Joint estimation for SDE driven by locally stable Lévy processes

Considering a class of stochastic differential equations driven by a locally stable process, we address the joint parametric estimation, based on high frequency observations of the process on a fixed time interval , of the drift coefficient, the scale coefficient and the jump activity of the process. This work extends [4] where the jump activity was assumed to be known and also [3] where the LAN property and the estimation of the three parameters are performed for a translated stable process. We propose an estimation method and show that the asymptotic properties of the estimators depend crucially on the form of the scale coefficient. If the scale coefficient is multiplicative: a(x, σ) = σa(x), the rate of convergence of our estimators is non diagonal and the asymptotic variance in the joint estimation of the scale coefficient and the jump activity is the inverse of the information matrix obtained in [3]. In the non multiplicative case, the results are better and we obtain a faster diagonal rate of convergence with a different asymptotic variance. In both cases, the estimation method is illustrated by numerical simulations showing that our estimators are rather easy to implement. MSC 2010 subject classifications: Primary 60G51, 60G52, 60J75, 62F12; secondary 60H07, 60F05 .