Abstract
In this paper, we first study the estimation of diffusion parameters for one-dimensional, ergodic stochastic processes observed at some discrete times, that is a solution of a given class of stochastic differential equations driven by stable processes. After recalling the non-parametric estimation framework of the drift functionn namely the Nadaraya-Watson estimation, we discuss an estimation method of the diffusion parameters (the scaling and the driving stable process parameters) based on the Euler–Maruyama scheme. The novelty of this paper which is our baseline is the combination of a characteristic sample function and the Least Squares Estimation (LSE) method. Secondly, we apply the diffusion parameters results for stable driven OrnsteinUhlenbeck (OU), Cox–Ingersoll–Ross (CIR) and Lotka–Volterra (LV) processes. We also consider the estimation of the drift coefficients in the linear case namely, the stable driven OU and CIR processes. Using the Ito formula and the linear statistical regression derived from the LSE method, we establish under certain conditions, the consistency of their drift coefficients estimators. We efficiently discuss our result with numerical simulations using synthetic data. A real data in finance, such as exchange rates is used to fit the parameters of a justified model amoung the stable driven OU and CIR processes. As a forthcoming work, we intend to study the rate of convergence of the estimators and to create a package on R software to handle this kind of estimation problem