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][0,T]. We consider the case where the sampling rate Δ=ΔT→0Δ=ΔT→0 as T→∞T→∞. We propose an adaptive wavelet threshold density estimator and study its performance for LpLp losses, p≥1p≥1, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates ΔTΔT 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.