Generalized moment estimators for \(\alpha\)-stable Ornstein-Uhlenbeck motions from discrete observations
The authors study the parameter estimation problem for discretely observed Ornstein-Uhlenbeck processes driven by \(\alpha\)-stable Lévy motions. The \(\alpha\)-stable Lévy motions is a stochastic process \((Z_t,t\geq 0)\) for which \(Z_1\in S_{\alpha}(1,\beta,0)\) where \(\alpha\) is the index of stablility and \(\beta\in(-1,1)\) is the skewness. Then Ornstein-Uhlenbeck processes driven by the \(\alpha\)-stable Lévy motion \(Z_t\) is presented by the stochastic equation \(dX_t=-\theta X_tdt+\sigma dZ_t,\; t\in[0,\infty),\;X_0=x_0,\) where \(\theta\), \(\sigma\) are some constants. If \(\theta>0,\) \(X_t\) is ergodic and it converges in law to the random variable \(X_0=\sigma\int_0^{\infty} e^{-\theta s}dZ_s.\) A method of moments via ergodic theory and via sample characteristic functions is proposed to estimate all the parameters involved in the Ornstein-Uhlenbeck processes. The strong consistency and the asymptotic normality of the proposed joint estimators when \(n \rightarrow \infty\) are demonstrated.
\(\alpha\)-stable Ornstein-Uhlenbeck motions
scientific article; zbMATH DE number 7186885
scientific article