Parameter estimation in multifactor uncertain differential equation

Abstract

In uncertainty theory, parameter estimation of uncertain differential equation is a very important research direction. The parameter estimation of multifactor uncertain differential equation needs to be solved. Multifactor uncertain differential equation is a differential equation driven by multiple Liu processes. The paper introduces two methods to solve the unknown parameters of the multifactor uncertain differential equation, they are the method of moment estimation and the method of least squares estimation. Several numerical examples are used to illustrate the proposed parameter estimation methods.

Keywords

Uncertainty theory multifactor uncertain differential equation parameter estimation

1 Introduction

Stochastic differential equations are very important in academic research, and they also have many applications in various industries. Generally speaking, there are many contents of the study of stochastic differential equations, and an important research topic is the parameter estimation of stochastic differential equations. It is very useful for the models that we build when we study practical problems. Many scholars have explored a large number of parameter estimation methods for us, which laid the foundation for our subsequent research. In terms of the parameters estimation of the stochastic differential system, Arato et al. [1] have made some important achievements at the earliest. Kailath [2] used the least squares estimation method to estimate the parameters of a linear stochastic system. Strasser [3] estimated parameters for a class of linear problems in stochastic systems and used the method of maximum likelihood estimation. He also concluded that the parameter result is asymptotically normal. Yoshida [4] studied a nonlinear stochastic system model with unknown parameters and proposed maximum likelihood estimation for parameter estimation. Dietz and Kutoyants [5] estimated the parameter of the stochastic system by the moment method. Bishwal [6] have comprehensively introduced how to solve the parameter estimation of stochastic differential equations.

When we build the model in the application, we will encounter some uncertain sudden situations. At this time we need to use uncertainty theory to solve these problems. Liu [7] created uncertainty theory and Liu [8] further perfected the uncertainty theory by establishing four axioms, that is, normality, duality, subadditivity and product axioms. In stochastic systems, we study stochastic differential equations based on the Wiener process. To correspond to it, Liu [8] proposed the Liu process. And Liu [9] proposed the concept of uncertain differential equations on the basis of Liu process. Sometimes, the uncertain differential equation is not only affected by one Liu process, it may be affected by multiple Liu processes. So Li et al. [10] gave us the concept of multifactor uncertain differential equation. In the study of stochastic differential equations, stability is an essential research content. Similarly, Liu [8] put forward the concept of stability in uncertainty theory. Yao et al. [11] introduced some stability theorems of uncertain differential equation. Yao et al. [12] proposed the stability in mean for uncertain differential equation. Sheng and Wang [13] also gave the stability in p-th moment for uncertain differential equation. Sheng and Shi [14] proposed the stability in mean of multi-dimensional uncertain differential equation. Zhang et al. [15] introduced the stability in measure and in mean for the solution of multifactor uncertain differential equation. Solving uncertain differential equations is also a research direction in uncertain differential equations. Chen and Liu [16] found the analytic solution of the linear uncertain differential equation. Yao and Chen [17] proposed a method for solving numerical solutions.

Nowadays, uncertain differential equations are widely used in various practical problems. Uncertain differential equations are most used to build models, Liu [8] first built a stock model based on uncertain differential equations. Then Peng and Yao [18] established the stock market model of uncertainty market by taking advantage of the uncertainty differential equation and got several option pricing formulas. Liu et al. [19] have established an uncertain currency model, and they have discussed the problem of uncertain currency options and derived the pricing formula for European and American currency options. Uncertain differential equations also have applications in optimal control. Sheng and Zhu [21] studied the optimistic value model for uncertain optimal control problems and proposed the optimal principle of the model. Sheng et al. [22] presented an uncertain partial differential equation model involving the age structure of population. Sun and Sheng [27] established an uncertain SIR rumor spreading model with Liu process. For the parameter estimation of uncertain differential equations that this paper is concerned about, Liu and Yao [20] gived the method of moment estimation and Sheng et al. [23] proposed the method of least squares estimation. Sheng et al. [24] also proposed three methods for uncertain differential equations to estimate parameters based on the different forms of solutions.

Based on the general uncertain differential equations, some scholars have also proposed several special uncertain differential equations, such as multi-dimensional uncertain differential equations [25], uncertain differential equations with jumps [26] and multifactor uncertain differential equations [10]. In this paper, we research the parameter estimation of multifactor uncertain differential equations. The method of moment estimation and the method of least square estimation to estimate the parameters are proposed.

The rest of this paper is organized as follows. In Section 2, some concepts of uncertain variables and uncertain differential equations are presented. Section 3 introduces two methods to estimate unknown parameters of uncertain differential equations. Several numerical examples about the methods of estimating unknown parameters are given in Section 4. Section 5 makes the conclusion in the end.

2 Preliminary

In this section, we give some definitions of relevant knowledge points in uncertain theory about the research topic.

Definition 1 (Liu [7]) Let ℒ be a σ-algebra on a nonempty set Γ. Each element Λ in ℒ is called a event in uncertainty theory and ℳ {Γ} is assigned to each event Λ to indicate the belief degree with which we believe Λ will happen. Liu suggested the following three axioms:

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

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

Axiom 3: (Subadditivity Axiom) For each countable sequence of events Λ₁, Λ₂, ⋯ , there is $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$ If the set function ℳ satisfies the normality, duality, and subadditivity axioms, we just call ℳ an uncertain measure.

Besides, Liu [8] defined the product uncertain measure as follows:

Axiom 4: (Product Axiom) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure ℳ is an uncertain measure, which needs to satisfy $ℳ {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}},$ where Λ_k is an event arbitrarily selected from ℒ_k for k = 1, 2, ⋯.

Definition 2 (Liu [7]) An uncertain variable is a measurable function from an uncertain space (Γ, ℒ, ℳ) to the set of real numbers. For any Borel set B of real numbers, the set {γ ∈ Γ|ξ (γ) ∈ B} is an event.

Definition 3 (Liu [7]) For any real number x, we define the uncertain distribution Φ of the uncertain variable ξ as $Φ (x) = ℳ {ξ \leq x}, x \in R .$

In particular, if the distribution of uncertain variable is normally uncertain distribution $Φ (x) = {(1 + exp (\frac{π (e - x)}{\sqrt{3} σ}))}^{- 1}, x \in R .$ Then ξ is a normal uncertain variable, expressed by $N$ (e, σ), where e and σ are real numbers, and σ > 0.

Definition 4 (Liu [7]) Let ξ be an uncertain variable, then the expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} ℳ {ξ \geq x} d x - \int_{- \infty}^{0} ℳ {ξ \leq x} d x .$ At least one of the two integrals in the formula exists. And the variance of ξ is defined by $V [ξ] = E [{(ξ - E [ξ])}^{2}] .$

In particular, if the distribution of uncertain variable is ξ normal uncertain distribution, i.e., ξ ∼ $N$ (e, σ), then its expected value is E [ξ] = e, and its variance is V [ξ] = σ² .

