Asymptotic Theory for Power Variation of LSS Processes

Abstract

This chapter provides an in-depth study of power variation and its asymptotics for Brownian and Levy semistationary (BSS and LSS) processes. Power variation techniques are used to draw inference on the integrated variance process. The theory is rather well-developed for semimartingales, in particular for the Brownian case, but some theory can also be developed for Levy-driven models. Beyond the semimartingale framework, the asymptotic theory for power variation for LSS processes turns out to be even harder and the corresponding proofs rely on different techniques, e.g. using concepts from Malliavin calculus. We present the key results in the semimartingale and the nonsemimartingale case. The latter, particularly in the context of LSS rather than BSS processes, is still a relatively open area.