Multiscale stochastic dynamics in finance

The estimation of volatility processes underlying observed financial returns is stochastically characterized in a semimartingale probabilistic setting through two related measures: realised and integrated volatility. According to the quadratic variation principle, the convergence of realised to integrated volatility can be verified both in time and frequency coordinates. We show from an experimental standpoint that consistent estimators for the integrated volatility hold when the scale coordinate is considered and wavelet-based estimators are adopted.