Moment estimations for parameters in uncertain delay differential equations

Abstract

As a type of differential equations driven by Liu process, uncertain delay differential equations (UDDEs) model dynamic systems with after-effects or memories in uncertain environment by incorporating time delay terms. Because it is natural for UDDEs to incorporate some unknown parameters, how to estimate them is a crucial problem in practice. This paper undertakes this issue by applying the method of moments based on discrete observations of solutions. With the Euler difference form of UDDEs, a function with respect to unknown parameters is proved to follow a standard normal uncertainty distribution. The moment estimations for unknown parameters are obtained by solving a system of equations which uses sample moments to approximate population moments. Analytic solutions for some types of UDDEs are derived. Numerical examples show that estimations give small biases and standard deviations as long as time steps are not too large. Applications to population growth models further illustrate the practicability of our method.

Keywords

Uncertain differential equation parameter estimation moments method uncertainty theory

1 Introduction

Depicting white noises as Wiener processes, stochastic differential equations (SDEs) describe time evolution of dynamic systems whose rates of changes depend only on their present states and some random noises. Noticing that SDEs usually incorporate unknown parameters, some scholars [5 , 26] were devoted to estimating these parameters based on observations of the systems. However in many fields such as medicine and economics [25], rates of changes depend both on present states and past states of systems. In such cases, stochastic delay differential equations (SDDEs) which incorporate past states using time delay terms were developed and provided important tools to describe and analyze these systems [24]. Later some numerical techniques for SDDEs [3, 6, 11] were widely investigated because usually there are no analytical form solutions.

Although there exists a thorough study of SDEs and SDDEs, they are both considered under the framework of probability theory. A prerequisite for probability theory is that the obtained distribution function is close enough to the real frequency. However, in many cases we have to use belief degrees given by some domain experts which has a much larger range than the real frequency [10]. As a result, probability theory is invalid and some attempts have been made in order to adapt to these situations. Zadeh [38] proposed fuzzy set theory which has been spread broadly. For example, Allahviranloo [2] applied uncertain information or data in several types to analyze linear systems. Phu et al. [27] investigated two new binary operations for intuitionistic fuzzy numbers and presented its application to an AIDs model, and Abbasi and Allahviranloo [1] proposed the new fuzzy arithmetic operations on pseudo-octagonal fuzzy numbers which model fuzzy system reliability in a more intelligent manner. Significant recent developments in fuzzy logic and fuzzy sets can be found in [29]. What’s more, fuzzy differential equations (FDEs) and fuzzy delay differential equations (FDDEs) [23] were investigated by many scholars. And fractional calculus were applied to an arbitrary-order inventory control problem [28]. However, Liu pointed out that uncertainty theory [13, 15] based on normality, duality, subadditivity and product axioms can deal with belief degrees better, which has been successfully applied to many fields. For example, uncertain regression analysis [12 , 36] models relationships among variables with imprecisely observed data, and uncertain time series analysis [22 , 34] predicts future values based on preciously imprecise observations. Noticing that many time varying systems influenced neither by random factors nor fuzzy factors, Liu proposed [14] a concept of uncertain process to better model uncertain phenomena varying with time. An important and useful uncertain process called Liu process is a type of stationary independent increment process whose increments are normal uncertain variables. Based on Liu process, uncertain differential equations (UDEs) [14] model time evolution of dynamic systems influenced by uncertain noises. Following that, many extensions of UDEs have been proposed such as high-order UDEs [35], multifactor UDEs [18], uncertain partial differential equations [32], uncertain fractional differential equation [39] and uncertain delay differential equation (UDDEs) [4]. Nowadays, UDEs have achieved fruitful results in many fields such as finance [16], population growth [30] and heat conduction [32]. More information about UDEs can be found in Yao’s book [35]. Later, Yao and Liu [37] suggested moment estimations for unknown parameters in UDEs, and Liu and Yang [21] estimated unknown parameters in high-order UDEs. As a type of generalizations of UDEs, UDDEs [4] incorporate the past states of systems to model phenomena with after-effect or memory in uncertain environment. These equations are more realistic because in general future states of systems are dependent on both the past states and present states. For example, cell division is not instantaneous once activated and patients show symptoms of illnesses until some time after the infection. The existence and uniqueness theorem of solutions for UDDEs were proved by Ge and Zhu [8]. Besides some types of stability for UDDEs were discussed later [9, 31].

