Purely discontinuous Levy processes and power variation: inference for integrated volatility and the scale parameter

Jeannette H.C. Woerner

Abstract

This paper provides consistency and a distributional result for an estimate of the integrated volatility in different Levy type stochastic volatility models based on high frequency data. As an estimator we consider the p-th power variation, i.e. the sum of the p-th power of the absolute value of the log-price returns, allowing irregularly spaced data. Furthermore, we derive conditions on the mean process under which it is negligible. This allows us more flexibility in modelling, namely to include further jump components or even to leave the framework of semimartingales by adding a certain fractional Brownian motion. As a special case our method includes  an estimating procedure for the scale parameter of discretely observed Levy processes.