Definition 5 (Liu [7]) Let ξ be an uncertain variable, and k be a positive integer. Then the k-th moment of ξ is defined by $E [ξ^{k}] = \int_{0}^{+ \infty} ℳ {ξ^{k} \geq r} d r - \int_{- \infty}^{0} ℳ {ξ^{k} \leq r} d r$ provided that at least one of the two integrals is finite.

If ξ has an inverse uncertainty distribution Φ^-1 (α), then $E [ξ^{k}] = \int_{0}^{1} {(Φ^{- 1} (α))}^{k} d α .$ Hence, for a standard normal uncertain variable ξ ∼ $N$ (0, 1) , we have $E [ξ^{k}] = {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α .$

Definition 6 (Liu [8]) If for any Borel set B₁, B₂, ⋯ , B_n, the uncertain variables ξ₁, ξ₂, ⋯ , ξ_n satisfy $ℳ {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} ℳ {ξ_{i} \in B_{i}},$ we call these uncertain variables independent.

Definition 7 (Liu [8]) The Liu process C_t is an uncertain process that satisfies the following three conditions:

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

(ii) C_t has stationary and independent increments,

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

Definition 8 (Liu [8]) Let C_t be a Liu process, f and g are two measurable real functions. X_t is an unknown uncertain process. Then we call $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (1) an uncertain differential equation.

The integral form of uncertain differential equation (1) is $X_{s} = X_{0} + \int_{0}^{s} f (t, X_{t}) d t + \int_{0}^{s} g (t, X_{t}) d C_{t} .$

Definition 9 (Li et al. [10]) Let C_1t, C_2t, ⋯ , C_nt be independent Liu processes, and f and g_i are given functions, i = 1, 2, ⋯ , n. Then $d X_{t} = f (t, X_{t}) d t + \sum_{i = 1}^{n} g_{i} (t, X_{t}) d C_{it}$ (2) is called a multifactor uncertain differential equation with respect to C_it, i = 1, 2, ⋯ , n. A solution is an uncertain process X_t that satisfies (2) identically at each time t.

A multifactor uncertain differential equation is essentially a type of differential equation driven by multiple Liu processes. The uncertain differential (2) and the uncertain integral $X_{t} = X_{0} + \int_{0}^{t} f (t, X_{s}) d s + \sum_{i = 1}^{n} \int_{0}^{t} g_{i} (t, X_{s}) d C_{is}$ is equivalent.

3 Parameter estimation

3.1 Moment estimation

In this part, we will use the method of moment estimation to estimate the parameters of multifactor uncertain differential equation.

First, we consider the simple multifactor uncertain differential equation. The coefficient function of each Liu process is the same.

Theorem 1. Suppose the multifactor uncertain differential equation $d X_{t} = f (t, X_{t}; μ) d t + \sum_{j = 1}^{m} g (t, X_{t}; σ) d C_{jt},$ (3) where the parameter μ and σ are to be estimated. Let x_{t
_i} be the observed value at time t_i, i = 1, 2, ⋯ , n. Then the moment estimation value $\hat{μ}$ and $\hat{σ}$ satisfy $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; \hat{μ}) (t_{i + 1} - t_{i})}{mg (t_{i}, x_{t_{i}}; \hat{σ}) (t_{i + 1} - t_{i})})}^{k}$ $= {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α,$ k = 1, 2, ⋯ , K, where K is the number of unknown parameters.

Proof. The difference form of equation (3) is $\begin{matrix} X_{t_{i + 1}} & = X_{t_{i}} + f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i}) \\ + \sum_{j = 1}^{m} g (t_{i}, X_{t_{i}}; σ) (C_{{jt}_{i + 1}} - C_{{jt}_{i}}) . \end{matrix}$ After transforming the above equation, we get $\begin{matrix} \frac{X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i})}{g (t_{i}, X_{t_{i}}; σ)} \\ = \sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}}) . \end{matrix}$ (4) Because the increment C_{jt
_i+1} - C_{jt
_i} ∼ $N$ (0, t_i+1 - t_i) are independent, respectively, j = 1, 2, ⋯ , m. Then the sum $\sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}})$ is also a normal uncertain variable $N$ (0, m (t_i+1 - t_i)). We divide the left term and the right term of equation (4) by m (t_i+1 - t_i) to get $\begin{matrix} \frac{X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i})}{g (t_{i}, X_{t_{i}}; σ) \cdot m (t_{i + 1} - t_{i})} \\ = \frac{\sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}})}{m (t_{i + 1} - t_{i})} \sim N (0, 1) . \end{matrix}$ We suppose that the observed value at time t_i is x_{t
_i}, i = 1, 2, ⋯ , n - 1, and substitute the observed values into the left term in above formula, i.e., $\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i})}{g (t_{i}, x_{t_{i}}; σ) \cdot m (t_{i + 1} - t_{i})} \sim N (0, 1) .$ Then we have n-1 values of the formulas with the unknown parameters, and these values can be regard as the samples of the standard normal uncertain distribution $N$ (0, 1). The k-th sample moment is $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i})}{mg (t_{i}, x_{t_{i}}; σ) (t_{i + 1} - t_{i})})}^{k},$ k = 1, 2, ⋯. The standard normal uncertain variable $N$ (0, 1) has a k-th moment ${(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α, k = 1, 2, \dots .$ This is also the k-th population moment. It follows that the sample moment is equal to the population moment, we can establish a system of equations to estimate unknown parameters μ and σ , denoted by $\hat{μ}$ and $\hat{σ}$ . We obtain $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; \hat{μ}) (t_{i + 1} - t_{i})}{mg (t_{i}, x_{t_{i}}; \hat{σ}) (t_{i + 1} - t_{i})})}^{k}$ $= {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α,$ k = 1, 2, ⋯ , K, where K is the number of unknown parameters. The theorem is proved.

Next, we have to consider a more general situation. That is, the coefficient functions of Liu processes are different.

Theorem 2. Suppose the multifactor uncertain differential equation $d X_{t} = f (t, X_{t}; μ) d t + \sum_{j = 1}^{m} g_{j} (t, X_{t}; σ) d C_{jt},$ (5) where the parameter μ and σ are to be estimated and σ_p ∈ σ , p = 1, 2, ⋯ , s. Let x_{t
_i} be the observed value at time t_i, i = 1, 2, ⋯ , n. Then the moment estimation value $\hat{μ}$ and ${\hat{σ}}_{p}$ satisfy $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; \hat{μ}) (t_{i + 1} - t_{i})}{\sum_{j = 1}^{m} ω_{j} g_{j} (t, x_{t}; \sum_{p = 1}^{s} ν_{p} {\hat{σ}}_{p}) \cdot m (t_{i + 1} - t_{i})})}^{k}$ $= {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α,$ k = 1, 2, ⋯ , K, where K is the number of unknown parameters, ω_j is the weight of measurable function g_j satisfying $\sum_{j = 1}^{m} ω_{j} = 1$ , j = 1, 2, ⋯ , m, and ν_p is the weight of unknown parameter σ_p satisfying $\sum_{p = 1}^{s} ν_{p} = 1$ , p = 1, 2, ⋯ , s.

