Joint estimation for SDE driven by locally stable Lévy processes

Abstract

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. Extending the methodology proposed in [6], where the jump activity was assumed to be known, we obtain two different rates of convergence in estimating simultaneously the scale parameter and the jump activity, depending on the scale coefficient. If the scale coefficient is multiplicative: $a(x,\sigma )=\sigma \overline{a}(x)$, the joint estimation of the scale coefficient and the jump activity behaves as for the translated stable process studied in [5] and the rate of convergence of our estimators is non diagonal. In the non multiplicative case, the results are different and we obtain a diagonal and faster rate of convergence which coincides with the one obtained in estimating marginally each parameter. In both cases, the estimation method is illustrated by numerical simulations showing that our estimators are rather easy to implement.