However, in many cases parameters in UDDEs are unknown, and need to be estimated based on observations of solutions. How to estimate them is a vital problem in the wide applications of UDDEs. So this paper undertakes this issue using the method of moments based on discrete observations of solutions for UDEEs. The remainder of this paper is organized as follows. The method of moments for UDDEs will be proposed in Section 2. Section 3 is going to give some theorems about this method and Section 4 will give some numerical examples. Following that applications to population growth models will be presented in Section 5. Finally Section 6 will make some conclusions and discuss future works.

2 Parameter estimation

Consider an UDDE ${\begin{matrix} d X (t) = f (t, X (t), X (t - τ), θ) d t + \\ g (t, X (t), X (t - τ), θ) d C_{t}, \\ 0 \leq t \leq T \\ X (t) = ξ (t), - τ \leq t \leq 0 \end{matrix}$ (1) where f and g are some known functions satisfying Lipschitz condition and linear growth condition with a set of unknown parameters θ , ξ (t) is a given function, τ > 0 is a given constant called time delay, and C_t is a Liu process. We define a mesh with step h such that there exists an integer N_τ satisfying τ = N_τh, and denote t_n = nh, n = 0, ⋯ , N with t_N ≤ T. Then Equation (1) has the following Euler difference form ${\begin{matrix} X (t_{n - N_{τ}}) = ξ (t_{n} - τ), n - N_{τ} \leq 0 \\ X (t_{n + 1}) = X (t_{n}) + \\ hf (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ) + \\ (C_{t_{n + 1}} - C_{t_{n}}) g (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ) \\ 0 \leq n \leq N - 1 . \end{matrix}$ (2) Rewriting Equation (2) as ${\begin{matrix} X (t_{n - N_{τ}}) = ξ (t_{n} - τ), n - N_{τ} \leq 0 \\ \frac{X (t_{n + 1}) - X (t_{n})}{g (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ) h} - \\ \frac{hf (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ)}{g (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ) h} \\ = \frac{C_{t_{n + 1}} - C_{t_{n}}}{h}, 0 \leq n \leq N - 1, \end{matrix}$ (3) we observe that $\begin{matrix} \frac{X (t_{n + 1}) - X (t_{n})}{g (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ) h} - \\ \frac{hf (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ)}{g (t_{n}, X (t_{n}), X (t_{n - N_{τ}}), θ) h} \\ \sim N (0, 1) \end{matrix}$ (4) whose uncertainty distribution is $Φ (x) = {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1}$ for each n (n = 0, 1, ⋯ , N - 1) according to Liu process’s properties. It follows from the moment of uncertain variables [17] that the p-th population moments for a normal uncertain variable ξ ∼ $N$ (0, 1) can be calculated as $\begin{matrix} E [ξ^{p}] & = \int_{0}^{1} {(Φ^{- 1} (α))}^{p} d α \\ = {(\frac{\sqrt{3}}{π})}^{p} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{p} d α . \end{matrix}$ (5) Give N + 1 observations x (t₀) , x (t₁) , ⋯ , x (t_N) of the solution X (t) at times (0, h, ⋯ , Nh), respectively and denote the estimations for θ as $\hat{θ}$ . Then $\begin{matrix} m_{n} (\hat{θ}) = \\ \frac{x (t_{n + 1}) - x (t_{n}) - hf (t_{n}, x (t_{n}), x (t_{n - N_{τ}}), \hat{θ})}{g (t_{n}, x (t_{n}), x (t_{n - N_{τ}}), \hat{θ}) h}, \end{matrix}$ n = 0, 1, ⋯ , N - 1, can be regarded as N samples for a standard normal uncertain variable $N$ (0, 1). Obviously the p-th sample moments $\frac{1}{N} \sum_{i = 0}^{N - 1} {(m_{n} (\hat{θ}))}^{p}$ are supposed to be good estimations for population methods. As a result, the moment estimations $\hat{θ}$ of unknown parameters θ in the UDDE (1) can be obtained by solving the following system of equations $\begin{matrix} \frac{1}{N} \sum_{n = 0}^{N - 1} {(m_{n} (θ))}^{p} \\ = & {(\frac{\sqrt{3}}{π})}^{p} \int_{0}^{1} {(ln \frac{α}{1 - α})}^{p} d α, \end{matrix}$ (6)p = 1, 2, ⋯ , P, where P is the number of unknown parameters.

