Statistical inference for time-changed Lévy processes via Mellin transform approach

Given a Lévy process (Lt)t≥0(Lt)t≥0 and an independent nondecreasing process (time change) (T(t))t≥0(T(t))t≥0, we consider the problem of statistical inference on TT based on low-frequency observations of the time-changed Lévy process LT(t)LT(t). Our approach is based on the genuine use of Mellin and Laplace transforms. We propose a consistent estimator for the density of the increments of TT in a stationary regime, derive its convergence rates and prove the optimality of the rates. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of TT. Finally, the performance of the estimator is analysed via a Monte Carlo simulation study.