Density estimation for compound Poisson processes from discrete data

In this article we investigate the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over [ 0 , T ] . We consider the case where the sampling rate Δ = Δ T → 0 as T → ∞ . We propose an adaptive wavelet threshold density estimator and study its performance for L p losses, p ≥ 1 , over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates Δ T that vanish slowly. The estimation procedure is based on the explicit inversion of the operator giving the law of the increments as a nonlinear transformation of the jump density.