Recursive ECF identification of linear systems driven by L\'evy processes

In the literature the empirical characteristic function method is presented as an off-line identification method. While the results of the off-line methods are attractive, the proposed algorithms are ill-conditioned in many cases so that they requires special attention. As an alternative to the off-line method in this paper we propose and analyze on-line empirical characteristic function methods. Such recursive methods enables us to carry out real-time statistical analysis as new data points are processed instantly. In constructing these algorithms we follow the general framework proposed by Djereveckii and Fradkov. On-line methods are also used to complement a computationally expensive off-line identification method. Namely, it would be uneconomical to re-estimate $\theta^*$ using the off-line method when a new data point is received. Instead, we can argue that only a refinement of the estimate $\hat{\theta}_N$ should be computed using the newly received data point. This scenario not only shows a motivation behind the study of recursive algorithms but also suggests that it is reasonable to suppose that an initial guess of the parameter is close to $\theta^*.$