Parameter estimation of simple uncertain differential equations

Abstract

Uncertain differential equation has become an important tool to deal with uncertain dynamic systems such as finance, control and medical fields. The paper aims to study the problem of estimating unknown parameters in uncertain differential equations (UDEs). Least-square method is introduced to estimate unknown parameters of a class of simple UDEs. Further, two least-square estimators of a simple UDE are obtained. The simulation results show that the proposed method is feasible for estimating unknown parameters of some UDEs.

Keywords

Uncertain measure uncertain differential equation parameter estimation least-square method

1 Introduction

Uncertain differential equation (UDE) proposed by Liu [2] is a type of differential equations defined by Liu process. It is an important tool to deal with uncertain dynamic systems in finance, control and medical fields. Liu [3] proposed the first uncertain stock model. Chen [12] obtained the American option pricing formula in uncertain financial market. Uncertain interest rate model was introduced by Chen and Gao [14]. Yao [11] provided a no-arbitrage theorem for uncertain stock model. Liu [18] studied uncertain currency model. Yao [8] applied UDE to a stock model with floating interest rate. Besides, Zhu [20] discussed uncertain optimal control model. Recently, Li et al. [22, 23] and Fang et al. [6] established uncertain SIS epidemic models to study the dynamic properties of epidemic. In order to investigate properties of UDEs, many researchers analyzed the analytic solution of UDEs. Chen and Liu [13] provided the analytic solution of a linear UDE. Liu [17] introduced an analytic method for solving UDEs. Yao [7] studied a type of UDEs with analytic solution. More details about the topic to see Wang [21], Yao [7], Yang and Sheng [15], Sheng et al. [19], Zhou and Li [5].

However, there exist always some unknown parameters in uncertain differential equations. In order to analyze the dynamic properties of UDEs, it is key to obtain the estimators of unknown parameters. Classical methods such as the methods of least-square and moment have been commonly employed in the estimation of unknown parameters in uncertainty distribution of uncertain variable. For example, Liu [4] applied the least-square method to estimate the unknown parameters in the uncertainty distribution. Wang and Peng [16] used the method of moments to estimate those of uncertainty distributions. Yao and Liu [10] introduced uncertain regression analysis to estimate the relationships among the response variable and the explanatory variables by the principle of least squares.

Different from uncertain variable, uncertain differential equation is the evolution of uncertain phenomenon indexed by time or spaces. Until now, estimating unknown parameters of UDEs has not been done and analyses have not been made. In this paper, we provide a method to estimate unknown parameters of a class of simple uncertain differential equations. The method can be extended to solve such uncertain differential equation, which its solution has the explicit expression of uncertainty distribution.

The rest is organized as follows. Definitions and notation of uncertainty theory are reviewed in Section 2. In Section 3, we introduce a class of simple uncertain differential equations with unknown parameters and obtain the corresponding solution and uncertainty distribution. Based on the least-square method, we obtain the estimators of unknown parameters. As an application, two least-square estimators of a simple uncertain differential equation are obtained in Section 4. Discussions are given in Section 5.

2 Preliminaries

Let ℒ be a σ-algebra on a non-empty set Γ. Every element Λ ∈ ℒ is called an event.

Definition 1. [1] A set function ℳ: ℒ → [0, 1] is called an uncertain measure if it is supposed to satisfy four 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, . . . , $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$ The triplet (Γ, ℒ, ℳ) is called an uncertainty space.

Axiom 4 (Product Axiom) Let (Γ_i, ℒ_i, ℳ_i) be uncertainty spaces and Λ_i be arbitrarily chosen events from ℒ_i for i = 1, 2, . . . , then the product uncertain measure ℳ is an uncertain measure satisfying $ℳ {\prod_{i = 1}^{\infty} Λ_{i}} = ⋀_{i = 1}^{\infty} ℳ_{i} {Λ_{i}},$ where a ∧ b = min {a, b}.

Definition 2. [1] An uncertain variable ξ is a measurable function from uncertainty space (Γ, ℒ, ℳ) to the set of real numbers $ℝ$ ; that is, for any Borel set B of $ℝ$ , the set {ξ ∈ B} = {γ ∈ Γ|ξ (γ) ∈ B} is an event.

For any real number $x \in ℝ$ , uncertainty distribution Φ of uncertain variable ξ is defined by Φ (x) = ℳ {ξ ≤ x}. 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 $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to + \infty} Φ (x) = 1 .$ If the inverse function Φ^-1 exists and is unique for each α ∈ (0, 1), then it is called inverse uncertainty distribution of ξ, denoted by Φ^-1 (α).

An uncertain variable ξ is said to be normal if it has uncertainty distribution $\begin{matrix} Φ (x) = (1 + exp (\frac{π (e - x)}{\sqrt{3} σ}))^{- 1}, x \in ℝ, \end{matrix}$ denoted by ξ ∼ 𝒩 (e, σ), where e and σ are real numbers with σ > 0. Evidently, the inverse uncertain distribution of ξ is $\begin{matrix} Φ^{- 1} (α) = e + \frac{\sqrt{3} σ}{π} ln \frac{α}{1 - α}, α \in (0, 1) . \end{matrix}$ Especially, if e = 0 and σ = 1, ξ is called standard normal, denoted by ξ ∼ 𝒩 (0, 1).