Proof. In order to calculate unknown parameters more conveniently, we can let ω_j be the weight of measurable function g_j satisfying $\sum_{j = 1}^{m} ω_{j} = 1$ , j = 1, 2, ⋯ , m, and let ν_p be the weight of unknown parameter σ_p satisfying $\sum_{p = 1}^{s} ν_{p} = 1$ , p = 1, 2, ⋯ , s. Then we replace every unknown parameter σ_p with $\sum_{p = 1}^{s} ν_{p} σ_{p}$ and replace every coefficient function g_j with $\sum_{j = 1}^{m} ω_{j} g_{j} (t, X_{t}; \sum_{p = 1}^{s} ν_{p} σ_{p})$ . So the multifactor uncertain differential equation (4d) can be approximately written as $\begin{matrix} d X_{t} = f (t, X_{t}; μ) d t + \\ \sum_{j = 1}^{m} \sum_{j = 1}^{m} ω_{j} g_{j} (t, X_{t}; \sum_{p = 1}^{s} ν_{p} σ_{p}) d C_{jt}, \end{matrix}$ the difference form is $\begin{matrix} X_{t_{i + 1}} = X_{t_{i}} + f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i}) \\ + \sum_{j = 1}^{m} \sum_{j = 1}^{m} ω_{j} g_{j} (t, X_{t}; \sum_{p = 1}^{s} ν_{p} σ_{p}) (C_{{jt}_{i + 1}} - C_{{jt}_{i}}) . \end{matrix}$ After transforming the above equation, we get $\begin{matrix} \frac{X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i})}{\sum_{j = 1}^{m} ω_{j} g_{j} (t, X_{t}; \sum_{p = 1}^{s} ν_{p} σ_{p})} \\ = \sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}}) . \end{matrix}$ (6) According to the concept of Liu process C_t, the sum $\sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}})$ is a normal uncertain variable $N$ (0, m (t_i+1 - t_i)). We divide the left term and the right term of equation (6) by m (t_i+1 - t_i) to get $\begin{matrix} \frac{X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i})}{\sum_{j = 1}^{m} ω_{j} g_{j} (t, X_{t}; \sum_{p = 1}^{s} ν_{p} σ_{p}) \cdot m (t_{i + 1} - t_{i})} \\ = \frac{\sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}})}{m (t_{i + 1} - t_{i})} \sim N (0, 1) . \end{matrix}$ We suppose that the observed value at time t_i is x_{t
_i}, i = 1, 2, ⋯ , n. Then the k-th sample moment is $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i})}{\sum_{j = 1}^{m} ω_{j} g_{j} (t, x_{t}; \sum_{p = 1}^{s} ν_{p} σ_{p}) \cdot m (t_{i + 1} - t_{i})})}^{k},$ k = 1, 2, ⋯ . The k-th population moment is ${(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α, k = 1, 2, \dots .$ It follows that the sample moment is equal to the population moment, we can establish a system of equations to estimate unknown parameters μ and σ_p, denoted by $\hat{μ}$ and ${\hat{σ}}_{p}$ . That is $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; \hat{μ}) (t_{i + 1} - t_{i})}{\sum_{j = 1}^{m} ω_{j} g_{j} (t, x_{t}; \sum_{p = 1}^{s} ν_{p} {\hat{σ}}_{p}) \cdot m (t_{i + 1} - t_{i})})}^{k}$ $= {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α,$ k = 1, 2, ⋯ , K, where K is the number of unknown parameters. The theorem is proved.

The method of solving multifactor uncertain differential equation through the above two theorems is the method of moment estimation.

Example 1. Consider the uncertain differential equation $d X_{t} = f (t, X_{t}; μ) d t + g (t, X_{t}; σ) d C_{1 t} + g (t, X_{t}; σ) d C_{2 t},$ where the parameter μ and the parameter σ are to be estimated. Then we get $\begin{matrix} \frac{X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i})}{g (t_{i}, X_{t_{i}}; σ) \cdot 2 (t_{i + 1} - t_{i})} \\ = \frac{(C_{1 t_{i + 1}} - C_{1 t_{i}}) + (C_{2 t_{i + 1}} - C_{2 t_{i}})}{2 (t_{i + 1} - t_{i})} \sim N (0, 1) . \end{matrix}$ We suppose that the observed value at time t_i is x_{t
_i}, i = 1, 2, ⋯ , n. Then the k-th sample moment is $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i})}{2 g (t_{i}, x_{t_{i}}; σ) (t_{i + 1} - t_{i})})}^{k},$ k = 1, 2, ⋯. It follows that the sample moment is equal to the population moment, we can establish a system of equations to estimate unknown parameters μ and σ, denoted by $\hat{μ}$ and $\hat{σ}$ . That is $\begin{matrix} \frac{1}{n - 1} \sum_{i = 1}^{n - 1} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; \hat{μ}) (t_{i + 1} - t_{i})}{2 g (t_{i}, x_{t_{i}}; \hat{σ}) (t_{i + 1} - t_{i})})}^{k} \\ = {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α, \end{matrix}$ (7)k = 1, 2, ⋯ , K, where K is the number of unknown parameters.

Example 2. Consider the uncertain differential equation $d X_{t} = f (t, X_{t}; μ) d t + g_{1} (t, X_{t}; σ_{1}) d C_{1 t} + g_{2} (t, X_{t}; σ_{2}) d C_{2 t} .$ After expressing the related functions and parameters with the weight, we have $\begin{matrix} d X_{t} = & f (t, X_{t}; μ) d t + ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{1 t} \\ + ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{1 t} \\ + ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{2 t} \\ + ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{2 t}, \end{matrix}$ where the parameter μ, σ₁ and σ₂ are to be estimated. Then we get $\frac{X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}; μ) (t_{i + 1} - t_{i})}{(ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) + ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}))}$ $\cdot \frac{1}{2 (t_{i + 1} - t_{i})} = \frac{(C_{1 t_{i + 1}} - C_{1 t_{i}}) + (C_{2 t_{i + 1}} - C_{2 t_{i}})}{2 (t_{i + 1} - t_{i})}$ $\sim N (0, 1) .$ We suppose that the observed value at time t_i is x_{t
_i}, i = 1, 2, ⋯ , n. Then the k-th sample moment is $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} [\frac{x_{t_{i + 1}} - x_{t_{i}}) (t_{i + 1} - t_{i})}{(ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2})}$ $\frac{- f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i})}{+ ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2})) \cdot 2 (t_{i + 1} - t_{i})}]^{k},$ k = 1, 2, ⋯ . It follows that the sample moment is equal to the population moment, we can establish a system of equations to estimate unknown parameters μ , σ₁ and σ₂, denoted by $\hat{μ}$ , $\hat{σ_{1}}$ and $\hat{σ_{2}}$ . That is $\frac{1}{n - 1} \sum_{i = 1}^{n - 1} [\frac{x_{t_{i + 1}} - x_{t_{i}}) (t_{i + 1} - t_{i})}{(ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2})}$ $\frac{- f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i})}{+ ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2})) \cdot 2 (t_{i + 1} - t_{i})}]^{k},$ $= {(\frac{\sqrt{3}}{π})}^{k} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{k} d α,$ (8)k = 1, 2, ⋯ , K, where K is the number of unknown parameters.

