Abstract
Stochastic Lotka–Volterra model driven by small -stable noises is used to describe population dynamics perturbed by random environment. However, parameters in the model are always unknown. The contrast function is given to obtain least squares estimators. The consistency and the rate of convergence of the least squares estimators are proved, and the asymptotic distribution of the estimators are derived by Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality. Some numerical examples are provided to verify the effectiveness of the estimators.
1. Introduction
Stochastic differential equations are basic tools for modeling random phenomena in the financial field. Due to the widespread application of the stochastic differential equations in the field of financial economics, it has attracted a large number of scholars to devote themselves to research in this field [1–3]. However, the parameters in stochastic model are always unknown. In the past few decades, the parameter estimation problem for economical models has been studied by many authors. For example, Yu and Phillips [4] utilized the Gaussian method to estimate the parameters of continuous time short-term interest rate models. Faff and Gray [5] discussed the estimation and comparison of short-rate models by the generalised method of moments. Rossi [6] applied particle filters and maximum likelihood estimation to solve the parameter estimation for Cox–Ingersoll–Ross model. Wei et al. [7] utilized the Gaussian estimation method to investigate the parameter estimation for discretely observed Cox–Ingersoll–Ross model. However, it is well known that many financial processes exhibit discontinuous sample paths and heavy tailed properties (e.g., certain moments are infinite). These features cannot be captured by Brownian motion [8, 9]. Therefore, it is natural to replace the driving Brownian motion by Lévy noises. In recent years, with the development of Lévy process theory and its application in the fields of engineering systems, economic systems, and management systems, it has attracted great attention from scholars. Therefore, some authors considered parameter estimation for stochastic differential equations driven by Lévy noises. For example, Li and Ma [10] discussed the asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Long [11] analyzed the least squares estimator for discretely observed 2Ornstein–Uhlenbeck processes with small Lévy noises. Then, Long et al. [8] tackled the least squares estimators for discretely observed stochastic processes driven by small Lévy noises. Singh et al. [12] utilized a randomized response model under Poisson distribution to estimate a rare sensitive attribute in two-stage sampling. Wei [13] used least squares estimation to discuss the discretely observed CoxCIngersollCRoss model driven by small symmetrical stable noises and studied the consistency and asymptotic distribution of the estimators. There have been many applications of small noise asymptotics to mathematical finance [14, 15]. From a practical point of view in parametric inference, it is more realistic and interesting to consider asymptotic estimation for diffusion processes with small noise based on discrete observations.
Lotka–Volterra model is often used to model population growth of a single species. However, systems are more or less influenced by random factors. Thus, stochastic Lotka–Volterra equation, being a reasonable and popular approach to model population dynamics perturbed by random environment, has recently been studied by many authors both from a mathematical perspective and in the context of real biological dynamics [16–19] to explain the change of biodiversity also over time [20–23]. For example, Mao et al. [24] investigated a multidimensional stochastic Lotka–Volterra system driven by one-dimensional standard Brownian motion. They revealed that the environmental noise could suppress population explosion. Later, Mao [25] discussed a finite second moment of the stationary distribution under Brownian noise, which is very important in application. Bao et al. [26] and Bao and Yuan [27] considered a competitive Lotka–Volterra population model with Lévy jumps. Zhao et al. [28] studied the parameter estimation for stochastic Lotka–Volterra model by using the maximum likelihood method from continuous time observations. However, due to the limitation of instrument precision, it is impossible to observe the system from continuous time. Moreover, few literatures considered the consistency and asymptotic distribution of parameter estimators for stochastic Lotka–Volterra driven by -stable noises. We consider the parameter estimation problem for discretely observed stochastic Lotka–Volterra model with small -stable noises. The contrast function is given to obtain the least squares estimators. The consistency and asymptotic distribution of the estimators are discussed by Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality.
The structure is that the stochastic Lotka–Volterra model driven by small -stable noises is introduced and the contrast function is given to obtain the least squares estimators in Section 2. In Section 3, the consistency of the estimators is proved and the asymptotic distribution of the estimators is studied. In Section 4, some simulations are made. The conclusion is given in Section 5.
2. Problem Formulation and Preliminaries
is a basic probability space equipped with a right continuous and increasing family of -algebras and is a strictly symmetric -stable Lévy motion.