Let T be an index set, and (Γ, ℒ, ℳ) be an uncertainty space. An uncertain process X_t is a measurable function from T × (Γ, ℒ, ℳ) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {X_t ∈ B} = {γ|X_t (γ) ∈ B} is an event. An uncertain process X_t is said to have an uncertainty distribution Φ_t (x), if at each time t uncertain variable X_t has uncertainty distribution Φ_t (x).

Definition 3. [2] An uncertain process X_t is said to have independent increments if X_{t
₁}, X_{t
₂} - X_{t
₁}, …, X_{t
_u} - X_{t
_u-1} are independent uncertain variables, where t₁, t₂, …, t_u are any times with t₁ < t₂ < ⋯ < t_u.

Definition 4. [2] An uncertain process is said to be a stationary independent increment process if it has not only stationary increments but also independent increments.

Definition 5. [3] An uncertain process C_t is said to be Liu process, if

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

C_t has stationary and independent increments with C₀ = 0.

Every increment C_t+s - C_s is a normal uncertain variable with an uncertainty distribution $Φ_{t} (x) = (1 + exp (- \frac{π x}{\sqrt{3} t}))^{- 1}, x \in ℝ .$

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_u+1 = b, the mesh is written as $△ = max_{1 \leq i \leq u} | t_{i + 1} - t_{i} |$ . Then, Liu integral of X_t is defined by $\int_{a}^{b} X_{t} {dC}_{t} = lim_{△ \to 0} \sum_{i = 1}^{u} X_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite. Let f and g be two given functions. Then

${dX}_{t} = f (t, X_{t}) dt + g (t, X_{t}) {dC}_{t}$ (1) is called an uncertain differential equation (UDE). An uncertain process X_t is called a solution of UDE (1), if X_t satisfies the UDE (1) for any time t.

Definition 6. [3] An uncertain differential equation is said to be stable if for any two solutions X_t and Y_t, we have $lim_{| X_{0} - Y_{0} | \to 0} ℳ {| X_{t} - Y_{t} | < ɛ for all t \geq 0} = 1$ for any given number ɛ > 0.

3 Parameter estimation

Consider the parameter estimation problem of a class of simple uncertain differential equations

${dX}_{t} = (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (t)) dt + β {dC}_{t}$ (2) with X₀ = x₀, where α_l (l = 0, 1, …, p) and β (≠0) are unknown parameters. In order to get the solution of the UDE (14), it is reasonable to provide the following assumption.

Assumption 1: For any t > 0, each F_l (t) (l = 1, 2, …, p) is a time integrable and differentiable function.

For a time interval [0, t], by integration, we have $\begin{matrix} X_{t} & = & x_{0} + \int_{0}^{t} (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (s)) ds + \int_{0}^{t} β {dC}_{s} \\ = & x_{0} + α_{0} t + \sum_{l = 1}^{p} α_{l} \int_{0}^{t} F_{l} (s) ds + β C_{t} . \end{matrix}$ Based on Appsumption 1, denote $G_{l} (t) = \int_{0}^{t} F_{l} (s) ds$ . Then,

$X_{t} = x_{0} + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t} .$ (3) Obviously, the solution X_t is an uncertain process, which is essentially a function of the variable t and Liu process C_t.

Theorem 1. The UDE (2) is stable.

Proof. From (3), it is easy to get two solutions of the UDE (2) with initial values x₀ and y₀, that is, $\begin{matrix} X_{t} = x_{0} + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t}, \\ Y_{t} = y_{0} + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t} . \end{matrix}$ For any given number ɛ > 0, we have $\begin{matrix} lim_{| x_{0} - y_{0} | \to 0} ℳ {| X_{t} - Y_{t} | < ɛ for all t \geq 0} \\ = & lim_{| x_{0} - y_{0} | \to 0} ℳ {| x_{0} - y_{0} | < ɛ} = 1 . \end{matrix}$

By Definition 6, the UDE (2) is stable.□

Theorem 2. Let Ψ_t (x) be uncertainty distribution of the solution X_t in (3) for $x \in ℝ$ and t > 0.