3.2 Least squares estimation

In this part, we will use the method of least squares estimation to estimate the parameters of multifactor uncertain differential equations.

First, we consider the simple multifactor uncertain differential equation. The coefficient function of each Liu process is the same.

Theorem 3. Suppose the multifactor uncertain differential equation $d X_{t} = f (t, X_{t}, μ) d t + \sum_{j = 1}^{m} g (t, X_{t}, σ) d C_{jt},$ (9) where the parameter μ and σ are to be estimated. Let x_{t
_i} be the observed value at time t_i, i = 1, 2, ⋯ , n. Then the least squares estimation value μ ^* is the solution of the following optimal problem $min_{μ} \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ) (t_{i + 1} - t_{i}))^{2},$ and the least squares estimation value σ ^* is the solution of the following equation $\begin{matrix} \sum_{i = 1}^{n - 1} g (t_{i}, x_{t_{i}}; σ)^{2} \cdot m^{2} (t_{i + 1} - t_{i})^{2} \\ = \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))^{2} . \end{matrix}$

Proof. The difference form of equation (9) is $\begin{matrix} X_{t_{i + 1}} - X_{t_{i}} = f (t_{i}, X_{t_{i}}, μ) (t_{i + 1} - t_{i}) \\ + \sum_{j = 1}^{m} g (t_{i}, X_{t_{i}}, σ) (C_{{jt}_{i + 1}} - C_{{jt}_{i}}) . \end{matrix}$ (10) The equation (10) could be written as $\begin{matrix} \sum_{j = 1}^{m} g (t_{i}, X_{t_{i}}, σ) (C_{{jt}_{i + 1}} - C_{{jt}_{i}}) \\ = & X_{t_{i + 1}} - X_{t_{i}} - f (t_{i}, X_{t_{i}}, μ) (t_{i + 1} - t_{i}) . \end{matrix}$ (11) The left side of the equation (11) contains a noise term, so it should be as small as possible. This can be achieved by finding the minimum value on the right side of the equation (11). That is to solve the following optimal problem. Supposed that the observed data is x_{t
_i} at t_i, i = 1, 2, ⋯ , n, $min_{μ} \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ) (t_{i + 1} - t_{i}))^{2} .$ Then, according to the above optimization problem, the estimate of μ can be solved, denoted by μ ^*. We can get the estimate of σ from the following equation $\begin{matrix} E [\sum_{i = 1}^{n - 1} (\sum_{j = 1}^{m} g (t_{i}, x_{t_{i}}; σ) \cdot (C_{{jt}_{i + 1}} - C_{{jt}_{i}}))^{2}] \\ = & min_{μ} \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))^{2} . \end{matrix}$ (12) The left term of the equation (refpoi) is written as $\begin{matrix} E [\sum_{i = 1}^{n - 1} (\sum_{j = 1}^{m} g (t_{i}, x_{t_{i}}; σ) \cdot (C_{{jt}_{i + 1}} - C_{{jt}_{i}}))^{2}] \\ = & \sum_{i = 1}^{n - 1} E [(\sum_{j = 1}^{m} g (t_{i}, x_{t_{i}}; σ) \cdot (C_{{jt}_{i + 1}} - C_{{jt}_{i}}))^{2}] \\ = & \sum_{i = 1}^{n - 1} g (t_{i}, x_{t_{i}}; σ)^{2} \cdot E [(\sum_{j = 1}^{m} (C_{{jt}_{i + 1}} - C_{{jt}_{i}}))^{2}] \\ = & \sum_{i = 1}^{n - 1} g (t_{i}, x_{t_{i}}; σ)^{2} \cdot E [[(C_{1 t_{i + 1}} - C_{1 t_{i}}) \\ + (C_{2 t_{i + 1}} - C_{2 t_{i}}) + \dots + (C_{{mt}_{i + 1}} - C_{m t_{i}})]^{2}] . \end{matrix}$ Then we can expand the squares of sum in the above formula, that is $\begin{matrix} [(C_{1 t_{i + 1}} - C_{1 t_{i}}) + (C_{2 t_{i + 1}} - C_{2 t_{i}}) + \dots \\ + (C_{{mt}_{i + 1}} - C_{m t_{i}})]^{2} \\ = & (C_{1 t_{i + 1}} - C_{1 t_{i}})^{2} + \dots + (C_{{mt}_{i + 1}} - C_{{mt}_{i}})^{2} \\ + 2 [(C_{1 t_{i + 1}} - C_{1 t_{i}}) (C_{2 t_{i + 1}} - C_{2 t_{i}}) + \dots \\ + (C_{(m - 1) t_{i + 1}} - C_{(m - 1) t_{i}}) (C_{{mt}_{i + 1}} - C_{{mt}_{i}})] . \end{matrix}$ Since C_jt is an independent increment process, and the increment C_{jt
_i+1} - C_{jt
_i} is a normal uncertain variable with an expected value 0 and variance (t_i+1 - t_i) ², so we have E [(C_{jt
_i+1} - C_{jt
_i}) ²] = (t_i+1 - t_i) ². The inverse uncertainty distribution of the increment C_{jt
_i+1} - C_{jt
_i} is $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α} \cdot (t_{i + 1} - t_{i})$ . Then we have $\begin{matrix} E [(C_{{jt}_{i + 1}} - C_{{jt}_{i}}) (C_{(j + 1) t_{i + 1}} - C_{(j + 1) t_{i}})] \\ = & \int_{0}^{1} [\frac{\sqrt{3}}{π} ln \frac{α}{1 - α} \cdot (t_{i + 1} - t_{i})]^{2} d α \\ = & (t_{i + 1} - t_{i})^{2} \int_{0}^{1} \frac{3}{π^{2}} (ln \frac{α}{1 - α})^{2} d α \\ = & (t_{i + 1} - t_{i})^{2} . \end{matrix}$ So we can get $\begin{matrix} E [[(C_{1 t_{i + 1}} - C_{1 t_{i}}) + (C_{2 t_{i + 1}} - C_{2 t_{i}}) + \dots \\ + (C_{{mt}_{i + 1}} - C_{{mt}_{i}})]^{2}] \\ = & E [(C_{1 t_{i + 1}} - C_{1 t_{i}})^{2} + \dots + (C_{{mt}_{i + 1}} - C_{{mt}_{i}})^{2} \\ + 2 [(C_{1 t_{i + 1}} - C_{1 t_{i}}) (C_{2 t_{i + 1}} - C_{2 t_{i}}) + \dots \\ + (C_{(m - 1) t_{i + 1}} - C_{(m - 1) t_{i}}) (C_{{mt}_{i + 1}} - C_{{mt}_{i}})]] \\ = & m (t_{i + 1} - t_{i})^{2} + 2 \cdot \frac{m (m - 1)}{2} \cdot (t_{i + 1} - t_{i})^{2} \\ = & m^{2} (t_{i + 1} - t_{i})^{2} . \end{matrix}$ Hence, the estimate of σ is a solution of the following equation $\begin{matrix} \sum_{i = 1}^{n - 1} g (t_{i}, x_{t_{i}}; σ)^{2} \cdot m^{2} (t_{i + 1} - t_{i})^{2} \\ = & \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))^{2} . \end{matrix}$ The theorem is proved.

