Limit Theorems for Power Variations of Pure-Jump Processes with Application to Activity Estimation

We define a new concept termed the activity signature function, which is constructed from discrete observations of a process evolving continuously in time. Under quite general regularity conditions, we derive the asymptotic properties of the function as the sampling frequency increases and show that it is a useful device for making inferences about the activity level of an Ito semimartingale. Monte Carlo work confirms the theoretical results. One empirical application is from finance. It indicates that the classical model comprised of a continuous component plus jumps is more plausible than a pure-jump model for the spot $/DM exchange rate over 1986-1999. A second application pertains to internet traffic data at NASA servers. We find that a pure-jump model with no continuous component and paths of infinite variation is appropriate for modeling this data set. In both cases the evidence obtained from the signature functions is quite convincing, and these two very disparate empirical outcomes illustrate the discriminatory power of the methodology.