_t = φ (t, t₀, X₀) and Y_t = ψ (t, t₀, X₀ + ΔX₀) are two solution of the uncertain differential equation with initial value (t₀, X₀) and (t₀, X₀ + ΔX₀) (ΔX₀ ≤ α, α → 0). So, we have ${\begin{matrix} φ = X_{0} + \int_{0}^{t} f (t, φ) dt + \int_{0}^{t} g (t, φ) {dC}_{t}, \\ ψ = (X_{0} + Δ X_{0}) + \int_{0}^{t} f (t, ψ) dt + \int_{0}^{t} g (t, ψ) {dC}_{t} . \end{matrix}$ Then $\begin{matrix} ψ - φ & = Δ X_{0} + \int_{0}^{t} f (t, ψ) - f (t, φ) dt \\ + \int_{0}^{t} g (t, ψ) - g (t, φ) {dC}_{t} \\ = Δ X_{0} + \int_{0}^{t} \frac{\partial f (t, φ + θ (φ - ψ))}{\partial x} (ψ - φ) dt \\ + \int_{0}^{t} \frac{\partial g (t, φ + θ (φ - ψ))}{\partial x} (ψ - φ) {dC}_{t} \end{matrix}$ Due to $\frac{\partial f}{\partial X}$ and $\frac{\partial g}{\partial X}$ are continuous. So, we obtain $\frac{\partial f (t, φ + θ (φ - ψ))}{\partial x} = \frac{\partial f (t, φ)}{\partial x} + γ_{1},$ and $\frac{\partial g (t, φ + θ (φ - ψ))}{\partial x} = \frac{\partial g (t, φ)}{\partial x} + γ_{2} .$ where γ₁ → 0, γ₂ → 0 (when ΔX₀ → 0) and γ₁ = γ₂ = 0 (when ΔX₀ = 0).

Then, when ΔX₀ ≠ 0, we have $\begin{matrix} \frac{ψ - φ}{Δ X_{0}} & = 1 + \int_{0}^{t} (\frac{\partial f (t, φ)}{\partial x} + γ_{1}) \cdot \frac{(ψ - φ)}{Δ X_{0}} dt \\ + \int_{0}^{t} (\frac{\partial g (t, φ)}{\partial x} + γ_{2}) \cdot \frac{(ψ - φ)}{Δ X_{0}} {dC}_{t} \end{matrix}$ Take $Z = \frac{ψ - φ}{Δ X_{0}}$ , we can obtain the uncertain differential equation $\frac{dZ}{dX} = [(\frac{\partial f (t, φ)}{\partial x} + γ_{1}) + K_{γ} (\frac{\partial g (t, φ)}{\partial x} + γ_{2})] \cdot Z$ with initial value Z₀ = 1. Where ΔX₀ ≠ 0 as a parameter of the function. It follows from Theorem 6 that, $Z = \frac{ψ - φ}{Δ X_{0}}$ is continuous about the parameters t, t₀, Z₀ and ΔX₀. Thus $lim_{Δ X_{0} \to 0} \frac{ψ - φ}{Δ X_{0}} = \frac{\partial φ}{\partial X_{0}} .$ Furthermore, we can get $\frac{\partial φ}{\partial X_{0}} = exp (\int_{0}^{t} \frac{\partial f}{\partial X} + \int_{0}^{t} \frac{\partial g}{\partial X})$ is the solution of Eq (9). Clearly, $\frac{\partial φ}{\partial X_{0}}$ is continuous function about parameters t, t₀ and X₀.

4. Conclusions

This paper presented some continuity theorems on solutions of uncertain differential equations in measure. Firstly, we introduced some properties of continuous dependence on solution, and presented some theorems of continuous dependence theorems on solutions in measure. Secondly, we discussed how the solution depends on parameter is continuous and dependence of the uncertain differential equation.

Footnotes

Acknowledgments

This research was supported by the Beijing Municipal Education Commission Foundation of China (No. KM201810038001), the Great Wall Scholar Training Program of Beijing Municipality (CIT&TCD20190338).

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