Next, we have to consider a more general situation. That is, the coefficient functions of Liu processes are different.

Theorem 4. Suppose the multifactor uncertain differential equation $d X_{t} = f (t, X_{t}; μ) d t + \sum_{j = 1}^{m} g_{j} (t, X_{t}; σ) d C_{jt},$ (13) where the parameter μ and σ are to be estimated, σ_p ∈ σ , p = 1, 2, ⋯ , s . Let x_{t
_i} be the observed value at time t_i, i = 1, 2, ⋯ , n. Then the least squares estimation value μ ^* is the solution of the following optimal problem $min_{μ} \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ) (t_{i + 1} - t_{i}))^{2},$ and the least squares estimation value $σ_{p}^{*}$ is the solution of the following equation $\begin{matrix} \sum_{i = 1}^{n - 1} {[\sum_{j = 1}^{m} ω_{j} g_{j} (t_{i}, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p})]}^{2} \cdot m^{2} {(t_{i + 1} - t_{i})}^{2} \\ = & \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))}^{2}, \end{matrix}$ where ω_j is the weight of measurable function g_j satisfying $\sum_{j = 1}^{m} ω_{j} = 1$ , j = 1, 2, ⋯ , m, and ν_p is the weight of unknown parameter σ_p satisfying $\sum_{p = 1}^{s} ν_{p} = 1$ , p = 1, 2, ⋯ , s.

$E {\sum_{i = 1}^{n - 1} [\sum_{j = 1}^{m} \sum_{j = 1}^{m} ω_{j} g_{j} (t, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p})$ $\cdot (C_{{jt}_{i + 1}} - C_{{jt}_{i}})]^{2}}$ $= min_{μ} \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))^{2} .$ (14) The left term of the equation (refpyi) is written as $\begin{matrix} E {\sum_{i = 1}^{n - 1} {[\sum_{j = 1}^{m} \sum_{j = 1}^{m} ω_{j} g_{j} (t, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p}) \cdot (C_{{jt}_{i + 1}} - C_{{jt}_{i}})]}^{2}} \\ = & \sum_{i = 1}^{n - 1} E {{[\sum_{j = 1}^{m} \sum_{j = 1}^{m} ω_{j} g_{j} (t, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p}) \cdot (C_{{jt}_{i + 1}} - C_{{jt}_{i}})]}^{2}} \\ = & \sum_{i = 1}^{n - 1} {[\sum_{j = 1}^{m} ω_{k} g_{j} (t_{i}, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p})]}^{2} \cdot E [\sum_{j = 1}^{m} {(C_{{jt}_{i + 1}} - C_{{jt}_{i}})}^{2}] \\ = & \sum_{i = 1}^{n - 1} {[\sum_{j = 1}^{m} ω_{j} g_{j} (t_{i}, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p})]}^{2} \cdot m^{2} {(t_{i + 1} - t_{i})}^{2} . \end{matrix}$ Hence, the estimates of σ_p is the solution of the following equation $\begin{matrix} \sum_{i = 1}^{n - 1} {[\sum_{j = 1}^{m} ω_{j} g_{j} (t_{i}, x_{t_{i}}; \sum_{p = 1}^{s} ν_{p} σ_{p})]}^{2} \cdot m^{2} {(t_{i + 1} - t_{i})}^{2} \\ = \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))}^{2}, \end{matrix}$ denoted by $σ_{p}^{*}$ . The theorem is proved.

The method of solving multifactor uncertain differential equation through the above two theorems is the method of least squares estimation.

Example 3. Consider the uncertain differential equation $d X_{t} = f (t, X_{t}; μ) d t + g (t, X_{t}; σ) d C_{1 t} + g (t, X_{t}; σ) d C_{2 t} .$ Assume that the observed value at time t_i is x_{t
_i}, i = 1, 2, ⋯ , n. We can estimate the unknown parameter μ by solving the following optimization problem $min_{μ} \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ) (t_{i + 1} - t_{i}))}^{2},$ (15) denoted by μ ^*. In order to estimate σ, we substitute μ ^* into the following equation $\begin{matrix} E [\sum_{i = 1}^{n - 1} [g (t_{i}, x_{t_{i}}; σ) ((C_{1 t_{i + 1}} - C_{1 t_{i}}) \\ + (C_{2 t_{i + 1}} - C_{2 t_{i}}))]^{2}] \\ = & \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ^{*}) (t_{i + 1} - t_{i}))}^{2} . \end{matrix}$ Because the increment (C_{1t_i+1} - C_{1t_i}) and the increment (C_{2t_i+1} - C_{2t_i}) are independent and (C_{jt
_i+1} - C_{jt
_i}) ∼ $N$ (0, t_i+1 - t_i), j = 1, 2. Then we have $\begin{matrix} E [\sum_{i = 1}^{n - 1} {[g (t_{i}, x_{t_{i}}; σ) ((C_{1 t_{i + 1}} - C_{1 t_{i}}) + (C_{2 t_{i + 1}} - C_{2 t_{i}}))]}^{2}] \\ = \sum_{i = 1}^{n - 1} g^{2} (t_{i}, X_{t_{i}}; σ) \cdot 4 (t_{i + 1} - t_{i})^{2} . \end{matrix}$

Therefore, the estimated value of σ is derived from the following equation $\begin{matrix} \sum_{i = 1}^{n - 1} g^{2} (t_{i}, x_{t_{i}}; σ) \cdot 4 (t_{i + 1} - t_{i})^{2} \\ = & \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ^{*}) (t_{i + 1} - t_{i}))}^{2} . \end{matrix}$ In this way, the estimated values of the unknown parameters can be obtained, that is μ ^* and σ^*.

