Generalized moment estimation of multi-dimensional uncertain differential equations

Abstract

Multi-dimensional uncertain differential equations (MUDEs) are often used to describe complex systems that vary with time. In this paper, the generalized moment estimation method is employed to estimate the MUDEs’ unknown parameters. A method to optimize parameters with multiple estimation results is proposed. The hypothesis test and α-path are proposed to verify the feasibility of the parameter results. Several examples of parameter estimation for MUDEs are given, as well as two numerical examples to verify the feasibility of the method.

Keywords

Uncertainty theory multi-dimensional uncertain differential equation generalized moment estimation parameter estimation

1 Introduction

In order to model dynamic systems vulnerable to random noise, Ito [1] introduced a class of stochastic differential equations (SDEs) driven by Wiener processes in 1944. SDEs have seen tremendous expansion and are now employed in a wide range of academic fields, such as physics, biology, and economics. A significant area of study for many academics was the parameter estimation of SDEs. Numerous parameter estimation techniques have been developed, including the least square method (Kailath) [2], maximum likelihood method (Strasser) [3], moment estimation method (Gallant) [4], and others. However, probability theory is unsuitable when the distribution function is not close to the real frequency. To address this issue, Liu [5] established a framework for uncertainty theory based on normality, duality, subadditivity, and product axioms in 2007.

Liu [7] first presented uncertain differential equations (UDEs) in 2008. It is a class of differential equations used to model and analyze dynamic systems with uncertainties that are driven by the Liu process. Various forms of UDEs have been proposed due to the increasing interest and research in UDEs by researchers. Backward uncertain differential equations were presented by Zhu and Ge [8], who also established existence theorems for their solutions. In 2015, Yao [9] proposed uncertain differential equations with jumps driven by renewal process and Liu process, and provided sufficient conditions for the uniqueness of the equations’ solutions. Yao [10] defined high-order uncertain differential equations as the UDEs with high-order derivatives. In 2014, Yao [11] proposed a multi-dimensional uncertain differential equations (MUDEs) driven by a multi-dimensional Liu process.

As the core problem of UDEs, parameter estimation is being thoroughly researched. Sheng et al. [12] presented a least squares estimation method for estimating unknown parameters. Yao and Liu [13] proposed a moment estimation method based on a difference form. Liu [14] proposed the generalized moment estimation method to deal with the case where the moment estimation equation cannot be solved. Using the principle of moment estimation, Lio and Liu [15] first solved the time-varying parameters in UDEs. In addition, Liu et al. [16] suggested the maximum likelihood method to estimate the parameters. Yang et al. [17] proposed the α-path method to estimate unknown parameters of UDEs from discrete sample data. Recently, the parameters of some special types of UDEs have also been estimated. Sheng et al. [18] introduced three methods for parameter estimation based on different types of solutions. Zhang et al. [19] obtained the unknown parameters of the multi-factor uncertain differential equation by using the least squares method and the moment estimation. Zhang et al. [20] estimated the parameters of high-order uncertain differential equations.

In this paper, the problem of parameter estimation for MUDE is solved by using generalized moment estimation. The remaining chapters of this article are organized as follows. In Section 2, some fundamental definitions and theorems of uncertainty theory are introduced. In Section 3, generalized moment estimates for one-dimensional uncertain differential equations are first proved and then extended to high-dimensional uncertain differential equations. To ensure that the parameter estimation results are optimal and unique, a notion of residuals is proposed, and parameters with numerous estimation outcomes are optimized. Both α-path and hypothesis testing approaches are employed to evaluate the validity of the parameter estimates. In Section 4, two numerical examples are given to verify the generalized moment estimation method. Section 5 is the summary of this paper.

2 Preliminaries

This section will demonstrate a number of concepts and theorems of uncertainty theory that are critical to this paper.

Definition 1. (Liu [5]) The expected value E [ξ] of uncertain variable ξ is $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} d x - \int_{- \infty}^{0} M {ξ \leq x} d x$ for any real number x.

The inverse function Φ^-1 (α) is called the inverse uncertainty distribution of uncertain variable ξ with regular uncertainty distribution Φ (x). The expected value can be equivalently expressed as $E [ξ] = \int_{0}^{1} Φ^{- 1} (α) d α .$

Definition 2. (Liu [6]) An uncertain process C_t is called a Liu process if

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

(ii) C_t has stationary and independent increments,

(iii) the increment C_s+t - C_s has a normal uncertainty distribution $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Definition 3. (Liu [5]) A real-valued function $X_{t}^{α}$ is called the α-path of the uncertain differential equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ if it solves the corresponding ordinary differential equation $d X_{t}^{α} = f (t, X_{t}^{α}) d t + | g (t, X_{t}^{α}) | Φ^{- 1} (α) d t,$ where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}, α \in (0, 1)$ is the inverse uncertainty distribution of a standard normal uncertain variable.

Definition 4. (Yao [11]) Supposing C_it, (i = 1, 2, ⋯ , n) be indepentent Liu processes, the form of n-dimensional Liu process is C_t = (C_1t, C_2t, ⋯ , C_nt) ^⊤.

Definition 5. (Yao [11]) Suppose that C_t is an n-dimensional Liu process, f (t, x) is a vector-valued function from T $\times R^{⊤}$ to $R^{m}$ , and g (t, x) is a matrix-valued function from T $\times R^{⊤}$ to the set of m × n matrices. Then $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (1) is called an m-dimensional uncertain differential equation driven by an n-dimensional Liu process.

Theorem 1. (Ye [21]) Let ξ be a population with regular uncertainty distribution Φ_θ where θ is an unknown parameter with θ ∈ Θ. The test for two-sided hypotheses H₀ : θ = θ₀ versus H₁ : θ ≠ θ₀ at significance level α is W= {(z₁, z₂, ⋯ , z_n): there are at least α of indexes i’s with 1 ≤ i ≤ n such as $z_{i} < Φ_{θ_{0}}^{- 1} (\frac{α}{2})$ or $z_{i} > Φ_{θ_{0}}^{- 1} (1 - \frac{α}{2})$ }.

Remark: For all observations, if the number of z_i satisfying $z_{i} \notin [Φ_{θ_{0}}^{- 1} (\frac{α}{2}), Φ_{θ_{0}}^{- 1} (1 - \frac{α}{2})]$ is at least nα, then (z₁, z₂, ⋯ , z_n) ∈ W and the uncertainty distribution Φ_θ is judged not to fit the observations.