Remark 2.1. For some uncertain delay differential equations with given observations, the system of equations (6) may have no solution. In these cases moment estimations of unknown parameters do not exist, and moment method in this paper needs to be improved.

3 Some theorems

In this section, we present some theorems for parameter estimation in UDDEs to further illustrate our method.

Theorem 3.1. Consider the following UDDE ${\begin{matrix} d X (t) = (f_{1} (t, X (t), X (t - τ)) + \\ θ f_{2} (t, X (t), X (t - τ))) d t \\ + f_{3} (t, X (t), X (t - τ)) d C_{t}, \\ 0 \leq t \leq T \\ X (t) = ξ (t), - τ \leq t \leq 0 \end{matrix}$ (7) where f_i, i = 1, 2, 3 and ξ are given functions such that (7) has a unique solution, τ is a given constant and θ is an unknown parameter to be estimated. Observe X (t) at a mesh t_n = nh, n = 0, 1, ⋯ , N, t_N ≤ T with step h such that there exists an integer N_τ satisfying τ = N_τh and get x (t_n), n = 0, 1, ⋯ , N. Then the moment estimation $\hat{θ}$ for θ is $\begin{matrix} \hat{θ} = \sum_{n = 0}^{N - 1} (\frac{x (t_{n + 1}) - x (t_{n})}{{hf}_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} \\ - \frac{f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}) \\ / \sum_{n = 0}^{N - 1} \frac{f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} \end{matrix}$ where x (t_{n-N_τ}) = ξ (t_n - τ), n - N_τ ≤ 0.

Proof: According to Equation (222), we observe that $\begin{matrix} \frac{X (t_{n + 1}) - X (t_{n}) - f_{1} (t_{n}, X (t_{n}), X (t_{n - N_{τ}})) h}{f_{3} (t_{n}, X (t_{n}), X (t_{n - N_{τ}})) h} - \\ \frac{θ f_{2} (t_{n}, X (t_{n}), X (t_{n - N_{τ}}))}{f_{3} (t_{n}, X (t_{n}), X (t_{n - N_{τ}}))} \sim N (0, 1) \end{matrix}$ for the UDDE (7). Thus it follows from Equation (6) that the moment estimation $\hat{θ}$ for θ is the solution of the following equation $\frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (θ)) = 0$ where $\begin{matrix} m_{n} (θ) = \\ \frac{x (t_{n + 1}) - x (t_{n}) - f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h}{f_{3} (t_{n}, X (t_{n}), X (t_{n - N_{τ}})) h} \\ - \frac{θ f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, X (t_{n}), X (t_{n - N_{τ}}))} . \end{matrix}$ As a result, it is obtained that $\begin{matrix} \hat{θ} = & \sum_{n = 0}^{N - 1} (\frac{x (t_{n + 1}) - x (t_{n})}{{hf}_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} \\ - \frac{f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}) \\ / \sum_{n = 0}^{N - 1} \frac{f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} \end{matrix}$ where x (t_{n-N_τ}) = ξ (t_n - τ), n - N_τ ≤ 0. The theorem is proved immediately.

