Estimation for the Discretely Observed Cox–Ingersoll–Ross Model Driven by Small Symmetrical Stable Noises

: This paper is concerned with the least squares estimation of drift parameters for the Cox–Ingersoll–Ross (CIR) model driven by small symmetrical  ‐ stable noises from discrete observations. The contrast function is introduced to obtain the explicit formula of the estimators and the error of estimation is given. The consistency and the rate of convergence of the estimators are proved. The asymptotic distribution of the estimators is studied as well. Finally, some numerical calculus examples and simulations are given.


Introduction
Stochastic differential equations driven by Brownian motion are always used to model the phenomena influenced by stochastic factors such as molecular thermal motion and short-term interest rate [1,2]. When establishing a pricing formula, the parameters in stochastic models describe the relevant asset dynamics. Nevertheless, in most cases, parameters are always unknown. Over the past few decades, many authors have studied the parameter estimation problem by maximum likelihood estimation [3][4][5], least squares estimation [6][7][8], and Bayes estimation [9,10]. However, non-Gaussian noise such as  -stable noise can more accurately reflect the practical random perturbation. Therefore, stochastic differential equations driven by  -stable noise have been investigated by many authors in recent years. Particularly, the parameter estimation problem has been discussed as well [11,12].
The Cox-Ingersoll-Ross (CIR) model [13,14] introduced in 1985 is an extension of the Vasicek model [15]; it is mean-reverting and remains non-negative. As we all know, the parameter estimation problem for the CIR model has been well studied [16,17]. However, many financial processes exhibit discontinuous sample paths and heavy-tailed properties (e.g., certain moments are infinite). These features cannot be captured by the CIR model. Therefore, it is natural to replace the Brownian motion by an  -stable process. In recent years, parameter estimation problems for the Levy-type CIR model have been discussed in some literature studies. For example, Ma and Yang [18] used least squares methods to study the parameter estimation problem for the CIR model driven by  -stable noises. Li and Ma [19] derived the conditional least squares estimators for a stable CIR model. However, the asymptotic distribution of the estimators has not been discussed in the literature. Asymptotic properties of estimators such as consistency, asymptotic distribution of estimation errors, and hypothesis tests can reflect the effectiveness of estimators and estimation methods, which helps to obtain a more reasonable economic model structure and more accurately grasp the dynamics of related assets. Therefore, it is of great important to study these topics.
The parameter estimation problem for the discretely observed CIR model with small symmetrical  -stable noises is studied in this article. The contrast function is introduced to obtain the least squares estimators. The consistency and asymptotic distribution of the estimators are derived by Markov inequality, Cauchy-Schwarz inequality and Gronwall's inequality. Some numerical calculus examples and simulations are given as well.
The structure of this paper is as follows. In Section 2, we introduce the CIR model driven by small symmetrical  -stable noises and obtain the explicit formula of the least squares estimators.
In Section 3, the consistency and asymptotic distribution of the estimators are studied. In Section 4, some simulation results are made. The conclusions are given in Section 5.

Problem Formulation and Preliminaries
In this paper, notation " P  " is used to denote "convergence in probability" and notation " ⇒"is used to denote "convergence in distribution". We write d for equality in distribution.
In this paper, we study the parametric estimation problem for the Cox-Ingersoll-Ross Model driven by small  -stable noises described by the following stochastic differential equation: Before giving the theorems, we need to establish some preliminary results.

Lemma 1.[20]
Let Z be a strictly  -stable Levy process and By using the Cauchy-Schwarz inequality, we find According to Gronwall's inequality, we obtain Then, it follows that By the Markov inequality, for any given The proof is complete.□ Remark 1. In Lemma 2, the following moment inequalities for stable stochastic integrals has been used to obtain the results: The above moment inequalities for stable stochastic integrals were established in Theorems 3.1 and 3.2 of [21].
it is clear that Therefore, we obtain it is clear that According to Lemma 2, when 0, n   , we have Therefore, we obtain In the following theorem, the consistency of the least squares estimators is proved. The proof is complete.
According to Lemma 2, we have By the Markov inequality, we have By the Markov inequality, we have Together with the results that Using the Markov inequality and Holder's inequality, for any given 0   , we have The proof is complete.□

Simulation
In this experiment, we generate a discrete sample   , the size of the sample is represented as "Size n" and given in the first column of the table.
In Table 1, 0.1   , the size is increasing from 1000 to 5000. In Table 2, , the size is increasing from 10,000 to 50,000. Based on the ten-time average of the least squares estimation of the random number in the calculation model, the tables list the values of the least squares estimator (LSE) , the absolute error (AE), and the relative error (RE) of the least squares estimator.
The two tables indicate that the absolute error between the estimator and the true value depends on the size of the true value samples for any given parameter. In Table 1, when n = 5000, the relative error of the estimators does not exceed 7%. In Table 2, when n = 50,000, the relative error of the estimators does not exceed 0.2%. The estimators are good.
In Figure 1 ,respectively. The two figures indicate that when T is fixed and  is small, the obtained estimators are closer to the true parameter value compared to that of the large  . When T is large enough and  is small enough, the obtained estimators are very close to the true parameter value. If we let T convergeto infinity and  convergeto zero, the two estimators will converge to the true value.

Conclusions
The aim of this paper was to study the parameter estimation problem for the Cox-Ingersoll-Ross model driven by small symmetrical  -stable noises from discrete observations. The contrast function was introduced to obtain the explicit formula of the least squares estimators and the error of estimation was given. The consistency and the rate of convergence of the least squares estimators were proved by Markov inequality, Cauchy-Schwarz inequality, and Gronwall's inequality. The asymptotic distribution of the estimators were discussed as well.