If β > 0, then $\begin{matrix} Ψ_{t} (x) = (1 + exp (- \frac{π}{\sqrt{3} β t} (x - x_{0} \\ {- α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))))}^{- 1} . \end{matrix}$

If β < 0, then $\begin{matrix} Ψ_{t} (x) = (1 + exp (\frac{π}{\sqrt{3} β t} (x - x_{0} \\ {- α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))))}^{- 1} . \end{matrix}$

Proof. Based on the Equation (3), for any $x \in ℝ$ , $\begin{matrix} Ψ_{t} (x) & = & ℳ {X_{t} \leq x} \\ = & ℳ {β C_{t} \leq x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t)} . \end{matrix}$

(i) If β > 0, one has

$\begin{matrix} Ψ_{t} (x) & = & ℳ {C_{t} \leq \frac{1}{β} (x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))} \\ = & (1 + exp (- \frac{π}{\sqrt{3} β t} (x - x_{0} - α_{0} t \\ - \sum_{l = 1}^{p} α_{l} G_{l} (t))))^{- 1} . \end{matrix}$

(ii) If β < 0, one has

$\begin{matrix} Ψ_{t} (x) & = & ℳ {C_{t} \geq \frac{1}{β} (x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))} \\ = & 1 - ℳ {C_{t} < \frac{1}{β} (x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))} \\ = & 1 - (1 + exp (- \frac{π}{\sqrt{3} β t} (x - x_{0} - α_{0} t \\ - \sum_{l = 1}^{p} α_{l} G_{l} (t))))^{- 1} \\ = & \frac{exp (- \frac{π}{\sqrt{3} β t} (x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t)))}{1 + exp (- \frac{π}{\sqrt{3} β t} (x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t)))} \\ = & (1 + exp (\frac{π}{\sqrt{3} β t} (x - x_{0} - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))))^{- 1} . \end{matrix}$ Thus, the conclusion is established.□

Remark 1. From Theorem 2, the unified form of uncertainty distribution of the solution X_t is $\begin{matrix} Ψ_{t} (x) & = & (1 + exp (- \frac{π}{\sqrt{3} | β | t} (x - x_{0} \\ {- α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))))}^{- 1} \end{matrix}$ for $x \in ℝ$ and t > 0. It reveals that $X_{t} \sim 𝒩 (x_{0} + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t), | β | t) .$ Theorem 3. Suppose G_l (t - s) = G_l (t) - G_l (s) for l = 1, 2, …, p. Then, the solution X_t with the initial value x₀ = 0 is a stationary independent increment process. Further, every increment $X_{t + s} - X_{t} \sim 𝒩 (α_{0} s + \sum_{l = 1}^{p} α_{l} G_{l} (s), | β | s)$ for any t, s > 0.

Proof. Since C_t is an independent increment process, the uncertain process C_{t
₁}, C_{t
₂} - C_{t
₁}, …, C_{t
_u} - C_{t
_u-1} are independent for t₁ < t₂ < ⋯ < t_u. For i = 1, 2, …, u, we have $\begin{matrix} X_{t_{i}} - X_{t_{i - 1}} & = & α_{0} (t_{i} - t_{i - 1}) \\ + \sum_{l = 1}^{p} α_{l} (G_{l} (t_{i}) - G_{l} (t_{i - 1})) \\ + β (C_{t_{i}} - C_{t_{i - 1}}) \\ = & α_{0} (t_{i} - t_{i - 1}) + \sum_{l = 1}^{p} α_{l} G_{l} (t_{i} - t_{i - 1}) \\ + β (C_{t_{i}} - C_{t_{i - 1}}), \end{matrix}$ where X_{t
₀} = x₀ = 0. Thus, X_{t
_i} - X_{t
_i-1} (i = 1, 2, …, u) are independent, that is, X_t is an independent increment process. Moreover, the increment C_t+s - C_t ∼ 𝒩 (0, s) for all s > 0 and G (t - s) = G (t) - G (s). Thus, $X_{t + s} - X_{s} = α_{0} s + \sum_{l = 1}^{p} α_{l} G_{l} (s) + β (C_{t + s} - C_{s})$ are identically distributed uncertain variables for all s > 0, that is, X_t is a stationary increment process. Therefore, uncertain process X_t with x₀ = 0 is a stationary independent increment process. Based on Theorem 2, it is easy to get the uncertainty distribution of increment X_t+s - X_s for any t, s > 0.□

Yao and Liu [10] introduced uncertain regression analysis to estimate the relationships among uncertain response variable and the explanatory variables. Let y be an uncertain observed response and u = (u₁, u₂, …, u_k) be a vector of explanatory variables. Suppose that the functional relationship between y and u can be expressed by a regression model

$y = θ_{0} + θ_{1} u_{1} + \dots + θ_{k} u_{k} + ɛ,$ (4) where θ_l (l = 0, 1, …, k) are unknown parameters called the regression coefficients, and ɛ is a disturbance term. Then, (4) is called an uncertain linear regression model.

Let y_i be the ith observed response of y and u_il be the i observation or level of regressor u_l. Suppose that n > k observations are available. The functional relationship between y_i and u_il is written by $\begin{matrix} y_{i} = θ_{0} + \sum_{l = 1}^{k} θ_{l} u_{il} + ɛ_{i}, i = 1, 2, \dots, n . \end{matrix}$ In matrix form,

$Y = U θ + ɛ,$ (5) where $\begin{matrix} Y = (\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{matrix}), U = (\begin{matrix} 1 & u_{11} & u_{12} & \dots & u_{1 k} \\ 1 & u_{21} & u_{22} & \dots & u_{2 k} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & u_{n 1} & u_{n 2} & \dots & u_{nk} \end{matrix}), \end{matrix}$ and θ = (θ₀, θ₁, ⋯ , θ_k) ^T, ɛ = (ɛ₁, ɛ₂, ⋯ , ɛ_n) ^T . Hereafter, the superscript T represents transpose of a vector or matrix.