Theorem 3.2. Consider the following UDDE ${\begin{matrix} d X (t) = f_{1} (t, X (t), X (t - τ)) d t + \\ μ f_{2} (t, X (t), X (t - τ)) d t + \\ σ f_{3} (t, X (t), X (t - τ)) d C_{t}, \\ 0 \leq t \leq T \\ X (t) = ξ (t), - τ \leq t \leq 0 \end{matrix}$ (8)where f_i, i = 1, 2, 3 and ξ are given functions such that (8) has a unique solution, τ is a given constant, μ and σ are unknown parameters to be estimated. Observe X (t) at a mesh t_n = nh, n = 0, 1, ⋯ , N, t_N ≤ T with step h such that there exists an integer N_τ satisfying τ = N_τh and get x (t_n), n = 0, 1, ⋯ , N. Then the moment estimation $(\hat{μ}, \hat{σ})$ for (μ, σ) is ${\begin{matrix} \hat{μ} \\ = \sum_{n = 0}^{N - 1} (\frac{x (t_{n + 1}) - x (t_{n})}{{hf}_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} - \\ \frac{f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}) \\ / \sum_{n = 0}^{N - 1} \frac{f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} \\ \hat{σ} = \sqrt{\frac{1}{N} \sum_{n = 0}^{N - 1} F_{n} (\hat{μ})^{2}} \end{matrix}$ where $\begin{matrix} F_{n} (\hat{μ}) = \\ \frac{x (t_{n + 1}) - x (t_{n}) - f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h} \\ - \frac{\hat{μ} f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}, \end{matrix}$ x (t_{n-N_τ}) = ξ (t_n - τ), n - N_τ ≤ 0.

Proof: For the UDDE (8), it follows from Equation (222) that $\begin{matrix} \frac{X (t_{n + 1}) - X (t_{n}) - f_{1} (t_{n}, X (t_{n}), X (t_{n - N_{τ}})) h}{σ f_{3} (t_{n}, X (t_{n}), X (t_{n - N_{τ}})) h} \\ - \frac{μ f_{2} (t_{n}, X (t_{n}), X (t_{n - N_{τ}}))}{σ f_{3} (t_{n}, X (t_{n}), X (t_{n - N_{τ}}))} \sim N (0, 1) . \end{matrix}$ According to Equation (6), we can obtain the moment estimation $(\hat{μ}, \hat{σ})$ for (μ, σ) by solving the following system of equations ${\begin{matrix} \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (μ, σ)) = 0 \\ \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (μ, σ))^{2} = 1 \end{matrix}$ where $\begin{matrix} m_{n} (μ, σ) = \\ \frac{x (t_{n + 1}) - x (t_{n}) - f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h}{σ f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h} \\ - \frac{μ f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{σ f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} . \end{matrix}$ As a result, we get ${\begin{matrix} \hat{μ} \\ = \sum_{n = 0}^{N - 1} (\frac{x (t_{n + 1}) - x (t_{n})}{{hf}_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} - \\ \frac{f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}) \\ / \sum_{n = 0}^{N - 1} \frac{f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))} \\ \hat{σ} = \sqrt{\frac{1}{N} \sum_{n = 0}^{N - 1} F_{n} (\hat{μ})^{2}} \end{matrix}$ where $\begin{matrix} F_{n} (\hat{μ}) = \\ \frac{x (t_{n + 1}) - x (t_{n}) - f_{1} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}})) h} \\ - \frac{\hat{μ} f_{2} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}{f_{3} (t_{n}, x (t_{n}), x (t_{n - N_{τ}}))}, \end{matrix}$ x (t_{n-N_τ}) = ξ (t_n - τ), n - N_τ ≤ 0, and the theorem is proved.

4 Numerical examples

In order to illustrate the rationality of our method, this section gives some numerical examples documented as follows to estimate unknown parameters in some UDDEs with different time steps and true values of unknown parameters, calculate corresponding biases and standard deviations to evaluate our method.