Example 4. Consider the uncertain differential equation $\begin{matrix} d X_{t} = f (t, X_{t}, μ) d t + g_{1} (t, X_{t}, σ_{1}) d C_{1 t} \\ + g_{2} (t, X_{t}, σ_{2}) d C_{2 t} . \end{matrix}$ After expressing the related functions and parameters with the weight, we have $\begin{matrix} d X_{t} = & f (t, X_{t}; μ) d t + ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{1 t} \\ + ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{1 t} \\ + ω_{1} g_{1} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{2 t} \\ + ω_{2} g_{2} (t, X_{t}; ν_{1} σ_{1} + ν_{2} σ_{2}) d C_{2 t} . \end{matrix}$

Assume that the observed value at time t_i is x_{t
_i}, i = 1, 2, ⋯ , n. We can estimate the unknown parameter μ by solving the following optimization problem $min_{μ} \sum_{i = 1}^{n - 1} (x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ) (t_{i + 1} - t_{i}))^{2} .$ (16) denoted by μ ^*. In order to estimate σ, we substitute μ ^* into the following equation $\begin{matrix} E [\sum_{i = 1}^{n - 1} [ω_{1} g_{1} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2}) \\ + ω_{2} g_{2} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2})]^{2} [(C_{1 t_{i + 1}} - C_{1 t_{i}}) \\ + (C_{2 t_{i + 1}} - C_{2 t_{i}})]^{2}] \\ = & \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}; μ^{*}) (t_{i + 1} - t_{i}))}^{2} . \end{matrix}$ Because the increment (C_{1t_i+1} - C_{1t_i}) and the increment (C_{2t_i+1} - C_{2t_i}) are independent and (C_{jt
_i+1} - C_{jt
_i}) ∼ $N$ (0, t_i+1 - t_i), j = 1, 2. Then we have $\begin{matrix} E [\sum_{i = 1}^{n - 1} [ω_{1} g_{1} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2}) \\ + ω_{2} g_{2} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2})]^{2} \\ \cdot [(C_{1 t_{i + 1}} - C_{1 t_{i}}) + (C_{2 t_{i + 1}} - C_{2 t_{i}})]^{2}] \\ = & \sum_{i = 1}^{n - 1} [ω_{1} g_{1} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2}) \\ + ω_{2} g_{2} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2})]^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2} . \end{matrix}$

Hence, the estimates of σ₁ and σ₂ are the solution of the following equation $\begin{matrix} \sum_{i = 1}^{n - 1} [ω_{1} g_{1} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2}) \\ + ω_{2} g_{2} (t, x_{t_{i}}; ν_{1} σ_{1} + ν_{2} σ_{2})]^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2} \\ = & \sum_{i = 1}^{n - 1} {(x_{t_{i + 1}} - x_{t_{i}} - f (t_{i}, x_{t_{i}}, μ^{*}) (t_{i + 1} - t_{i}))}^{2} . \end{matrix}$ In this way, the estimated values of the unknown parameters can be obtained, that is μ ^*, $σ_{1}^{*}$ and $σ_{2}^{*}$ .s

4 Numerical experiments

In this part, we will use actual data to demonstrate the above parameter estimation methods.

Example 5. For the uncertain differential equation $d X_{t} = μ X_{t} d t + σ X_{t} d C_{1 t} + σ X_{t} d C_{2 t} .$ There are two parameters μ and σ to be estimated.

Assume that we have observed data as shown in Table 1.

Table 1
Observed Data in Example 5

i 1 2 3 4 5 6 7 8

t _i 0.5 0.8 1.1 1.4 1.8 2.2 2.5 2.9

x _{t
_i} 1.21 1.35 1.67 2.21 1.89 2.58 2.35 1.52

i 9 10 11 12 13 14 15 16

t _i 3.2 3.5 3.9 4.2 4.5 4.8 5.0 5.3

x _{t
_i} 1.48 1.75 2.26 2.47 2.32 2.65 2.73 2.49

i	1	2	3	4	5	6	7	8
t _i	0.5	0.8	1.1	1.4	1.8	2.2	2.5	2.9
x _{t _i}	1.21	1.35	1.67	2.21	1.89	2.58	2.35	1.52
i	9	10	11	12	13	14	15	16
t _i	3.2	3.5	3.9	4.2	4.5	4.8	5.0	5.3
x _{t _i}	1.48	1.75	2.26	2.47	2.32	2.65	2.73	2.49

We use the equation (7) to get the following equations ${\begin{matrix} \frac{1}{15} \sum_{i = 1}^{15} \frac{x_{t_{i + 1}} - x_{t_{i}} - \hat{μ} x_{t_{i}} (t_{i + 1} - t_{i})}{2 \hat{σ} x_{t_{i}} (t_{i + 1} - t_{i})} = 0 \\ \frac{1}{15} \sum_{i = 1}^{15} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - \hat{μ} x_{t_{i}} (t_{i + 1} - t_{i})}{2 \hat{σ} x_{t_{i}} (t_{i + 1} - t_{i})})}^{2} = 1 \end{matrix}$ which are $\hat{μ} = 0.2208, \hat{σ} = 0.2712 .$ Therefore, the multifactor uncertain differential equation is $d X_{t} = 0.2208 X_{t} d t + 0.2712 X_{t} d C_{1 t} + 0.2712 X_{t} d C_{2 t} .$

Example 6. For the uncertain differential equation $d X_{t} = μ X_{t} d t + σ_{1} X_{t} d C_{1 t} + σ_{2} d C_{2 t} .$ There are three parameters μ, σ₁ and σ₂ to be estimated.

Assume that we have observed data as shown in Table 2.