An important objective of uncertain regression analysis is to estimate unknown parameters in regression model. One of these techniques is the method of least-square, which is used to estimate parameters in model (5). The least-square function is $\begin{matrix} Q (θ) & = & \sum_{i = 1}^{n} ɛ_{i}^{2} = ɛ^{T} ɛ = (Y - U θ)^{T} (Y - U θ) \\ = & Y^{T} Y - θ^{T} U^{T} Y - Y^{T} U θ + θ^{T} U^{T} U θ \\ = & Y^{T} Y - 2 θ^{T} U^{T} Y + θ^{T} U^{T} U θ . \end{matrix}$ Let $\hat{θ} = ({\hat{θ}}_{0}, {\hat{θ}}_{1}, \dots, {\hat{θ}}_{k})^{T}$ be the estimator of parameter vector θ. With respect to θ, the function Q must be minimized and satisfies $\begin{matrix} \frac{\partial Q}{\partial θ} |_{\hat{θ}} = - 2 U^{T} Y + 2 U^{T} U \hat{θ} = 0 . \end{matrix}$ If the inverse matrix (U^TU) ^-1 exists, the least-square estimator $\hat{θ}$ of θ is

$\hat{θ} = (U^{T} U)^{- 1} U^{T} Y .$ (6)

Case 1: β > 0.

From Theorem 2 (i), uncertainty distribution Ψ_t (x) of X_t becomes

$\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{Ψ_{t} (x)}{1 - Ψ_{t} (x)} & = & - \frac{α_{0}}{β} - \sum_{l = 1}^{p} \frac{α_{l}}{β} \cdot \frac{G_{l} (t)}{t} \\ + \frac{1}{β} \cdot \frac{x - x_{0}}{t} . \end{matrix}$ (7) Let (x_i, t_j, Ψ_{t
_j} (x_i)) be observations and take X₀ = x₀, then $\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{Ψ_{t_{j}} (x_{i})}{1 - Ψ_{t_{j}} (x_{i})} & = & - \frac{α_{0}}{β} - \sum_{l = 1}^{p} \frac{α_{l}}{β} \cdot \frac{G_{l} (t_{j})}{t_{j}} \\ + \frac{1}{β} \cdot \frac{x_{i} - x_{0}}{t_{j}} \end{matrix}$ for i = 1, 2, …, n and j = 1, 2, …, m.

Denote $y_{ij} = \frac{\sqrt{3}}{π} ln \frac{Ψ_{t_{j}} (x_{i})}{1 - Ψ_{t_{j}} (x_{i})}$ for i = 1, 2, …, n, j = 1, 2, …, m. In order to estimate unknown parameters of (7), the elements of Y, U, ɛ and θ of (5) are replaced by Y, U, ε and ϑ as follows

$\begin{matrix} Y & = {(y_{11}, \dots, y_{1 m}, y_{21}, \dots, y_{2 m}, \dots, y_{n 1}, \dots, y_{nm})}^{T}, \\ U & = (\begin{matrix} 1 & G_{1} (t_{1}) / t_{1} & \dots & G_{p} (t_{1}) / t_{1} & (x_{1} - x_{0}) / t_{1} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 1 & G_{1} (t_{m}) / t_{m} & \dots & G_{p} (t_{m}) / t_{m} & (x_{1} - x_{0}) / t_{m} \\ 1 & G_{1} (t_{1}) / t_{1} & \dots & G_{p} (t_{1}) / t_{1} & (x_{2} - x_{0}) / t_{1} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 1 & G_{1} (t_{m}) / t_{m} & \dots & G_{p} (t_{m}) / t_{m} & (x_{2} - x_{0}) / t_{m} \\ 1 & G_{1} (t_{1}) / t_{1} & \dots & G_{p} (t_{1}) / t_{1} & (x_{n} - x_{0}) / t_{1} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 1 & G_{1} (t_{m}) / t_{m} & \dots & G_{p} (t_{m}) / t_{m} & (x_{n} - x_{0}) / t_{m} \end{matrix}), \\ ε & = {(ε_{11}, \dots, ε_{1 m}, ε_{21}, \dots, ε_{2 m}, \dots, ε_{n 1}, \dots, ε_{nm})}^{T}, \\ ϑ & = (- \frac{α_{0}}{β}, - \frac{α_{1}}{β}, \dots, - \frac{α_{p}}{β}, \frac{1}{β})^{T} . \end{matrix}$ (8) Then, the Equation (5) is equivalent to the matrix form $Y = U ϑ + ε .$

Based the Equation (6), the following result is directly obtained.