3 Parameter estimation

In this section, the paper will introduce how to use generalized moment estimation to determine unknown parameters in MUDEs. And the hypothesis test is proposed to verify the reasonableness of the parameter estimation results.

The m-dimensional UDE with parameters driven by n-dimensional Liu process C_t is $d X_{t} = f (t, X_{t}; μ) d t + g (t, X_{t}; σ) d C_{t}$ (2) where μ and σ are the unknown constant parameters vectors to be solved.

3.1 One-dimensional uncertain differential equations

One-dimensional UDEs are also known as multi-factor UDEs. Consider a one-dimensional UDE with unknown parameters ( μ, σ) $d X_{t} = f (t, X_{t}; μ) d t + \sum_{j = 1}^{n} g_{j} (t, X_{t}; σ) d C_{jt} .$ (3) Let time step Δt_i = t_i - t_i-1. The equation (3) can be written in Euler difference form as: $\begin{matrix} X_{t_{i} + Δ t_{i}} = X_{t_{i}} + f (t_{i}, X_{t_{i}}; μ) Δ t_{i} \\ + \sum_{j = 1}^{n} g_{j} (t_{i}, X_{t_{i}}; σ) \cdot (C_{j (t_{i} + Δ t_{i})} - C_{{jt}_{i}}) . \end{matrix}$ (4) According to the properties of Liu process, for any j (j = 1, 2, ⋯ , n), we can get: $\frac{(C_{j (t_{i} + Δ t_{i})} - C_{{jt}_{i}})}{Δ t_{i}} \sim N (0, 1) .$ Hence, $\begin{matrix} \sum_{j = 1}^{n} g_{j} (t_{i}, X_{t_{i}}; σ) Δ t_{i} \cdot \frac{(C_{j (t_{i} + Δ t_{i})} - C_{{jt}_{i}})}{Δ t_{i}} \\ \sim N (0, \sum_{j = 1}^{n} g_{j} (t_{i}, X_{t_{i}}; σ) Δ t_{i}), \\ \Leftrightarrow X_{t_{i} + Δ t_{i}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) Δ t_{i} \\ \sim N (0, \sum_{j = 1}^{n} g_{j} (t_{i}, X_{t_{i}}; σ) Δ t_{i}) . \end{matrix}$ Therefore, $\begin{matrix} \frac{X_{t_{i} + Δ t_{i}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) Δ t_{i}}{\sum_{j = 1}^{n} g_{j} (t_{i}, X_{t_{i}}; σ) Δ t_{i}} \sim N (0, 1) . \end{matrix}$ (5) When the uncertain variable X_t has N discrete observations x_t, the parameter function k_i ( μ ; σ) is $k_{i} (μ; σ) = \frac{x_{t_{i} + Δ t_{i}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) Δ t_{i}}{\sum_{j = 1}^{n} g_{j} (t_{i}, x_{t_{i}}; σ) Δ t_{i}}$ where i = 1, 2, ⋯ , N - 1 and k_i ( μ ; σ) can be considered as N - 1 samples of standard normal uncertain variables.

Let P be the number of unknown parameters. According to the principle of moment estimation, the p-order sample moment of this group of samples k_i ( μ ; σ)(i = 1, 2, ⋯ , N - 1) is $\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{i} (μ; σ)^{p}$ and the corresponding p-order population moment is $\int_{0}^{1} Φ^{- 1} (α)^{p} d α,$ where p = 1, 2, ⋯ , P.

The generalized moment estimate $(\hat{μ}; \hat{σ})$ of the unknown parameters is the optimal solution of the following equation: $min_{(μ, σ)} \sum_{p = 1}^{P} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{i} (μ, σ)^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)}^{2}$ (6) where $k_{i} (μ; σ) = \frac{x_{t_{i} + Δ t_{i}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) Δ t_{i}}{\sum_{j = 1}^{n} g_{j} (t_{i}, x_{t_{i}}; σ) Δ t_{i}}$ and $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α} .$

In summary, the generalized moment estimation can solve the unknown parameters in the one-dimensional UDEs.

3.2 High-dimensional uncertain differential equations

The case of the one-dimensional UDEs is then extended to the high-dimensional UDEs using the same general logic.

The matrix form of the equation (2) is as follows: $d (\begin{matrix} X_{1 t} \\ X_{2 t} \\ ⋮ \\ X_{mt} \end{matrix}) = (\begin{matrix} f_{1} (t, X_{t}; μ) \\ f_{2} (t, X_{t}; μ) \\ ⋮ \\ f_{m} (t, X_{t}; μ) \end{matrix}) d t + d (\begin{matrix} C_{1 t} \\ C_{2 t} \\ ⋮ \\ C_{nt} \end{matrix})$ $\cdot (\begin{matrix} \begin{matrix} g_{11} (t, & X_{t}; σ) & g_{12} (t, X_{t}; σ) & \dots & g_{1 n} (t, X_{t}; σ) \\ g_{21} (t, & X_{t}; σ) & g_{22} (t, X_{t}; σ) & \dots & g_{2 n} (t, X_{t}; σ) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ g_{m 1} (t, & X_{t}; σ) & g_{m 2} (t, X_{t}; σ) & \dots & g_{mn} (t, X_{t}; σ) \end{matrix} \end{matrix})$ Due to $\begin{matrix} d (\begin{matrix} X_{1 t} \\ X_{2 t} \\ ⋮ \\ X_{mt} \end{matrix}) = (\begin{matrix} d X_{1 t} \\ d X_{2 t} \\ ⋮ \\ d X_{mt} \end{matrix}) and \\ d (\begin{matrix} C_{1 t} \\ C_{2 t} \\ ⋮ \\ C_{nt} \end{matrix}) = (\begin{matrix} d C_{1 t} \\ d C_{2 t} \\ ⋮ \\ d C_{nt} \end{matrix}), \end{matrix}$ the equation (2) can be expressed as: ${\begin{matrix} d X_{1 t} = f_{1} (t, X_{t}; μ) d t + \sum_{j_{1} = 1}^{n} g_{1 j_{1}} (t, X_{t}; σ) d C_{j_{1} t} \\ d X_{2 t} = f_{2} (t, X_{t}; μ) d t + \sum_{j_{2} = 1}^{n} g_{2 j_{2}} (t, X_{t}; σ) d C_{j_{2} t} \\ ⋮ \\ d X_{mt} = f_{m} (t, X_{t}; μ) d t + \sum_{j_{m} = 1}^{n} g_{{mj}_{m}} (t, X_{t}; σ) d C_{j_{m} t} . \end{matrix}$

