Threshold Models for Lévy Processes and Approximate Maximum Likelihood Estimation

Using the Lévy process (solution of the Ito–Skorokhod stochastic differential equation), we propose the structure of the model of a threshold process and an approximate maximum likelihood method based on the approximation of the logarithmic function of the likelihood of observations. We find the estimates for the parameters of the two-mode threshold jump process with discretely sampled data and show that it is possible to determine the presence of threshold effects by checking the likelihood ratio.