Table 2

Observed Data in Example 6

i	1	2	3	4	5	6	7	8
t _i	1.13	2.24	3.22	4.34	5.08	6.36	7.12	8.28
x _{t _i}	6.31	3.46	4.83	5.72	4.34	3.51	4.96	5.62
i	9	10	11	12	13	14	15	16
t _i	9.31	10.14	11.26	12.38	13.29	14.04	15.35	16.15
x _{t _i}	8.02	7.32	6.23	5.89	3.99	6.42	7.01	6.71

Let $ω_{1} = ω_{2} = \frac{1}{2}$ , $ν_{1} = \frac{1}{3}$ , $ν_{2} = \frac{2}{3}$ , we use the equation (8) to get the following equations ${\begin{matrix} \frac{1}{15} \sum_{i = 1}^{15} \frac{x_{t_{i + 1}} - x_{t_{i}} - \hat{μ} x_{t_{i}} (t_{i + 1} - t_{i})}{[\frac{1}{2} (\frac{1}{3} {\hat{σ}}_{1} + \frac{2}{3} {\hat{σ}}_{2}) x_{t_{i}} + \frac{1}{2} (\frac{1}{3} {\hat{σ}}_{1} + \frac{2}{3} {\hat{σ}}_{2})]} \\ \cdot \frac{1}{2 (t_{i + 1} - t_{i})} = 0 \\ \frac{1}{15} \sum_{i = 1}^{15} (\frac{x_{t_{i + 1}} - x_{t_{i}} - \hat{μ} x_{t_{i}} (t_{i + 1} - t_{i})}{[\frac{1}{2} (\frac{1}{3} {\hat{σ}}_{1} + \frac{2}{3} {\hat{σ}}_{2}) x_{t_{i}} + \frac{1}{2} (\frac{1}{3} {\hat{σ}}_{1} + \frac{2}{3} {\hat{σ}}_{2})]} \\ \cdot \frac{1}{2 (t_{i + 1} - t_{i})})^{2} = 1 \\ \frac{1}{15} \sum_{i = 1}^{15} (\frac{x_{t_{i + 1}} - x_{t_{i}} - \hat{μ} x_{t_{i}} (t_{i + 1} - t_{i})}{[\frac{1}{2} (\frac{1}{3} {\hat{σ}}_{1} + \frac{2}{3} {\hat{σ}}_{2}) x_{t_{i}} + \frac{1}{2} (\frac{1}{3} {\hat{σ}}_{1} + \frac{2}{3} {\hat{σ}}_{2})]} \\ \cdot \frac{1}{2 (t_{i + 1} - t_{i})})^{4} = \frac{21}{5} \end{matrix} .$ which are $\hat{μ} = 4.6359, {\hat{σ}}_{1} = 5.4001, {\hat{σ}}_{2} = 1.4911 .$ Therefore, the multifactor uncertain differential equation is $d X_{t} = 4.6359 X_{t} d t + 5.4001 X_{t} d C_{1 t} + 1.4911 X_{t} d C_{2 t} .$

Example 7. For the uncertain differential equation $d X_{t} = μ X_{t} d t + σ X_{t} d C_{1 t} + σ X_{t} d C_{2 t} .$ There are two parameters μ and σ to be estimated.

Assume that we have observed data as shown in Table 3, and we use the optimization problem (15) to get the estimate of μ is $μ^{*} = \frac{\sum_{i = 1}^{9} x_{t_{i}} (x_{t_{i + 1}} - x_{t_{i}}) (t_{i + 1} - t_{i})}{\sum_{i = 1}^{9} {x_{t_{i}}}^{2} {(t_{i + 1} - t_{i})}^{2}} .$ The parameter σ satisfies $\begin{matrix} \sum_{i = 1}^{9} σ^{2} \cdot {x_{t_{i}}}^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2} \\ = & \sum_{i = 1}^{9} {(x_{t_{i + 1}} - x_{t_{i}} - μ^{*} \cdot x_{t_{i}} (t_{i + 1} - t_{i}))}^{2} \end{matrix}$ which implies the estimate of σ is $σ^{*} = {(\frac{\sum_{i = 1}^{9} {(x_{t_{i + 1}} - x_{t_{i}} - μ^{*} x_{t_{i}} (t_{i + 1} - t_{i}))}^{2}}{\sum_{i = 1}^{9} {x_{t_{i}}}^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2}})}^{\frac{1}{2}} .$ So the estimated values of the unknown parameters are $μ^{*} = 0.1044, σ^{*} = 0.1680 .$ Therefore, the multifactor uncertain differential equation is $d X_{t} = 0.1044 X_{t} d t + 0.1680 X_{t} d C_{1 t} + 0.1680 X_{t} d C_{2 t} .$

Table 3

Observed Data in Example 7

i	1	2	3	4	5	6	7	8	9	10
t _i	1.5	2.1	2.9	3.5	4.1	4.7	5.0	5.4	6.1	6.5
x _{t _i}	1.21	1.12	1.45	1.62	1.35	1.78	1.32	1.52	1.65	1.87

Example 8. For the uncertain differential equation $d X_{t} = μ d t + σ_{1} X_{t} d C_{1 t} + σ_{2} X_{t} d C_{2 t} .$ There are three parameters μ, σ₁ and σ₂ to be estimated.

Assume that we have observed data as shown in Table 4, and we use the optimization problem (16) to get the estimate of μ is $μ^{*} = \frac{\sum_{i = 1}^{9} (x_{t_{i + 1}} - x_{t_{i}}) (t_{i + 1} - t_{i})}{\sum_{i = 1}^{9} {(t_{i + 1} - t_{i})}^{2}} .$ Let $ω_{1} = ω_{2} = \frac{1}{2}$ , $ν_{1} = \frac{1}{3}$ , $ν_{2} = \frac{2}{3}$ , the parameter σ₁ and σ₂ satisfy $\begin{matrix} \sum_{i = 1}^{9} {(\frac{1}{3} σ_{1} + \frac{2}{3} σ_{2})}^{2} \cdot x_{t_{i}}^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2} \\ = & \sum_{i = 1}^{9} {(x_{t_{i + 1}} - x_{t_{i}} - μ^{*} (t_{i + 1} - t_{i}))}^{2} \end{matrix}$ which implies the estimate of $\frac{1}{3} σ_{1} + \frac{2}{3} σ_{2}$ is $\frac{1}{3} σ_{1}^{*} + \frac{2}{3} σ_{2}^{*} = {(\frac{\sum_{i = 1}^{9} {(x_{t_{i + 1}} - x_{t_{i}} - μ^{*} (t_{i + 1} - t_{i}))}^{2}}{\sum_{i = 1}^{9} x_{t_{i}}^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2}})}^{\frac{1}{2}} .$ So the estimated values of the unknown parameters are $μ^{*} = 0.1614, σ_{1}^{*} = 0.0915, σ_{2}^{*} = 0.1831 .$ Therefore, the multifactor uncertain differential equation is Example $d X_{t} = 0.1614 X_{t} d t + 0.0915 X_{t} d C_{1 t} + 0.1831 X_{t} d C_{2 t} .$

Table 4

Observed Data in Example 8

i	1	2	3	4	5	6	7	8	9	10
t _i	0.32	0.68	1.12	1.56	1.98	2.21	2.57	3.01	3.42	4.00
x _{t _i}	1.32	1.65	1.72	2.31	1.98	2.64	2.11	1.69	2.45	2.27

Example 9. For the uncertain differential equation $d X_{t} = (μ_{1} - μ_{2} X_{t}) d t + σ X_{t} d C_{1 t} + σ X_{t} d C_{2 t}$ there are three parameters μ₁, μ₂ and σ to be estimated.