Theorem 4. Let $\hat{ϑ}$ , ${\hat{α}}_{l} (l = 0, 1, \dots, p)$ and $\hat{β}$ be the least-square estimators of ϑ, α_l and β in (7), respectively. Then, the estimators ${\hat{α}}_{l}$ and $\hat{β}$ satisfy $\begin{matrix} \hat{ϑ} & = (- \frac{{\hat{α}}_{0}}{\hat{β}}, - \frac{{\hat{α}}_{1}}{\hat{β}}, \dots, - \frac{{\hat{α}}_{p}}{\hat{β}}, \frac{1}{\hat{β}})^{T} \\ = (U^{T} U)^{- 1} U^{T} Y, \end{matrix}$ where Y and U are defined in (8).

Case 2: β < 0

Based on Theorem 2 (ii), uncertainty distribution Ψ_t (x) of X_t becomes

$\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{1 - Ψ_{t} (x)}{Ψ_{t} (x)} & = & - \frac{α_{0}}{β} - \sum_{l = 1}^{p} \frac{α_{l}}{β} \cdot \frac{G_{l} (t)}{t} \\ + \frac{1}{β} \cdot \frac{x - x_{0}}{t} . \end{matrix}$ (9) Let (x_i, t_j, Ψ_{t
_j} (x_i)) be observations and take X₀ = x₀, then $\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{1 - Ψ_{t_{j}} (x_{i})}{Ψ_{t_{j}} (x_{i})} & = & - \frac{α_{0}}{β} - \sum_{l = 1}^{p} \frac{α_{l}}{β} \cdot \frac{G_{l} (t_{j})}{t_{j}} \\ + \frac{1}{β} \cdot \frac{x_{i} - x_{0}}{t_{j}} \end{matrix}$ for i = 1, 2, …, n and j = 1, 2, …, m.

Similar to Theorem 4, we get the following result.

Theorem 5. Let ${\hat{α}}_{l} (l = 0, 1, \dots, p)$ and $\hat{β}$ be the least-square estimators of α_l and β in (7), respectively. Then, the estimators ${\hat{α}}_{l}$ and $\hat{β}$ satisfy $\begin{matrix} \hat{ϑ} & = (- \frac{{\hat{α}}_{0}}{\hat{β}}, - \frac{{\hat{α}}_{1}}{\hat{β}}, \dots, - \frac{{\hat{α}}_{p}}{\hat{β}}, \frac{1}{\hat{β}})^{T} \\ = (U^{T} U)^{- 1} U^{T} Z, \end{matrix}$ where U is defined in (8), and $Z = {(z_{11}, \dots, z_{1 m}, z_{21}, \dots, z_{2 m}, \dots, z_{n 1}, \dots, z_{nm})}^{T}$ with $z_{ij} = \frac{\sqrt{3}}{π} ln \frac{1 - Ψ_{t_{j}} (x_{i})}{Ψ_{t_{j}} (x_{i})}$ .

4 A simple uncertain differential equation

Consider a simple uncertain differential equation $\begin{matrix} {dX}_{t} = α dt + β {dC}_{t}, X_{0} = x_{0}, \end{matrix}$ where α and β (≠0) are unknown parameters. By integration, it is to get $\begin{matrix} X_{t} = x_{0} + α t + β C_{t} . \end{matrix}$ From Theorem 2, uncertainty distribution Ψ_t (x) of X_t is

$Ψ_{t} (x) = (1 + exp (- \frac{π (x - x_{0} - α t)}{\sqrt{3} | β | t}))^{- 1}$ (10) for $x \in ℝ$ and t > 0.

Example 1. Consider a simple UDE dX_t = αdt + βdC_t with X₀ = x₀. For x ∈ [-50, 50] and t ∈ (0, 10], take β = 0.5 and x₀ = 0, Fig. 1(a)–(c) respectively show the curves of uncertainty distribution Ψ_t (x) for α = -1, 3, 6.

Fig.1

Trajectories of uncertainty distribution Ψ_t (x) with α = -1, 3, 6.

Theorem 6. Let (x_i, t_j, Ψ_{t
_j} (x_i)) be observations from (10) for i = 1, 2, …, n and j = 1, 2, …, m.

If β > 0, then the least-square estimators $\hat{α}$ and $\hat{β}$ of α and β in (10) are $\begin{matrix} {\begin{matrix} \hat{α} = & \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i} - x_{0}}{t_{j}} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}}}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}} - nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i} - x_{0}}{t_{j}}}, \\ \hat{β} = & \frac{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - (\sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}})^{2}}{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i} - x_{0}}{t_{j}} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}}}, \end{matrix} \end{matrix}$ where $y_{ij} = \frac{\sqrt{3}}{π} ln \frac{Ψ_{t_{j}} (x_{i})}{1 - Ψ_{t_{j}} (x_{i})}$ for i = 1, 2, …, n, j = 1, 2, …, m .