Give the true value of parameter θ = θ ₀ and a time step h such that there exists an integer N_τ satisfying τ = N_τh, where the constant τ > 0 is the time delay.

For each n = 0, 1, 2, ⋯ , N, produce linear uncertain variable η_n ∼ $L$ (0, 1) whose uncertainty distribution is $Φ (x) = {\begin{matrix} 0, & if x \leq 0 \\ x - 1, & if 0 < x \leq 1 \\ 1, & if x > 1 \end{matrix}$ and obtain the normal uncertain variable ξ_n ∼ $N$ (0, 1) by ξ_n = Φ^-1 (η_n) where Φ^-1 (x) is the inverse uncertainty distribution of the standard normal uncertain variable $N$ (0, 1), i.e., $Φ^{- 1} (x) = \frac{\sqrt{3}}{π} ln \frac{x}{1 - x} .$

Generate corresponding x_t
₀, x_t
₁, x_t
₂, ⋯ , x_{t
_N} according to the Euler differential form (222) using normal uncertain variables obtained in Step 2.

With observations (t_n, x (t_n)), n = 0, 1, ⋯, N in Step 3, calculate the moment estimation $\hat{θ}$ of θ₀ by our method.

In order to evaluate estimations, we replicate Steps 2, 3, 4 for 500 times to get moment estimations ${\hat{θ}}_{j}$ for θ₀ , j = 1, 2, ⋯ , 500 and calculate the corresponding bias, i.e., $b = \frac{1}{500} \sum_{j = 1}^{500} {\hat{θ}}_{j} - θ_{0},$ (9) and the standard deviation, i.e., $sd = \sqrt{\frac{1}{500} \sum_{j = 1}^{500} {({\hat{θ}}_{j} - \frac{1}{500} \sum_{j = 1}^{500} {\hat{θ}}_{j})}^{2}} .$ (10)

Example 4.1. Consider the following UDDE ${\begin{matrix} d X (t) = μ X (t - 2) d t + σ d C_{t}, t \in [0, 4] \\ X (t) = 1, t \in [- 2, 0] \end{matrix}$ (11) where μ and σ are unknown parameters to be estimated. With observations x (t_n) at t_n = hn = 0.2n, n = 0, 1, ⋯ , 20 generated in Step 3, it follows from Theorem 3.2 that the moment estimation $(\hat{μ}, \hat{σ})$ of (μ, σ) is ${\begin{matrix} \hat{μ} = 5 (x (t_{20}) - x (t_{0})) / \sum_{n = 0}^{19} x (t_{n - 10}) \\ \hat{σ} = \sqrt{\frac{1}{20} \sum_{n = 0}^{19} {(\frac{x (t_{n + 1}) - x (t_{n}) - 0.2 \hat{μ} x (t_{n - 10})}{0.2})}^{2}} . \end{matrix}$ For a set of true values (μ₀, σ₀), we calculate biases and standard deviations for $(\hat{μ}, \hat{σ})$ which are shown in Tables 1 and 2.

Obviously, the smaller the absolute value of the bias b shown in Equation (9) and the standard deviation sd shown in Equation (10), the better the estimation. As we can see from Table 1, with a fixed time step h = 0.2 and different true values (μ₀, σ₀), absolute values of biases for estimations μ and σ are small. And Table 2 shows that with different true values (μ₀, σ₀), standard deviations for estimations μ and σ are also small. Thus our method can obtain ideal estimation results for unknown parameters in UDDEs with different initial values.

