Lévy processes with infinite jump activity present intricate stochastic dynamics. In this work, we focus on the non-parametric estimation of the increment density for both the entire process and the small-jump component. We develop spectral estimators, establish their convergence rates,

Empirical evidence shows that calibrating exponential Lévy models by options with different maturities leads to conflicting information. In other words, the stationarity implicitly assumed in the exponential Lévy model is not satisfied. An identifiable time-inhomogeneous Lévy model is proposed that