If β < 0, then the least-square estimators $\hat{α}$ and $\hat{β}$ of α and β in (10) are $\begin{matrix} {\begin{matrix} \hat{α} = & \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} z_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} z_{ij} \frac{x_{i} - x_{0}}{t_{j}} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}}}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} z_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}} - nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} z_{ij} \frac{x_{i} - x_{0}}{t_{j}}}, \\ \hat{β} = & \frac{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - (\sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}})^{2}}{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} z_{ij} \frac{x_{i} - x_{0}}{t_{j}} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} z_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}}}, \end{matrix} \end{matrix}$ where $z_{ij} = \frac{\sqrt{3}}{π} ln \frac{1 - Ψ_{t_{j}} (x_{i})}{Ψ_{t_{j}} (x_{i})}$ for i = 1, 2, …, n, j = 1, 2, …, m.

Proof. (i) By the Equation (10),

$\frac{\sqrt{3}}{π} ln \frac{Ψ_{t} (x)}{1 - Ψ_{t} (x)} = - \frac{α}{β} + \frac{1}{β} \cdot \frac{x - x_{0}}{t} .$ (11) Thus, $\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{Ψ_{t_{j}} (x_{i})}{1 - Ψ_{t_{j}} (x_{i})} = - \frac{α}{β} + \frac{1}{β} \cdot \frac{x_{i} - x_{0}}{t_{j}} \end{matrix}$ for i = 1, 2, …, n and j = 1, 2, …, m. From Theorem 4, take $\begin{matrix} U & = & {(\begin{matrix} 1 & \dots & 1 & \dots & 1 & \dots & 1 \\ \frac{x_{1} - x_{0}}{t_{1}} & \dots & \frac{x_{1} - x_{0}}{t_{m}} & \dots & \frac{x_{n} - x_{0}}{t_{1}} & \dots & \frac{x_{n} - x_{0}}{t_{m}} \end{matrix})}^{T} \end{matrix}$ and $θ = (- \frac{α}{β}, \frac{1}{β})^{T}$ , we have $\hat{θ} = (- \frac{\hat{α}}{\hat{β}}, \frac{\hat{α}}{\hat{β}}) = {(U^{T} U)}^{- 1} U^{T} Y = \frac{1}{| U^{T} U |} \times (\begin{matrix} \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{i j} \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{i j} \frac{x_{i} - x_{0}}{t_{j}} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}} \\ - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{i j} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}} + n m \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{i j} \frac{x_{i} - x_{0}}{t_{j}} \end{matrix}),$ where $\begin{matrix} | U^{T} U | & = nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} \\ - (\sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}})^{2} . \end{matrix}$

Then,

$\begin{matrix} - \frac{\hat{α}}{\hat{β}} & = & \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i} - x_{0}}{t_{j}} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}}}{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - (\sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}})^{2}}, \\ \frac{1}{\hat{β}} & = & \frac{- \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}} + nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i} - x_{0}}{t_{j}}}{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i} - x_{0}}{t_{j}})^{2} - (\sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i} - x_{0}}{t_{j}})^{2}} . \end{matrix}$ The estimators $\hat{α}$ and $\hat{β}$ follow. The proof of (ii) is similar to that of (i). □

Example 2. Similar to Example 1, for a simple UDE dX_t = αdt + βdC_t with X_t = x₀, take α = -1, β = 0.5, x₀ = 0. Thus, the uncertainty distribution Ψ_t (x) of X_t is $\begin{matrix} Ψ_{t} (x) = (1 + exp (- \frac{π (x + t)}{0.5 \sqrt{3} t}))^{- 1} \end{matrix}$ for $x \in ℝ$ and t > 0. Take x ∈ [-50, 50], t ∈ (0, 10]. Figure 1(a) shows the trajectories of uncertainty distribution Ψ_t (x).

Denote

$y_{ij} = \frac{\sqrt{3}}{π} ln \frac{Ψ_{t_{j}} (x_{i})}{1 - Ψ_{t_{j}} (x_{i})} .$ From the Equation (11) and the above parameters, we have

$y_{ij} = - \frac{α}{β} + \frac{1}{β} \frac{x_{i} - x_{0}}{t_{j}} = 2 + 2 x_{i} / t_{j}$ (12)

for i = 1, 2, …, n and j = 1, 2, …, m. Figure 2(a) provides the curves of the above equation y_ij and the corresponding data set.

Fig.2

Trajectories of the true values y_ij and fitted values ${\hat{y}}_{ij}$ .