Similar to the case of one-dimensional UDEs, the above system of equations can be written in Euler difference form as: ${\begin{matrix} X_{1 (t_{i} + Δ t_{i})} = X_{1 t_{i}} + f_{1} (t_{i}, X_{t_{i}}; μ) Δ t_{i} \\ + \sum_{j_{1} = 1}^{n} g_{1 j_{1}} (t_{i}, X_{t_{i}}; σ) (C_{j_{1} (t_{i} + Δ t_{i})} - C_{j_{1} t_{i}}) \\ X_{2 (t_{i} + Δ t_{i})} = X_{2 t_{i}} + f_{2} (t_{i}, X_{t_{i}}; μ) Δ t_{i} \\ + \sum_{j_{2} = 1}^{n} g_{2 j_{2}} (t_{i}, X_{t_{i}}; σ) (C_{j_{2} (t_{i} + Δ t_{i})} - C_{j_{2} t_{i}}) \\ ⋮ \\ X_{m (t_{i} + Δ t_{i})} = X_{{mt}_{i}} + f_{m} (t_{i}, X_{t_{i}}; μ) Δ t_{i} \\ + \sum_{j_{m} = 1}^{n} g_{{mj}_{m}} (t_{i}, X_{t_{i}}; σ) (C_{j_{m} (t_{i} + Δ t_{i})} - C_{j_{m} t_{i}}) . \end{matrix}$ And we can get $\frac{X_{z (t_{i} + Δ t_{i})} - X_{{zt}_{i}} - f_{z} (t_{i}, X_{t_{i}}; μ) Δ t_{i}}{\sum_{j_{m} = 1}^{n} g_{{zj}_{m}} (t, X_{t_{i}}; σ) Δ t_{i}} \sim N (0, 1),$ where z = 1, 2, ⋯ , m.

When the uncertain variable X_t has N discrete observations x_t, a set of parameter functions can be obtained: ${\begin{matrix} k_{1 i} (μ, σ) = \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - f_{1} (t_{i}, x_{t_{i}}; μ) Δ t_{i}}{\sum_{j_{1} = 1}^{n} g_{1 j_{1}} (t_{i}, x_{t_{i}}; σ) Δ t_{i}} \\ k_{2 i} (μ, σ) = \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - f_{2} (t_{i}, x_{t_{i}}; μ) Δ t_{i}}{\sum_{j_{2} = 1}^{n} g_{2 j_{2}} (t_{i}, x_{t_{i}}; σ) Δ t_{i}} \\ ⋮ \\ k_{mi} (μ, σ) = \frac{x_{m (t_{i} + Δ t_{i})} - x_{{mt}_{i}} - f_{m} (t_{i}, x_{t_{i}}; μ) Δ t_{i}}{\sum_{j_{m} = 1}^{n} g_{{mj}_{m}} (t_{i}, x_{t_{i}}; σ) Δ t_{i}} \end{matrix}$ where i = 1, 2, ⋯ , N - 1.

Any set of k_zi ( μ ; σ) (z = 1, 2, ⋯ , m) can be viewed as a set of samples from a standard normal distribution. For each set of samples k_zi ( μ ; σ), the optimization equation to solve for the unknown parameters can be obtained by using the principle of generalized moment estimation, just like in one-dimension UDE: ${\begin{matrix} min_{(μ, σ)} \sum_{p = 1}^{P} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (μ, σ)^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)}^{2} \\ min_{(μ, σ)} \sum_{p = 1}^{P} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{2 i} (μ, σ)^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)}^{2} \\ ⋮ \\ min_{(μ, σ)} \sum_{p = 1}^{P} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{mi} (μ, σ)^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)}^{2} . \end{matrix}$

The above system of equations is challenging to solve directly, therefore we split it up into various minimization issues. For each minimization equation that is solved, we can estimate the unknown parameter $(\hat{μ}, \hat{σ})$ of the equation, resulting in m sets of estimates.

When different equations contain the same unknown parameter, it results in a situation where there are multiple estimates of a parameter. The primary issue to be addressed in the following will be how to determine the best results for parameter estimate.

Let μ = (μ₁, μ₂, ⋯ , μ_s) ^⊤ and σ = (σ₁, σ₂, ⋯ , σ_v) ^⊤. The parameter estimations $μ^{z} = (μ_{1}^{z}, μ_{2}^{z}, \dots, μ_{s}^{z})^{⊤}$ and $σ^{z} = (σ_{1}^{z}, σ_{2}^{z}, \dots, σ_{v}^{z})^{⊤}$ are obtained by solving the z-th minimization problem $min_{(μ; σ)} \sum_{p = 1}^{P} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (μ, σ)^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)}^{2}$ and $C_{z} = \sum_{p = 1}^{P} (\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (μ^{z}, σ^{z})^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)^{2}$ is defined as the residual resulting from the estimation result ( μ^z, σ^z) where z = 1, 2, ⋯ , m.

Therefore, the total residual arising from the solution of all minimization problems can be expressed as $R = C_{1} + C_{2} + \dots + C_{m} .$

It goes without saying that the optimization of parameter estimation results can directly reflect the total residual R. When optimizing a parameter with multiple estimation results, the best estimate should minimize the residual within the range of estimation results.

As an example, the component μ₁ of μ has m parameter estimates: $μ_{1}^{1}$ , $μ_{1}^{2}$ , $μ_{1}^{3}$ , ⋯, $μ_{1}^{m}$ . Therefore, the optimal estimation result $\hat{μ_{1}}$ of μ₁ should be within the range of estimation results: $\begin{matrix} min (μ_{1}^{1}, μ_{1}^{2}, \dots, μ_{1}^{m}) \leq \hat{μ_{1}} \\ \leq max (μ_{1}^{1}, μ_{1}^{2}, \dots, μ_{1}^{m}) . \end{matrix}$ And the other parameters maintain the outcomes of the initial estimation when optimizing for μ₁.