Table 1

Biases for the estimations $\hat{μ}$ and $\hat{σ}$ in Example 4.1 which are calculated as Step 5 for some different values (μ₀, σ₀)

	(μ₀, σ₀) = (2, 1.5)	(μ₀, σ₀) = (1, 0.5)	(μ₀, σ₀) = (3, 1)
μ	-0.006	0.001	-0.006
σ	-0.050	-0.017	-0.027

Table 2

Standard deviations for the estimations $\hat{μ}$ and $\hat{σ}$ in Example 4.1 which are calculated as Step 5 for some different values (μ₀, σ₀)

	(μ₀, σ₀) = (2, 1.5)	(μ₀, σ₀) = (1, 0.5)	(μ₀, σ₀) = (3, 1)
μ	0.187	0.082	0.093
σ	0.292	0.098	0.195

Next we consider a set of time steps h with the true value (μ₀, σ₀) = (3, 1) in Example 4.1, and calculate corresponding biases and standard deviations for estimations as Step 5, which are shown in Figures 1 and 2.

Fig. 1

Biases of $\hat{μ}$ and $\hat{σ}$ with different time steps h and (μ₀, σ₀) = (3, 1) in Example 4.1.

Fig. 2

Standard deviations of $\hat{μ}$ with different time steps h and (μ₀, σ₀) = (3, 1) in Example 4.1.

As we can see, as the time step h decreases, the absolute value of bias b and the standard deviation sd becomes smaller. Thus the estimation converges to the real value with the decrease of time step h. When using our method to estimate the unknown parameters in UDDEs, the time step of observations for uncertain delay differential equation should be as small as possible.

5 Application to population growth model

One of the important applications for UDDEs is population growth models, which help us understand how numbers change over time or in relation to each other. A population X (t) in a noisy environment with a constant death rate α, a constant birth rate β and an average development period τ can be depicted by the following UDDE, ${\begin{matrix} d X (t) = - α X (t) d t + β X (t - τ) d t \\ + σ d C_{t}, t \in [0, T] \\ X (t) = ξ (t), t \in [- τ, 0] . \end{matrix}$ (12) In order to estimate the set of unknown parameters (α, β, σ) in this model, we give the following theorem.

Theorem 5.1. The moment estimation $(\hat{α}, \hat{β}, \hat{σ})$ for the set of unknown parameters (α, β, σ) in the population growth model (12), i.e., ${\begin{matrix} d X (t) = - α X (t) d t + β X (t - τ) d t + σ d C_{t}, \\ t \in [0, T] \\ X (t) = ξ (t), t \in [- τ, 0] \end{matrix}$ is ${\begin{matrix} \hat{β} = (x (t_{N}) - x (t_{0}) + \hat{α} h \sum_{n = 0}^{N - 1} x (t_{n})) / h \sum_{n = 0}^{N - 1} x (t_{n - N_{τ}}) \\ \hat{σ} = \sqrt{\frac{\sum_{n = 0}^{N - 1} {(x (t_{n + 1}) - x (t_{n}) + \hat{α} x (t_{n}) - \hat{β} x (t_{n - N_{τ}}) h)}^{2}}{h^{2} N}} \end{matrix}$ with $\hat{α}$ solving the following equation $\begin{matrix} \sum_{n = 0}^{N - 1} (x (t_{n + 1}) - x (t_{n}) + α x (t_{n}) h - x (t_{n - N_{τ}}) \\ (x (t_{N}) - x (t_{0}) + α h \sum_{n = 0}^{N - 1} x (t_{n})) \\ {/ \sum_{n = 0}^{N - 1} x (t_{n - N_{τ}}))}^{3} = 0, \end{matrix}$ where x (t_n) are observations of X (t) at a mesh t_n = nh ≤ T with step h such that there exists an integer N_τ satisfying τ = N_τh, n = 0, 1, ⋯ , N, respectively, x (t_{n-N_τ}) = ξ (t_n - τ), n - N_τ ≤ 0.