A random variable is said to have a stable distribution with index of stability , scale parameter , skewness parameter , and location parameter if it has the following characteristic function:
We denote . When , we say is strictly -stable; if in addition , we call symmetrical -stable. Throughout this article, -stable motion is strictly symmetrical and .
We study the parametric estimation problem for stochastic Lotka–Volterra model driven by small -stable noises described by the following stochastic differential equation:where and are unknown parameters, .
Since the stochastic Lotka–Volterra model is driven by small -stable noises and due to the complexity of the -stable noises, it is difficult to obtain the likelihood function. Thus, the maximum likelihood estimation cannot be used. Therefore, the contrast function is given to obtain least squares estimators.
Consider the following contrast function:where .
We obtain the estimators:
The work will be carried out under the following assumptions.
Assumption 1. and are positive true values of the parameters.
Assumption 2. .
3. Main Results and Proofs
is the solution to the underlying ordinary differential equation under the true value of the parameters:
Discretizing equation (2), we obtain
Then, a more explicit decomposition for and can be given
Before giving the theorems, we need to establish some preliminary results.
Lemma 1 (see [29]). is a strictly -stable Lévy process and . Then,If is symmetric, that is, , then, there exists some -stable Lévy process , such that
Lemma 2. When and , we have
Proof. Integrating on both sides of equation (2) and (8), we haveBy using the Cauchy–Schwarz inequality, we obtainwhere is the upper bound of .
According to Gronwall’s inequality, we haveSquaring on both sides of equation (17),By the Markov inequality, for any given , when , we havenamely,The proof is complete.
Lemma 3. When and , we have
Proof. Aswe obtainAccording to Lemma 2, when and , we haveAccording to equations (23) and (24), we obtainThe proof is complete.
Applying the same methods used in Lemma 3, we haveIn the following theorem, the consistency of the least squares estimators is proved.
Theorem 1. When , , and , the least squares estimators and are consistent, namely,
Proof. According to Lemma 3, we haveWhen and , we obtainAccording to Lemma 2, we haveFrom equations (28) and (29), we obtainBy the Markov inequality, we havewhere is constant.
As , , and , we obtainAccording to Lemma 2, we obtainBy the Markov inequality, we haveAs , , and , we obtainAccording to Lemma 2, when and ,From equations (37) and (38), we haveAccording to equations (27), (30), (31), (33), and (39), when , , and ,Using the same methods in Theorem 1, we obtainTogether with the results thatwhen , , and , we haveThe proof is complete.
Theorem 2. When , , and ,
Proof. According to the explicit decomposition for , we haveFrom Lemma 2, when , , and ,From equations (45) and (46), we haveAccording to Lemma 2, we obtainUsing the Markov inequality and Holder’s inequality, for any given , we haveas , , and .
Moreover,where .
According to Lemma 1, we haveFrom equations (53) and (54),According to equations (45)–(51) and (55), we haveFrom equation (8), we obtainAccording to equations (25)–(27) and (55), we haveFrom equations (54)–(56), we haveThe proof is complete.
4. Simulation
In this experiment, we generate a discrete sample and compute and from the sample. Let . For every given true value of the parameters-, the size of the sample is represented as“Size ” and given in the first column of the table. In Table 1, , the size is increasing from 1000 to 5000. In Table 2, , the size is increasing from 10,000 to 50,000. The tables list the value of “”, “,” and the absolute errors of least squares estimators.
Two tables illustrate that when is large enough and is small enough, the obtained estimators are very close to the true parameter value. Therefore, the methods used in this paper are effective and the obtained estimators are good.
In Figure 1, ; when , the size is increasing from 1000 to 5000; when , the size is increasing from 10,000 to 50,000. In Figure 2, ; when , the size is increasing from 1000 to 5000; when , the size is increasing from 10,000 to 50,000. Two figures illustrate that when is large enough and is small enough, the obtained estimators are very close to the true parameter value.
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5. Conclusion
The parameter estimation problem for discretely observed stochastic Lotka–Volterra model driven by small -stable noises has been studied. The contrast function has been given to obtain the least squares estimators. The consistency and asymptotic distribution of the least squares estimators have been discussed by using the Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality. Further research topics will include parameter and state estimation for partially observed stochastic system driven by -stable noises.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61403248 and Innovative Training Program for College Students in Henan Province under Grant S202010479028.