Next we try to estimate the parameters α and β for the UDE from the observations of y_ij. Suppose x_i = -50 : 0.1 : 50 and t_j = 0.1 : 0.1 : 10, we have n = 1001 and m = 100. There exist 100, 100 observations y_ij from (x_i, t_j) in Fig. 2(a). From Theorem 6, the least-square estimators $\hat{α}$ and $\hat{β}$ of α and β are

$\begin{matrix} {\begin{matrix} \hat{α} = & \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i}}{t_{j}})^{2} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i}}{t_{j}} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i}}{t_{j}}}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i}}{t_{j}} - nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i}}{t_{j}}} = - 1.0625, \\ \hat{β} = & \frac{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} (\frac{x_{i}}{t_{j}})^{2} - (\sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i}}{t_{j}})^{2}}{nm \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \frac{x_{i}}{t_{j}} - \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{x_{i}}{t_{j}}} = 0.5 . \end{matrix} \end{matrix}$

Thus, by least-square method, we get the estimated parameters of the UDE, denoted by $d {\hat{X}}_{t} = \hat{α} dt + \hat{β} {dC}_{t} = - 1.0625 dt + 0.5 {dC}_{t} .$ Figure 2(b) shows the curves of estimated value ${\hat{y}}_{ij}$ of y_ij, that is,

${\hat{y}}_{ij} = \hat{α} + \hat{β} x_{i} / t_{j} = 2.125 + 2 x_{i} / t_{j} .$ (13)

Define the coefficient of determination $R^{2} = \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} ({\hat{y}}_{ij} - \bar{y})^{2}}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} (y_{ij} - \bar{y})^{2}},$ where $\bar{y} = \frac{1}{nm} \sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij}$ . From the Equations (12) and (13), we have R² = 0.98, which reveals the estimated values $\hat{α} = - 1.0625$ and $\hat{β} = 0.5$ are the good estimations of unknown parameters α and β.

5 Discussions

In the paper, we solve the estimation problem of a class of simple uncertain differential equations. The solution and uncertainty distribution of uncertain differential equation are first obtained. By the method of least-square, the estimators of unknown parameters are given. Further, we provide the analytic expressions of two unknown parameters for a simple uncertain differential equation. The conclusions were validated and illuminated by numerical simulation.

The above results can be extended such uncertain differential equations, which the explicit expressions of their solutions and uncertainty distribution can be obtained. For example, consider an uncertain differential equation $df (X_{t}) = (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (t)) f (X_{t}) dt + β f (X_{t}) {dC}_{t}$ (14) with the initial value f (x₀), where X_t is an uncertain process and f (x) is a non-negative function. If f (x) = x (x > 0), we get a linear UDE $\begin{matrix} {dX}_{t} = (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (t)) X_{t} dt + β X_{t} {dC}_{t} . \end{matrix}$ If f (x) = sin x (0 < x < π), we get a non-linear UDE $\begin{matrix} {dX}_{t} = (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (t)) tan X_{t} dt + β tan X_{t} {dC}_{t} . \end{matrix}$

For (14), since $\frac{df (X_{t})}{f (X_{t})} = (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (t)) dt + β {dC}_{t},$ it is equivalent to the following UDE $d ln f (X_{t}) = (α_{0} + \sum_{l = 1}^{p} α_{l} F_{l} (t)) dt + β {dC}_{t} .$ From (3), it is easy to get $\begin{matrix} ln f (X_{t}) = ln f (x_{0}) + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t}, \end{matrix}$ that is, the solution of the UDE (14) is $\begin{matrix} f (X_{t}) = exp (ln f (x_{0}) + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t}) . \end{matrix}$ If f (x) is a non-negative monotone function, then $\begin{matrix} X_{t} & = f^{- 1} (\exp (ln f (x_{0}) + α_{0} t \\ + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t})) . \end{matrix}$ Let Ψ_f (x, t) be the uncertainty distribution of f (X_t). Similar to Theorem 2, for any x > 0, we have $\begin{matrix} Ψ_{f} (x, t) & = & ℳ {f (X_{t}) \leq x} \\ = & ℳ {exp (ln f (x_{0}) + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t}) \leq x} \\ = & ℳ {ln f (x_{0}) + α_{0} t + \sum_{l = 1}^{p} α_{l} G_{l} (t) + β C_{t} \leq ln x} \\ = & ℳ {β C_{t} \leq ln x - ln f (x_{0}) - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t)} . \end{matrix}$ Then, $\begin{matrix} Ψ_{f} (x, t) & = & (1 + exp (- \frac{π}{\sqrt{3} | β | t} (ln x - ln f (x_{0}) \\ - α_{0} t - \sum_{l = 1}^{p} α_{l} G_{l} (t))))^{- 1} . \end{matrix}$ That is, $\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{Ψ_{f} (x, t)}{1 - Ψ_{f} (x, t)} & = & - \frac{α_{0}}{| β |} - \sum_{l = 1}^{p} \frac{α_{l}}{| β |} \cdot \frac{G_{l} (t)}{t} \\ + \frac{1}{| β |} \cdot \frac{ln x - ln f (x_{0})}{t} . \end{matrix}$

