In this paper, we consider the maximum likelihood estimation for the symmetric α-stable Ornstein–Uhlenbeck (SαS-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the SαS distribution. By means of transition function and Laplace transform, we construct an explicit approximate sequence of likelihood function, which converges to the likelihood function of SαS distribution. Based on the approximation of likelihood function we give an algorithm for computing maximum likelihood estimation. We also numerically simulate some experiments which demonstrate the accuracy and stability of the proposed estimator.

中文翻译：

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对称 α 稳定 Ornstein-Uhlenbeck 过程的最大似然估计

在本文中，我们考虑了对称的最大似然估计α-稳定的 Ornstein–Uhlenbeck (SαS-OU) 基于离散观察的过程。由于在 Lévy 情况下很难获得最大似然函数的封闭形式表达式，我们选择混合柯西和高斯分布来逼近 S 的概率密度函数 (PDF)αS 分布。通过转移函数和拉普拉斯变换，我们构造了一个显式的似然函数近似序列，它收敛于S的似然函数αS 分布。基于似然函数的近似，我们给出了一种计算最大似然估计的算法。我们还对一些实验进行了数值模拟，这些实验证明了所提出的估计器的准确性和稳定性。