Notation, Naming, and General Definitions

Abstract

The empirical studies included in this book are done over multiple time scales, and a large part with high-frequency time series. The usual notations need to be extended in order to denote accurately the time and time interval(s) dependencies in the computed time series like returns and volatilities. For inhomogeneous time series, a set of convenient operators is introduced, which makes it possible to estimate efficiently derived quantities like returns, volatilities, and volatility changes, with definitions that are appropriate for random time series. The choice of a convenient normalization annualizes systematically the derived quantities in order to easily compare statistics across time horizons. The definitions of the average, expectation, and histogram complete the set of basic tools needed for the empirical analysis.