Let (x_i, t_j, Ψ_f (x_i, t_j)) be observations from the above equation. Then, $\begin{matrix} \frac{\sqrt{3}}{π} ln \frac{Ψ_{f} (x_{i}, t_{j})}{1 - Ψ_{f} (x_{i}, t_{j})} & = & - \frac{α_{0}}{| β |} - \sum_{l = 1}^{p} \frac{α_{l}}{| β |} \cdot \frac{G_{l} (t_{j})}{t_{j}} \\ + \frac{1}{| β |} \cdot \frac{ln x_{i} - ln f (x_{0})}{t_{j}} \end{matrix}$ for i = 1, 2, …, n and j = 1, 2, …, m. In (8), denote $\begin{matrix} y_{ij} = \frac{\sqrt{3}}{π} ln \frac{Ψ_{f} (x_{i}, t_{j})}{1 - Ψ_{f} (x_{i}, t_{j})} \end{matrix}$ for i = 1, 2, …, n, j = 1, 2, …, m, and the last column of the matrix U is replaced by $\begin{matrix} (\frac{ln x_{1} - ln f (x_{0})}{t_{1}}, \dots, \frac{ln x_{1} - ln f (x_{0})}{t_{m}}, \dots, \\ \frac{ln x_{n} - ln f (x_{0})}{t_{1}}, \dots, \frac{ln x_{n} - ln f (x_{0})}{t_{m}})^{T} . \end{matrix}$ Based on Theorems 4 and 5, we respectively obtain the least-square estimators ${\hat{α}}_{l}$ and $\hat{β}$ of α_l (l = 0, 1, …, p) and β in (14).

For some complicate uncertain differentialequations $\begin{matrix} {dX}_{t} = f (t, X_{t}, α) dt + g (t, X_{t}, β) {dC}_{t}, X_{0} = x_{0}, \end{matrix}$ where α = (α₀, α₁, …, α_p) and β = (β₀, β₁, …, β_q) are unknown parameter vectors, we can not obtain their analytic solution and uncertainty distribution. Thus, the method isn’t applicable for this type. In the future, we plan to develop new methods to solve the problem.

Footnotes

Acknowledgments

This research is funded by the National Natural Science Foundation of China (Grant No. 11661076) and the Natural Science Foundation of Xinjiang (Grant No. 2016D01C043).

References

Liu , Uncertainty Theory. Springer-Verlag, Berlin, 2007.

Liu , Fuzzy process, hybird process and uncertain process, J. Uncertain Syst 2 (2008), 3–16.

Liu , Some research problems in uncertainty theory, J. Uncertain Syst 3 (2009), 3–10.

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

Zhou and

Z.M.

Li , Analytic solutions of two types of nonlinear uncertain differential equations, J. Intell. Fuzzy Syst 35 (2018), 2413–2420.

Fang ,

Z.M.

Li ,

Yang and

M.Y.

Zhou , Solution and α-path of uncertain SIS epidemic model with standard incidence and demography, J. Intell. Fuzzy Syst 35 (2018), 927–935.

Yao , a type of uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications 1 (2013), 1–8.

Yao , Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optim Decis Ma 14 (2015), 399–42.

Yao and

Liu , Uncertain regression analysis: an approach for imprecise observations, Soft computing 22 (2018), 5579–5582.

10.

Yao , Multi-dimensional uncertain calculus with Liu process, J. Uncertain Syst 8 (2014), 244–254.

11.

Chen , American option pricing formula for uncertain financial market, International Journal of Operations Research 8 (2011), 32–37.

12.

Chen and

Liu , Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim Decis Ma 19 (2010), 69–81.

13.

Chen and

Gao , Uncertain term structure model of interest rate, Soft Computing 17(4) (2013), 597–604.

14.

X.F.

Yang and

Y.Y.

Shen , Runge-Kutta method for solving uncertain differential equations. Journal of Uncertainty Analysis and Applications (2015), 3–17.

15.

X.S.

Wang and

Z.X.

Peng , Method of moments for estimating uncertainty distribution. Journal of Uncertainty Analysis and Applications, (2014), 1–10.

16.

Y.H.

Liu , An analytic method for solving uncertain differential equations. J. Uncertain Syst 16 (2012), 244–249.

17.

Y.H.

Liu , Uncertain currency model and currency option pricing, International Journal of Intelligent Systems 30 (2015), 40–51.

18.

Y.H.

Sheng ,

Shi and

Cui , Almost sure stability for multifactor uncertain differential equation, J. Intell. Fuzzy Syst 32 (2017), 2187–2194.

19.

Zhu , Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems 41 (2010), 535–547.

20.

Wang , Analytic solution for a general type of uncertain differential equation. Information: An International Interdisciplinary Journal 16 (2013), 1003–1010.

21.

Z.M.

Li ,

Y.H.

Sheng ,

Z.D.

Teng and

Miao , An uncertain differential equation for SIS epidemic model. J. Intell. Fuzzy Syst 33 (2017), 2317–2327.

22.

Z.M.

Li ,

Teng ,

Hong and

Shi , Comparison of three SIS epidemic models: deterministic, stochastic and uncertain. J. Intell. Fuzzy Syst 35 (2018), 5785–5796.