Proof: For the UDDE (12), it follows from Equation (222) that $\begin{matrix} \frac{X (t_{n + 1}) - X (t_{n}) + α X (t_{n}) h - β X (t_{n - N_{τ}}) h}{σ h} \\ \sim N (0, 1) . \end{matrix}$ According to Equation (6), we can obtain the moment estimation $(\hat{α}, \hat{β}, \hat{σ})$ for (α, β, σ) by solving the following system of equations ${\begin{matrix} \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (α, β, σ)) = 0 \\ \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (α, β, σ))^{2} = 1 \\ \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (α, β, σ))^{3} = 0 \end{matrix}$ where $\begin{matrix} m_{n} (α, β, σ) = \\ \frac{x (t_{n + 1}) - x (t_{n}) + α x (t_{n}) h - β x (t_{n - N_{τ}}) h}{σ h} . \end{matrix}$ The theorem is followed immediately.

In the above population growth model, if the noisy behaviour is related to the system itself, we get the following UDDE ${\begin{matrix} d X (t) = - α X (t) d t + β X (t - τ) d t + \\ σ X (t) d C_{t}, t \in [0, T] \\ X (t) = ξ (t), t \in [- τ, 0] . \end{matrix}$ (13) Similarly, to estimate the set of unknown parameters (α, β, σ) we have the following theorem.

Theorem 5.2. The moment estimation $(\hat{α}, \hat{β}, \hat{σ})$ for the set of unknown parameters (α, β, σ) in the population growth model (13), i.e., ${\begin{matrix} d X (t) = - α X (t) d t + β X (t - τ) d t + σ X (t) d C_{t}, \\ t \in [0, T] \\ X (t) = ξ (t), t \in [- τ, 0] \end{matrix}$ is ${\begin{matrix} \hat{β} = (\sum_{n = 0}^{N - 1} \frac{x (t_{n + 1})}{x (t_{n})} - N + α hN) / h \sum_{n = 0}^{N - 1} \frac{x (t_{n - N_{τ}})}{x (t_{n})} \\ \hat{σ} = \sqrt{\frac{1}{N} \sum_{n = 0}^{N - 1} {(\frac{x (t_{n + 1}) - x (t_{n}) + \hat{α} x (t_{n}) - \hat{β} x (t_{n - N_{τ}}) h}{hx (t_{n})})}^{2}} \end{matrix}$ with $\hat{α}$ solving the following equation $\begin{matrix} \sum_{n = 0}^{N - 1} (\frac{x (t_{n + 1}) - x (t_{n})}{x (t_{n})} + α h - \frac{x (t_{n - N_{τ}})}{x (t_{n})} \\ {(\sum_{n = 0}^{N - 1} \frac{x (t_{n + 1})}{x (t_{n})} - N + α hN) / \sum_{n = 0}^{N - 1} \frac{x (t_{n - N_{τ}})}{x (t_{n})})}^{3} \\ = 0, \end{matrix}$ where x (t_n) are observations of X (t) at a mesh t_n = nh ≤ T with step h such that there exists an integer N_τ satisfying τ = N_τh, n = 0, 1, ⋯ , N, respectively, x (t_{n-N_τ}) = ξ (t_n - τ), n - N_τ ≤ 0.

Proof: For the UDDE (13), it follows from Equation (222) that $\begin{matrix} \frac{X (t_{n + 1}) - X (t_{n}) + α X (t_{n}) h - β X (t_{n - N_{τ}}) h}{σ X (t_{n}) h} \\ \sim N (0, 1) . \end{matrix}$ According to Equation (6), we can obtain the moment estimation $(\hat{α}, \hat{β}, \hat{σ})$ for (α, β, σ) by solving the following system of equations ${\begin{matrix} \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (α, β, σ)) = 0 \\ \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (α, β, σ))^{2} = 1 \\ \frac{1}{N} \sum_{n = 0}^{N - 1} (m_{n} (α, β, σ))^{3} = 0 \end{matrix}$ where $\begin{matrix} m_{n} (α, β, σ) = \\ \frac{x (t_{n + 1}) - x (t_{n}) + α x (t_{n}) h - β x (t_{n - N_{τ}}) h}{σ x (t_{n}) h} . \end{matrix}$ The theorem is followed immediately.