Process the following objective optimization problem to obtain the optimal estimate of μ₁: ${\begin{matrix} min_{(μ_{1})} \sum_{z = 1}^{m} \sum_{p = 1}^{P} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (μ_{1})^{p} - \int_{0}^{1} Φ^{- 1} (α)^{p} d α)}^{2} \\ subject to : \\ max (μ_{1}^{1}, μ_{1}^{2}, \dots, μ_{1}^{m}) \geq μ_{1} \\ min (μ_{1}^{1}, μ_{1}^{2}, \dots, μ_{1}^{m}) \leq μ_{1} \end{matrix}$ where k_zi (μ₁) refers to the parameter function containing the unknown parameter μ₁ and $k_{zi} (μ_{1}) = k_{zi} (μ_{1}, μ_{2}^{z}, \dots, μ_{s}^{z}, σ_{1}^{z}, \dots, σ_{v}^{z}) .$

After optimization, both the best estimate of μ₁ is obtained and the sum of the residuals associated with μ₁ is minimized. The best estimate of all parameters and the minimum residual R can be attained by improving each parameter with multiple estimated results in turn. When optimizing other parameters, the optimum values of the parameters are applied to those that have already been improved. For example, when estimating the optimal parameters for μ₂, $μ_{1} = \hat{μ_{1}}$ instead of $μ_{1} = μ_{1}^{z}$ (z = 1, 2, ⋯ , m).

The final optimal and unique parameter estimation results are obtained as follows: $\hat{μ} = ({\hat{μ}}_{1}, {\hat{μ}}_{2}, \dots, {\hat{μ}}_{s})^{⊤}$ and $\hat{σ} = ({\hat{σ}}_{1}, {\hat{σ}}_{2}, \dots, {\hat{σ}}_{v})^{⊤}$ .

Example 1. Consider the following UDE: $d X_{t} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) μ d t + (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}) d C_{t}$ where μ = (μ₁, μ₂) ^TDSFTDT are two parameters to be estimated. This two-dimensional UDE can be expressed as ${\begin{matrix} d X_{1 t} = μ_{1} d t + d C_{1 t} + d C_{2 t} \\ d X_{2 t} = μ_{2} d t + d C_{1 t} + d C_{2 t} . \end{matrix}$ Assume that there are N observations. The parameter equations can be obtained ${\begin{matrix} k_{1 i} (μ_{1}) = \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - μ_{1} Δ t_{i}}{2 Δ t_{i}} \\ k_{2 i} (μ_{2}) = \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - μ_{2} Δ t_{i}}{2 Δ t_{i}} . \end{matrix}$ According to the equation (6), two minimization problems can be obtained: ${\begin{matrix} min_{μ_{1}} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (μ_{1}))}^{2} \\ min_{μ_{2}} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{2 i} (μ_{2}))}^{2} . \end{matrix}$ By calculation, the final estimate of μ₁ and μ₂ is ${\begin{matrix} {\hat{μ}}_{1} = \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}}}{Δ t_{i}} \\ {\hat{μ}}_{2} = \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}}}{Δ t_{i}} . \end{matrix}$ Example 2. Consider the two-dimensional UDE: $d X_{t} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) μ X_{t} d t + (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) σ X_{t} d C_{t}$ with two parameters μ and σ > 0 to be estimated. The form of the equation is ${\begin{matrix} d X_{1 t} = μ X_{1 t} d t + σ X_{1 t} d C_{t} \\ d X_{2 t} = μ X_{2 t} d t + σ X_{2 t} d C_{t} . \end{matrix}$ Assume that there are N observations. The parameter equations can be obtained: ${\begin{matrix} k_{1 i} (μ; σ) = \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - μ x_{1 t_{i}} Δ t_{i}}{σ x_{1 t_{i}} Δ t_{i}} \\ k_{2 i} (μ; σ) = \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - μ x_{2 t_{i}} Δ t_{i}}{σ x_{2 t_{i}} Δ t_{i}} . \end{matrix}$ According to the Equation (6), the generalized moment estimation $(\hat{μ}, \hat{σ})$ is the optimal solution of ${\begin{matrix} min_{(μ, σ)} {{(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (μ, σ))}^{2} \\ + {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (μ, σ)^{2} - 1)}^{2}} \\ min_{(μ, σ)} {{(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (μ, σ))}^{2} \\ + {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (μ, σ)^{2} - 1)}^{2}} . \end{matrix}$ By solving the above two minimization problems respectively, we can get: ${\begin{matrix} μ_{z} & = \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{z (t_{i} + Δ t_{i})} - x_{{zt}_{i}}}{x_{{zt}_{i}} Δ t_{i}} \\ (σ_{z})^{2} & = \frac{1}{N - 1} \sum_{i = 1}^{N - 1} {(\frac{x_{z (t_{i} + Δ t_{i})} - x_{{zt}_{i}}}{x_{{zt}_{i}} Δ t_{i}})}^{2} \\ - {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{z (t_{i} + Δ t_{i})} - x_{{zt}_{i}}}{x_{{zt}_{i}} Δ t_{i}})}^{2} \end{matrix}$ where z = 1, 2. Since two parameter estimates are available for both μ and σ, the following optimization is required: ${\begin{matrix} min_{(μ)} \sum_{z = 1}^{2} K_{z} (μ) \\ subject to : \\ max (μ_{1}, μ_{2}) \geq μ \\ min (μ_{1}, μ_{2}) \leq μ \end{matrix}$ where $\begin{matrix} K_{z} (μ) & = {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (μ, σ_{z}))}^{2} \\ + {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (μ, σ_{z})^{2} - 1)}^{2} \end{matrix}$ and σ_z is the result of the initial estimation. The optimal estimate of μ can be obtained as $\hat{μ}$ . The optimization of σ is shown as follows: ${\begin{matrix} min_{(σ)} \sum_{z = 1}^{2} K_{z} (σ, \hat{μ}) \\ subject to : \\ max (σ_{1}, σ_{2}) \geq σ \\ min (σ_{1}, σ_{2}) \leq σ \end{matrix}$ where $\begin{matrix} K_{z} (σ, \hat{μ}) & = {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (σ, \hat{μ}))}^{2} \\ + {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{zi} (σ, \hat{μ})^{2} - 1)}^{2} . \end{matrix}$

