2010/07/08 by Rosenbaum, Mathieu, Tankov, Peter
#60G51 #60G52 #62M05 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1007.1414
We provide asymptotic results and develop high frequency statistical procedures for time-changed Lévy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with respect to the starting point is equal to ε. This setting can be seen as a first step towards a model for tick-by-tick financial data allowing for large jumps. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. In particular, these results allow us to construct consistent estimators of the time change and of the Blumenthal-Getoor index of the underlying Lévy process. Convergence rates and a central limit theorem are established under additional assumptions.