Non Parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

We consider a stochastic process driven by a diffusion and jumps. We devise a technique, which is based on a discrete record of observations, for identifying the times when jumps larger than a suitably defined threshold occurred. The technique allows also jump size estimation. We prove the consistency of a nonparametric estimator of the integrated infinitesimal variance of the process continuous part when the jump component with infinite activity is Lévy. Central limit results are proved in the case where the jump component has finite activity. Some simulations illustrate the reliability of the methodology in finite samples.1