Example 3. Consider the following UDE: ${\begin{matrix} d X_{1 t} = (X_{1 t} + X_{2 t}) d t + σ_{1} d C_{1 t} \\ d X_{2 t} = X_{2 t} d t + σ_{2} d C_{2 t} \end{matrix}$ where σ₁, σ₂ > 0 are parameters to be estimated. According to the equation (6), the estimates ${\hat{σ}}_{1}$ and ${\hat{σ}}_{2}$ solve the optimization problems ${\begin{matrix} min_{σ_{1}} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (σ_{1}))}^{2} \\ min_{σ_{2}} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{2 i} (σ_{2}))}^{2} \end{matrix}$ where ${\begin{matrix} k_{1 i} (σ_{1}) = \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - (x_{1 t_{i}} + x_{2 t_{i}}) Δ t_{i}}{σ_{1} Δ t_{i}} \\ k_{2 i} (σ_{2}) = \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - x_{2 t_{i}} Δ t_{i}}{σ_{2} Δ t_{i}} . \end{matrix}$ Since when ${\begin{matrix} \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - (x_{1 t_{i}} + x_{2 t_{i}}) Δ t_{i}}{σ_{1} Δ t_{i}} = 0 \\ \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - x_{2 t_{i}} Δ t_{i}}{σ_{2} Δ t_{i}} = 0, \end{matrix}$ it can be deduced that ${\begin{matrix} \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - (x_{1 t_{i}} + x_{2 t_{i}}) Δ t_{i}}{Δ t_{i}} = 0 \\ \frac{1}{N - 1} \sum_{i = 1}^{N - 1} \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - x_{2 t_{i}} Δ t_{i}}{Δ t_{i}} = 0 \end{matrix}$ Therefore the above optimization equation cannot find the estimates of σ₁ and σ₂. A new optimization problem is obtained based on the fact that the 2-order sample moments are equal to the 2-order population moments: ${\begin{matrix} min_{σ_{1}} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{1 i} (σ_{1})^{2} - 1)}^{2} \\ min_{σ_{2}} {(\frac{1}{N - 1} \sum_{i = 1}^{N - 1} k_{2 i} (σ_{2})^{2} - 1)}^{2} . \end{matrix}$ The final estimation results is ${\begin{matrix} {\hat{σ}}_{1} = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N - 1} {(\frac{x_{1 (t_{i} + Δ t)} - 2 x_{1 t_{i}} - x_{2 t_{i}} Δ t}{Δ t})}^{2}} \\ {\hat{σ}}_{2} = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N - 1} {(\frac{x_{2 (t_{i} + Δ t)} - 2 x_{2 t_{i}} Δ t}{Δ t})}^{2}} . \end{matrix}$

3.3 Reasonableness test of estimated results

By means of the generalized moment estimation, we have obtained estimates $(\hat{μ}, \hat{σ})$ of the unknown parameters, as well as $k_{z 1} (\hat{μ}, \hat{σ})$ , $k_{z 2} (\hat{μ}, \hat{σ})$ , $\dots, k_{zN - 1} (\hat{μ}, \hat{σ})$ which obey the standard normal distribution N (0, 1) for any z. In the following, we will test the estimation results using the α-path method and hypothesis testing methods.

The α-path method is an intuitive way of determining whether parameter estimates are reasonable through images. When the results of the parameter estimation are obtained, we can determine the multi-dimensional uncertain differential equation and the corresponding ordinary differential equation. The parameter estimates are reasonable if two α-paths can be found to sandwich all observations when α ∈ (0, 1).

Hypothesis testing is an objective and rigorous method of discerning the reasonableness of the parameter estimates. When the observations fall into the rejection domain, the original hypothesis is rejected. The detailed steps are shown below:

For any set of k_zi(i = 1, 2, ⋯ , N - 1), the test for the hypotheses: $\begin{matrix} H_{0} : μ = \hat{μ} & and σ = \hat{σ} \\ vs \\ H_{1} : μ \neq \hat{μ} & or σ \neq \hat{σ} \end{matrix}$ at significance level α is

W= {(k_z1, k_z2, ⋯ , k_zN-1): there are at least α of indexes i’s with 1 ≤ i ≤ n such as $z_{i} < Φ^{- 1} (\frac{α}{2})$ or $z_{i} > Φ^{- 1} (1 - \frac{α}{2})$ },

where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α} .$ If the number of k_zi satisfying $k_{zi} \notin [Φ^{- 1} (\frac{α}{2}), Φ^{- 1} (1 - \frac{α}{2})]$ is at least (N - 1) α, then (k_z1, k_z2, ⋯ , k_zN-1) ∈ W. And the original hypothesis H₀ is rejected, meaning that the parameter estimate result is erroneous.

According to the same method, the hypothesis testing is performed on the data k_zi when z = 1, 2, ⋯ , m in turn. When all the original hypothesis hold, the estimation results are judged to be reasonable for MUDEs with unknown parameters.

4 Numerical examples

In this section, two numerical examples with observed data are used to validate the above parameter estimation method. The α-path and hypothesis testing are used to verify the reasonableness of the parameter estimation results.

Example 4. Suppose the two-dimensional UDE with 25 observations shown in Table 1: ${\begin{matrix} d X_{1 t} = μ_{1} d t + σ_{1} d C_{1 t} \\ d X_{2 t} = μ_{2} d t + σ_{1} d C_{1 t} + σ_{2} d C_{2 t} \end{matrix}$

