With Jumps: An Introduction to Power Variations
This chapter studies the simplest possible process having both a non-trivial continuous part and jumps. It starts with the asymptotic behavior of power variations when the model is nonparametric, that is, without specifying the law of the jumps. This is done in the same spirit as in Chapter 3: the ideas for the proofs are explained in detail, but technicalities are omitted. Then, it considers the use of these variations in a parametric estimation setting based on the generalized method of moments. There, it considers the ability of certain moment functions, corresponding to power variations, to achieve identification of the parameters of the model and the resulting rate of convergence. It shows that the general nonparametric results have a parametric counterpart in terms of which values of the power p are better able to identify parameters from either the continuous or jump part of the model.