Abstract
We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. This estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, but it is also applicable in other areas. We develop a probabilistic asymptotic theory for realised relative power variations of Brownian semistationary processes, and introduce inference methods based on the theory. We also discuss how to extend the asymptotic theory to other classes of processes exhibiting stochastic volatility/intermittency. As an empirical application, we study relative energy dissipation in data of atmospheric turbulence.
Highlights
IntroductionThe concept of volatility expresses the ubiquitous phenomenon that observational fields exhibit more variation than expected; that is, more than the most basic type of random influence envisaged
The concept of volatility expresses the ubiquitous phenomenon that observational fields exhibit more variation than expected; that is, more than the most basic type of random influence1 envisaged.volatility is a relative concept, and its meaning depends on the particular setting under investigation
We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular
Summary
The concept of volatility expresses the ubiquitous phenomenon that observational fields exhibit more variation than expected; that is, more than the most basic type of random influence envisaged. We introduce the notion of relative volatility/intermittency and the closely related statistics, realised relative power variations. They pave the way for practical applications of some recent advances in the asymptotic theory of power variations of non-semimartingales Realised relative power variations are self-scaling and, admit a statistically feasible central limit theorem, which can be used, e.g., to construct confidence intervals for the realised relative volatility/intermittency. Appendices contain a discussion of extending the theory beyond Brownian semistationary processes (Appendix A), an alternative method of assessing the volatility/intermittency of a Brownian semistationary process (Appendix B), and some supporting results (Appendix C)
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