where μ₁, μ₂, σ₁, and σ₂ are unknown parameters. According to equation (6), we have ${\begin{matrix} min_{(μ_{1}, σ_{1})} {{(\frac{1}{24} \sum_{i = 1}^{24} k_{1 i} (μ_{1}, σ_{1}))}^{2} \\ + {(\frac{1}{24} \sum_{i = 1}^{24} k_{1 i} (μ_{1}, σ_{1})^{2} - 1)}^{2}} \\ min_{(μ_{2}, σ_{1}, σ_{2})} {{(\frac{1}{24} \sum_{i = 1}^{24} k_{2 i} (μ_{2}, σ_{1}, σ_{2}))}^{2} \\ + {(\frac{1}{24} \sum_{i = 1}^{24} k_{2 i} (μ_{2}, σ_{1}, σ_{2})^{2} - 1)}^{2} \\ + {(\frac{1}{24} \sum_{i = 1}^{24} k_{2 i} (μ_{2}, σ_{1}, σ_{2})^{3})}^{2}} . \end{matrix}$ and ${\begin{matrix} k_{1 i} (μ_{1}, σ_{1}) = \frac{x_{1 (t_{i} + Δ t_{i})} - x_{1 t_{i}} - μ_{1} Δ t_{i}}{σ_{1} Δ t_{i}} \\ k_{2 i} (μ_{2}, σ_{1}, σ_{2}) = \frac{x_{2 (t_{i} + Δ t_{i})} - x_{2 t_{i}} - μ_{2} Δ t_{i}}{(σ_{1} + σ_{2}) Δ t_{i}}, \end{matrix}$ Thus, the generalized moment estimation of the unknown parameters are: estimation results: ${\begin{matrix} ({\hat{μ}}_{1}, σ_{1}^{1}) = (0.075, 12.0239) \\ ({\hat{μ}}_{2}, σ_{1}^{2}, {\hat{σ}}_{2}) = (1.1745, 8.8083, 3.4835) . \end{matrix}$ The following optimization is performed for σ₁ with two parameter estimation results: ${\begin{matrix} min_{(σ_{1})} \sum_{z = 1}^{2} K_{z} (σ_{1}) \\ subject to : \\ max (σ_{1}^{1}, σ_{1}^{2}) \geq σ_{1} \\ min (σ_{1}^{1}, σ_{2}^{2}) \leq σ_{1}, \end{matrix}$ where $\begin{matrix} K_{1} (σ_{1}) & = {{(\frac{1}{24} \sum_{i = 1}^{24} k_{1 i} (\hat{μ_{1}}, σ_{1}))}^{2} \\ + {(\frac{1}{24} \sum_{i = 1}^{24} k_{1 i} (\hat{μ_{1}}, σ_{1})^{2} - 1)}^{2}} \end{matrix}$ and $\begin{matrix} K_{2} (σ_{1}) & = {{(\frac{1}{24} \sum_{i = 1}^{24} k_{2 i} (\hat{μ_{2}}, σ_{1}, \hat{σ_{2}}))}^{2} \\ + {(\frac{1}{24} \sum_{i = 1}^{24} k_{2 i} (\hat{μ_{2}}, σ_{1}, \hat{σ_{2}})^{2} - 1)}^{2} \\ + {(\frac{1}{24} \sum_{i = 1}^{24} k_{2 i} (\hat{μ_{2}}, σ_{1}, \hat{σ_{2}})^{3})}^{2}} . \end{matrix}$

Table 1
Observed datas in Example 4

t _i 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

x _{1t_i} 3.12 2.10 3.08 2.11 2.22 2.3 4.28 3.16 2.26 3.26

x _{2t_i} 2.22 3.16 3.08 4.11 3.22 4.12 5.10 3.04 3.11 2.72

t _i 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

x _{1t_i} 4.30 5.38 3.50 1.36 3.34 2.43 3.43 5.55 4.54 3.54

x _{2t_i} 5.12 4.13 3.58 2.91 5.42 4.62 3.13 2.68 2.61 3.22

t _i 3.1 3.2 3.3 3.4 3.5

x _{1t_i} 3.56 4.56 4.60 3.55 3.3

x _{2t_i} 5.12 2.88 3.48 2.61 2.82

The final optimal estimate of σ₁ is: 11.2623. In conclusion, the estimated value of the unknown parameter is $({\hat{μ}}_{1}, {\hat{μ}}_{2}, {\hat{σ}}_{1}, {\hat{σ}}_{2}) = (0.075, 1.1745, 11.2623, 3.4835) .$

Hence the resulting two-dimensional UDE is ${\begin{matrix} d X_{1 t} = 0.075 d t + 11.2623 d C_{1 t} \\ d X_{2 t} = 1.1745 d t + 11.2623 d C_{1 t} + 3.4835 d C_{2 t}, \end{matrix}$ and the corresponding ordinary differential equation is ${\begin{matrix} d X_{1 t}^{α} = 0.075 d t + 11.2623 Φ^{- 1} (α) d t \\ d X_{2 t}^{α} = 1.1745 d t + 14.7458 Φ^{- 1} (α) d t . \end{matrix}$

Fig. 1

Observations and α-path of X_1t in Example 4.

Fig. 2

Observations and α-path of X_2t in Example 4.

After graphing the observed values, we can visually see that the observations are all sandwiched by two α-paths. Therefore, it can be intuitively judged that the parameter estimation results are reasonable.

Table 2

The data for the parameter functions in Example 4

i	1	2	3	4	5	6	7	8
k _1i	-0.9123	0.8635	-0.8679	0.0910	0.0644	1.7510	-1.0010	-0.8058
k _2i	0.5578	-0.1339	0.6189	-0.6832	0.5307	0.5849	-1.4770	-0.0322
i	9	10	11	12	13	14	15	16
k _1i	0.8813	0.9168	0.9523	-1.6760	-1.9070	1.7510	-0.8147	0.8813
k _2i	-0.3441	1.5480	-0.7510	-0.4526	-0.5340	1.6230	-0.6222	-1.0900
i	17	18	19	20	21	22	23	24
k _1i	1.8760	-0.9035	-0.8946	0.0111	0.8813	0.0289	-0.9390	-0.2286
k _2i	-0.3848	-0.1271	0.3340	1.2090	-1.5990	0.3272	-0.6696	0.0628

In order to be objective and rigorous, we perform hypothesis tests on the parameter results. For the parameter function k_1i (μ₁, σ₁), the test for the hypotheses: $\begin{matrix} H_{0} : μ_{1} = \hat{μ_{1}} & and σ_{1} = \hat{σ_{1}} \\ vs \\ H_{1} : μ_{1} \neq \hat{μ_{1}} & or σ_{1} \neq \hat{σ_{1}} \end{matrix}$ at significance level α = 0.05 is

W= {(k₁₍₁₎, k₁₍₂₎, ⋯ , k₁₍₂₄₎): there are at least 1.2 of indexes i’s with 1 ≤ i ≤ n such as $z_{i} < Φ^{- 1} (\frac{α}{2})$ or $z_{i} > Φ^{- 1} (1 - \frac{α}{2})$ },

where $Φ^{- 1} (\frac{0.05}{2}) = - 2.0198, Φ^{- 1} (1 - \frac{0.05}{2}) = 2.0198 .$

However, for each i = 1, 2, ⋯ , 24, k_1i∈ [-2.0198, 2.0198]. Therefore, ((k₁₍₁₎, k₁₍₂₎, ⋯ , k₁₍₂₄₎) ∉ W. The original hypothesis H₀ holds and $({\hat{μ}}_{1}, {\hat{σ}}_{1})$ is reasonable for the first equation.

