Wahib Arroum and Owen Jones, 
Estimation for time-changed self-similar stochastic processes. 

Proceedings SPIE Vol. 6039, Complex Systems, A. Bender Ed. Article 60390F (Jan. 16, 2006). 

We consider processes of the form X(t) = X~(theta(t)) where X~ is a self-similar process with stationary increments and theta is a deterministic subordinator with a periodic activity function a = theta' > 0.
Such processes have been proposed as models for high-frequency financial data, such as currency exchange rates, where there are known to be daily and weekly periodic fluctuations in the volatility, captured here by the periodic activity function.
We review an existing estimator for the activity function then propose three new methods for estimating it and present some experimental studies of their performance.
We finish with an application to some foreign exchange and FTSE100 futures data.