Example 5.1. Assume that we have observations at t_n = 0.1n, n = 0, 1, ⋯ , 30 shown in Table 3 for the population growth model ${\begin{matrix} d X (t) = - α X (t) d t + β X (t - 1) d t + σ d C_{t}, \\ t \in [0, 3] \\ X (t) = 0.5 t + 1, t \in [- 1, 0] . \end{matrix}$ Then we want to estimate unknown parameters (α, β, σ) in this population growth model. According to Theorem 5.1, we have ${\begin{matrix} \hat{β} = (x (t_{30}) - x (t_{0}) + 0.1 \hat{α} \sum_{n = 0}^{29} x (t_{n})) / h \sum_{n = 0}^{29} x (t_{n - 10}) \\ \hat{σ} = \sqrt{\frac{\sum_{n = 0}^{29} {(x (t_{n + 1}) - x (t_{n}) + \hat{α} x (t_{n}) - \hat{β} x (t_{n - 10}) h)}^{2}}{0.3}} \end{matrix}$ with $\hat{α}$ solving the following equation $\begin{matrix} \sum_{n = 0}^{29} (x (t_{n + 1}) - x (t_{n}) + 0.1 α x (t_{n}) - x (t_{n - 10}) \\ {(x (t_{30}) - x (t_{0}) + 0.1 α \sum_{n = 0}^{29} x (t_{n})) / \sum_{n = 0}^{29} x (t_{n - 10}))}^{3} \\ = 0, \end{matrix}$ where x (t_n-10) =0.5 (t_n - 1) +1, n ≤ 10. As a result, we get $(\hat{α}, \hat{β}, \hat{σ}) = (1.44, 2.94, 1.18),$ and the population growth model ${\begin{matrix} d X (t) = - 1.44 X (t) d t + 2.94 X (t - 1) d t + \\ 1.18 d C_{t}, t \in [0, 3] \\ X (t) = 0.5 t + 1, t \in [- 1, 0] . \end{matrix}$ According to this result, we know the population has a estimated death rate α = 1.44, a birth rate β = 2.94, and a noise level σ = 1.18.

Table 3

Observations for the population growth model in Example 5.1

n	0	1	2	3	4	5	6
t _i	0	0.1	0.2	0.3	0.4	0.5	0.6
x _{t _i}	1.00	1.06	1.07	1.13	1.22	1.24	1.31
n	7	8	9	10	11	12	13
t _i	0.7	0.8	0.9	1	1.1	1.2	1.3
x _{t _i}	1.41	1.41	1.57	1.59	1.69	1.76	2.13
n	14	15	16	17	18	19	20
t _i	1.4	1.5	1.6	1.7	1.8	1.9	2.0
x _{t _i}	2.20	2.46	2.50	2.40	2.45	2.65	2.70
n	21	22	23	24	25	26	27
t _i	2.1	2.2	2.3	2.4	2.5	2.6	2.7
x _{t _i}	2.52	2.67	2.74	2.90	3.09	3.30	3.26
n	28	29	30
t _i	2.8	2.9	3.0
x _{t _i}	3.65	3.75	3.86

6 Conclusion

It is natural for UDDEs to contain parameters whose values are to be estimated based on some observations of solutions, so how to estimate them is a core problem in the wide application of UDDEs. This paper undertook this issue using the method of moments. Approximating the UDDEs with Euler difference forms, moment estimations for unknown parameters were given as solutions of systems of equations. Some analytic solutions and numerical examples were documented to illustrate our method in detail. What’s more, applications to population growth models were presented. For the future work, estimations for unknown parameters in UDDEs when we observe at unequal time intervals will be studied. Besides, more general kinds of delays such as time or state dependent delays can be considered.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 61873329), the Young Academic Innovation Team of Capital University of Economics and Business (No. QNTD202002), and the special fund of basic scientific research business fees of Beijing Municipal University of Capital University of Economics and Business (XRZ2020016).

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