For the parameter function k_2i (μ₂, σ₁, σ₂), the test for the hypotheses: $\begin{matrix} H_{0} : μ_{2} = \hat{μ_{2}} and σ_{1} & = \hat{σ_{1}} and σ_{2} = \hat{σ_{2}} \\ vs \\ H_{1} : μ_{2} \neq \hat{μ_{2}} or σ_{1} & \neq \hat{σ_{1}} or σ_{2} \neq \hat{σ_{2}} \end{matrix}$ at significance level α = 0.05 is

W= {(k₂₍₁₎, k₂₍₂₎, ⋯ , k₂₍₂₄₎): there are at least 1.2 of indexes i’s with 1 ≤ i ≤ n such as z_i < -2.0198 or z_i > 2.0198}.

For any i = 1, 2, ⋯ , 24, k_2i∈ [-2.0198, 2.0198]. Therefore, (k₂₍₁₎, k₂₍₂₎, ⋯ , k₂₍₂₄₎) ∉ W. The original hypothesis H₀ holds and $({\hat{μ}}_{2}, {\hat{σ}}_{1}, {\hat{σ}}_{2})$ is reasonable for the second equation.

In summary, the parameter estimates obtained by generalised moment estimation are reasonable.

Example 5. Assume the uncertain stock price model with uncertain volatility with 45 observations shown in Table 3 as follows: ${\begin{matrix} d D_{t} = {rD}_{t} d t \\ d X_{t} = μ X_{t} d t + \sqrt{σ_{t}} X_{t} d C_{1 t} \\ d σ_{t} = k (θ - σ_{t}) d t + σ \sqrt{σ_{t}} d C_{2 t} \end{matrix}$ where C_1t and C_2t are independent Liu processes, X_t denotes the stock price at time t, D_t denotes the bond price at time t, σ_t denotes the volatility of the stock price, r denotes the price of the risk-free rate set to a constant 0.006, μ denotes the long-drift of the stock price, k denotes the rate of reversion to the long-term price variance, θ denotes the long-term price variance, σ denotes the volatility. μ, k, θ, and σ are unknown parameters.

Table 3

Observed datas in Example 5

t _i	1	2	3	4	5	6	7	8	9	10	11	12
x _{t _i}	9.12	9.10	9.08	9.11	9.22	9.31	9.28	9.16	9.26	9.26	9.30	9.38
σ_{t _i}	0.02	0.03	0.05	0.04	0.09	0.10	0.06	0.08	0.09	0.08	0.06	0.11
t _i	13	14	15	16	17	18	19	20	21	22	23	24
x _{t _i}	9.50	9.36	9.34	9.30	9.43	9.55	9.54	9.54	9.56	9.59	9.60	9.55
σ_{t _i}	0.16	0.12	0.04	0.05	0.13	0.21	0.06	0.04	0.1	0.12	0.18	0.22
t _i	25	26	27	28	29	30	31	32	33	34	35	36
x _{t _i}	9.63	9.77	9.86	9.73	9.96	9.84	10.02	10.06	10.10	9.96	10.01	10.30
σ_{t _i}	0.24	0.23	0.31	0.25	0.22	0.14	0.34	0.12	0.26	0.26	0.14	0.28
t _i	37	38	39	40	41	42	43	44	45
x _{t _i}	10.38	10.20	10.16	10.34	10.30	10.28	10.50	10.36	10.45
σ_{t _i}	0.31	0.40	0.42	0.34	0.36	0.21	0.41	0.34	0.35

Parameter estimation is required only for the second and third equations in the set of equations. For the second equation, the minimization problem can be obtained from the generalized moment estimate: $min_{μ} {(\frac{1}{44} \sum_{i = 1}^{44} \frac{x_{t_{i} + Δ t_{i}} - x_{t_{i}} - μ x_{t_{i}} Δ t_{i}}{\sqrt{σ_{t_{i}}} x_{t_{i}} Δ t_{i}})}^{2},$ and the parameter function is: $k_{1 i} = \frac{x_{t_{i} + Δ t_{i}} - x_{t_{i}} - μ x_{t_{i}} Δ t_{i}}{\sqrt{σ_{t_{i}}} x_{t_{i}} Δ t_{i}} .$ The optimal estimate of u is $\hat{μ} = \sum_{i = 1}^{44} \frac{\frac{X_{t_{i} + Δ t_{i}} - X_{t_{i}}}{\sqrt{σ_{t_{i}}} X_{t_{i}} Δ t_{i}}}{\frac{1}{\sqrt{σ_{t_{i}}}}} = 0.0032 .$

Fig. 3

Observations and α-path of X_t in Example 5.

According to equation (6), the corresponding minimization problem for the third equation is: $\begin{matrix} min_{(k; θ; σ)} {(\frac{1}{44} \sum_{i = 1}^{44} k_{2 i} (k, θ, σ))^{2} + (\frac{1}{44} \sum_{i = 1}^{44} k_{2 i} (k, \\ θ, σ)^{2} - 1)^{2} + (\frac{1}{44} \sum_{i = 1}^{44} k_{2 i} (k, θ, σ)^{3})^{2}} \end{matrix}$

where $k_{2 i} = \frac{σ_{t_{i} + Δ t_{i}} - σ_{t_{i}} - k (θ - σ_{t_{i}}) Δ t_{i}}{σ \sqrt{σ_{t_{i}}} Δ t_{i}} .$ The final generalized moment estimate obtained is $(\hat{k}, \hat{θ}, \hat{σ}) = (0.6574, 0.1583, 0.2438) .$

The differential equation to be evaluated is ${\begin{matrix} d D_{t} = 0.006 D_{t} d t \\ d X_{t} = 0.0032 X_{t} d t + \sqrt{σ_{t}} X_{t} d C_{1 t} \\ d σ_{t} = 0.6574 (0.1583 - σ_{t}) d t + 0.2438 \sqrt{σ_{t}} d C_{2 t}, \end{matrix}$ and the corresponding ordinary differential equation is

Fig. 4

Observations and α-path of σ_t in Example 5.