Assume that we have observed data as shown in Table 5.

Table 5

Observed Data in Example 9

i	1	2	3	4	5	6	7	8
t _i	0.4	0.9	1.5	1.9	2.1	2.6	2.9	3.2
x _{t _i}	2.65	3.21	2.98	3.45	3.74	4.12	3.52	4.26
i	9	10	11	12	13	14	15	16
t _i	3.6	4.0	4.5	4.9	5.3	5.8	6.2	6.7
x _{t _i}	4.54	3.95	4.89	4.65	5.01	4.36	3.65	4.21

First, we use the method of moment estimation. According to the equation (7), we get the following equations ${\begin{matrix} \frac{1}{15} \sum_{i = 1}^{15} \frac{x_{t_{i + 1}} - x_{t_{i}} - ({\hat{μ}}_{1} - {\hat{μ}}_{2} x_{t_{i}}) (t_{i + 1} - t_{i})}{2 \hat{σ} (t_{i + 1} - t_{i})} = 0 \\ \frac{1}{15} \sum_{i = 1}^{15} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - ({\hat{μ}}_{1} - {\hat{μ}}_{2} x_{t_{i}}) (t_{i + 1} - t_{i})}{2 \hat{σ} (t_{i + 1} - t_{i})})}^{2} = 1 \\ \frac{1}{15} \sum_{i = 1}^{15} {(\frac{x_{t_{i + 1}} - x_{t_{i}} - ({\hat{μ}}_{1} - {\hat{μ}}_{2} x_{t_{i}}) (t_{i + 1} - t_{i})}{2 \hat{σ} (t_{i + 1} - t_{i})})}^{3} = 0 \end{matrix}$ which are ${\hat{μ}}_{1} = 5.8868, {\hat{μ}}_{2} = 1.4356, \hat{σ} = 0.5835 .$ Therefore, the multifactor uncertain differential equation is $\begin{matrix} d X_{t} & = (5.8868 - 1.4356 X_{t}) d t + 0.5835 X_{t} d C_{1 t} \\ + 0.5835 X_{t} d C_{2 t} . \end{matrix}$ Then, we use the method of least square estimation. According to the optimization problem (15), we get the estimate of μ₁ and μ₂ are $\begin{matrix} (\begin{matrix} μ_{1}^{*} \\ μ_{2}^{*} \end{matrix}) & = {(\begin{matrix} \sum_{i = 1}^{15} {(t_{i + 1} - t_{i})}^{2}, - \sum_{i = 1}^{15} x_{t_{i}} {(t_{i + 1} - t_{i})}^{2} \\ \sum_{i = 1}^{15} x_{t_{i}} {(t_{i + 1} - t_{i})}^{2}, - \sum_{i = 1}^{15} {x_{t_{i}}}^{2} {(t_{i + 1} - t_{i})}^{2} \end{matrix})}^{- 1} \\ \times (\begin{matrix} \sum_{i = 1}^{15} (x_{t_{i + 1} - x_{t_{i}}}) (t_{i + 1} - t_{i}) \\ \sum_{i = 1}^{15} x_{t_{i}} (x_{t_{i + 1} - x_{t_{i}}}) (t_{i + 1} - t_{i}) \end{matrix}) . \end{matrix}$ The parameter σ satisfies $\begin{matrix} \sum_{i = 1}^{15} σ^{2} \cdot {x_{t_{i}}}^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2} \\ = & \sum_{i = 1}^{15} {(x_{t_{i + 1}} - x_{t_{i}} - (μ_{1}^{*} - μ_{2}^{*} x_{t_{i}}) (t_{i + 1} - t_{i}))}^{2}, \end{matrix}$ which implies the estimate of σ is $σ^{*} = {(\frac{\sum_{i = 1}^{15} {(x_{t_{i + 1}} - x_{t_{i}} - (μ_{1}^{*} - μ_{2}^{*} x_{t_{i}}) (t_{i + 1} - t_{i}))}^{2}}{\sum_{i = 1}^{15} {x_{t_{i}}}^{2} \cdot 4 {(t_{i + 1} - t_{i})}^{2}})}^{\frac{1}{2}} .$ So the estimated values of the unknown parameters are $μ_{1}^{*} = 3.6215, μ_{2}^{*} = 0.8689, σ^{*} = 0.1360 .$ Therefore, the multifactor uncertain differential equation is $\begin{matrix} d X_{t} & = (3.6215 - 0.8689 X_{t}) d t + 0.1360 X_{t} d C_{1 t} \\ + 0.1360 X_{t} d C_{2 t} . \end{matrix}$

5 Conclusions

Parameter estimation of uncertain differential equation is always a worthy question of discussion. In this paper, we mainly proposed the methods of moment estimation and least square estimation to estimate the unknown parameter of multifactor uncertain differential equations. We first solved the problem of simple forms of multifactor uncertain differential equations, and then extended to the general form. The paper also gave some related numerical examples to demonstrate the parameter estimation method. In the future, the methods of parameter estimation in multifactor uncertain differential equations can be applied to practical models, such as multifactor uncertain stock model with floating interest rate. We can also study the errors between the estimated results of the proposed methods and the actual situation. The research prospects of uncertain differential equations are still very broad, and we will next study the parameter estimation of other special uncertain differential equations.

Footnotes

Acknowledgments

This research is funded by the Natural Science Foundation of Xinjiang (Grant No. 2020D01C017) and the National Natural Science Foundation of China (Grant No. 12061072) and Intelligent manufacturing integrated standardization and new mode application project-Beryllium Copper alloy network collaborative manufacturing new mode application project (No. 2018YFC0825504).

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Parameter estimation in multifactor uncertain differential equation

Abstract

Keywords

1 Introduction

2 Preliminary

3.1 Moment estimation

Table 1 Observed Data in Example 5 i 1 2 3 4 5 6 7 8 t i 0.5 0.8 1.1 1.4 1.8 2.2 2.5 2.9 x t i 1.21 1.35 1.67 2.21 1.89 2.58 2.35 1.52 i 9 10 11 12 13 14 15 16 t i 3.2 3.5 3.9 4.2 4.5 4.8 5.0 5.3 x t i 1.48 1.75 2.26 2.47 2.32 2.65 2.73 2.49

Footnotes

Acknowledgments

References

Table 1
Observed Data in Example 5

i 1 2 3 4 5 6 7 8

t _i 0.5 0.8 1.1 1.4 1.8 2.2 2.5 2.9

x _{t
_i} 1.21 1.35 1.67 2.21 1.89 2.58 2.35 1.52

i 9 10 11 12 13 14 15 16

t _i 3.2 3.5 3.9 4.2 4.5 4.8 5.0 5.3

x _{t
_i} 1.48 1.75 2.26 2.47 2.32 2.65 2.73 2.49