Low-rank volatility estimation for high-dimensional Lévy processes and low frequency observations

Mathias Trabs (University Paris-Dauphine)
Wednesday, July 1, 2015 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Whenever the modelling of random processes in biology, finance or physics requires to incorporate jumps, Lévy processes are one of the building blocks under consideration. Consequently, their statistical analysis attracted much attention in the last decades. We first review some results on the nowadays well understood nonparametric estimation of the characteristic triplet of a univariate Lévy process based on low frequent observations. The underlying inverse problem is ill-posed, where the degree of ill-posedness is determined by the characteristic triplet itself. This strong interplay can be decoupled if we either observe a time-changed Lévy process at equidistant time points or, similarly, if we observe the Lévy process at random sampling times. Indeed, in the latter two observation schemes faster rates can be achieved, especially allowing for polynomial convergence rates for the volatility. With this knowledge at hand we study the estimation problem for multi- and high-dimensional (time-changed) Lévy processes, focusing on the estimation of the volatility matrix. To overcome the curse of dimensionality, a low-rank condition is imposed. Applying a spectral approach, we construct a weighted leastsquares estimator with `1-penalisation. For this estimator, oracle inequalities are proven which allow for separating the high-dimensional estimation problem from the ill-posed inverse problem. For the volatility matrix estimator we moreover derive convergence rates. These reflect surprising phenomena which do not occur in the one dimensional setting. The convergence rates are optimal in the minimax sense (at least in some settings). This talk is based on an ongoing joint work with Denis Belomestny.