${\begin{matrix} d D_{t}^{α} = 0.006 D_{t} d t \\ d X_{t}^{α} = 0.0032 X_{t} d t + \sqrt{σ_{t}} X_{t} Φ^{- 1} (α) d t \\ d σ_{t}^{α} = 0.6574 (0.1583 - σ_{t}) d t + 0.2438 \sqrt{σ_{t}} Φ^{- 1} (α) d t . \end{matrix}$

All of the observations are visible as falling between the two alpha-paths. The parameter estimates are then objectively tested using hypothesis testing. For the parameter function k_1i (μ), the test for the hypotheses: $H_{0} : μ = \hat{μ} vs H_{1} : μ \neq \hat{μ}$ at significance level α = 0.05 is

W= {(k₁₍₁₎, k₁₍₂₎, ⋯ , k₁₍₄₄₎): there are at least 2.2 of indexes i’s with 1 ≤ i ≤ n such as $z_{i} < Φ^{- 1} (\frac{α}{2})$ or $z_{i} > Φ^{- 1} (1 - \frac{α}{2})$ },

where $Φ^{- 1} (\frac{0.05}{2}) = - 2.0198, Φ^{- 1} (1 - \frac{0.05}{2}) = 2.0198 .$

However, for any i = 1, 2, ⋯ , 44, k_1i∈ [-2.0198, 2.0198]. Therefore, (k₁₍₁₎, k₁₍₂₎, ⋯ , k₁₍₄₄₎) ∉ W. The original hypothesis H₀ holds and $\hat{μ}$ is reasonable for the second equation.

Table 4

The data for the parameter functions in Example 5

t _i	1	2	3	4	5	6	7	8	9	10	11
k _1i	-0.0381	-0.0312	0.0005	0.0444	0.0219	-0.0203	-0.0659	0.0273	-0.0107	0.0040	0.0221
k _2i	-2.3470	-1.5240	-1.4890	-0.5695	-0.4772	-1.0160	-0.7472	-0.6015	-0.7506	-1.0370	-0.2449
t _i	12	13	14	15	16	17	18	19	20	21	22
k _1i	0.0289	-0.0449	-0.0154	-0.0374	0.0482	0.0264	-0.0093	-0.0131	-0.0055	-0.0002	-0.0062
k _2i	0.2257	-0.3987	-1.2450	-1.3900	0.1615	0.6984	-1.0380	-1.4170	-0.3644	-0.2377	0.4123
t _i	23	24	25	26	27	28	29	30	31	32	33
k _1i	-0.0198	0.0110	0.0231	0.0125	-0.0029	0.0409	-0.0325	0.0403	0.0014	0.0022	-0.0335
k _2i	0.5246	0.5296	0.3660	1.0874	0.2927	0.2484	-0.3449	2.0610	-0.7073	1.3600	0.5378
i	34	35	36	37	38	39	40	41	42	43	44
k _1i	0.0036	0.0689	0.0086	-0.0369	-0.0113	0.0224	-0.0121	-0.0086	0.0397	-0.0258	0.0094
k _2i	-0.4275	1.4030	0.8527	1.3980	1.1600	0.5825	0.9809	-0.1190	2.0940	0.6115	0.9106

For the parameter function k_2i (k, θ, σ), the test for the hypotheses: $\begin{matrix} H_{0} : k = \hat{k} and θ & = \hat{θ} and σ = \hat{σ} \\ vs \\ H_{1} : k \neq \hat{k} or θ & \neq \hat{θ} or σ \neq \hat{σ} \end{matrix}$ at significance level α = 0.10 is

W= {(k₂₍₁₎, k₂₍₂₎, ⋯ , k₂₍₄₄₎): there are at least 4.4 of indexes i’s with 1 ≤ i ≤ n such as $z_{i} < Φ^{- 1} (\frac{α}{2})$ or $z_{i} > Φ^{- 1} (1 - \frac{α}{2})$ },

where $Φ^{- 1} (\frac{0.10}{2}) = - 1.6234, Φ^{- 1} (1 - \frac{0.10}{2}) = 1.6234 .$

Due to the k₂₍₁₎, k₂₍₃₀₎ and k₂₍₄₂₎∉ [-1.6234, 1.6234], (k₂₍₁₎, k₂₍₂₎, ⋯ , k₂₍₄₄₎) ∉ W. Hence, the original hypothesis H₀ holds and $(\hat{k}, \hat{θ}, \hat{σ})$ is reasonable for the third equation.

In summary, the parameter estimates obtained by generalised moment estimation are reasonable.

5 Conclusions

The multi-dimensional uncertain differential equations (MUDEs) are very suitable choice when modeling uncertainty for some complex events. It is inevitable that the MUDEs contains some unknown parameters, and the method to estimate the value of the unknown parameters through the observed values is called the parameter estimation of the MUDEs. In this paper, the unknown parameters in MUDEs were estimated using generalized moment estimation, and the estimated cases of one-dimensional UDE were first derived and subsequently extended to the estimation of high-dimensional UDE. Optimization methods were proposed for parameters with multiple estimation results to ensure that the parameter estimation results were optimal and unique. To test the reasonableness of the estimates, we used both α-path and hypothesis testing methods. Analytical and numerical examples were given to illustrate the feasibility of the method.

Footnotes

Acknowledgments

This work was funded by the National Natural Science Foundation of China (Grant Nos. 12061072 and 62162059) and the Xinjiang Key Laboratory of Applied Mathematics (Grant No. XJDX1401).

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t _i	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2.0
x _{1t_i}	3.12	2.10	3.08	2.11	2.22	2.3	4.28	3.16	2.26	3.26
x _{2t_i}	2.22	3.16	3.08	4.11	3.22	4.12	5.10	3.04	3.11	2.72
t _i	2.1	2.2	2.3	2.4	2.5	2.6	2.7	2.8	2.9	3.0
x _{1t_i}	4.30	5.38	3.50	1.36	3.34	2.43	3.43	5.55	4.54	3.54
x _{2t_i}	5.12	4.13	3.58	2.91	5.42	4.62	3.13	2.68	2.61	3.22
t _i	3.1	3.2	3.3	3.4	3.5
x _{1t_i}	3.56	4.56	4.60	3.55	3.3
x _{2t_i}	5.12	2.88	3.48	2.61	2.82