Efficient Estimation of Integrated Volatility in Presence of Infinite Variation Jumps with Multiple Activity Indices

Abstract

In a recent paper [6], we derived a rate efficient (and in some cases variance efficient) estimator for the integrated volatility of the diffusion coefficient of a process in presence of infinite variation jumps. The estimation is based on discrete observations of the process on a fixed time interval with asymptotically shrinking equidistant observation grid. The result in [6] is derived under the assumption that the jump part of the discretely-observed process has a finite variation component plus a stochastic integral with respect to a stable-like Lévy process with index $\beta >1$. Here we show that the procedure of [6] can be extended to accommodate the case when the jumps are a mixture of finitely many integrals with respect to stable-like Lévy processes with indices $\beta _1>\cdots >\beta _M\ge 1$.