Nonparametric estimation for Levy processes from low-frequency observations

We suppose that a L'evy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the L'evy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific $C^2$-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.

💡 Research Summary

This paper addresses the problem of non‑parametric inference for the Lévy–Khintchine characteristics (drift γ, diffusion σ², and jump measure ν) of a Lévy process when the process is observed at equally spaced discrete times with a fixed sampling interval Δ. While most of the existing literature focuses on high‑frequency regimes (Δ → 0), the authors consider the low‑frequency setting where Δ remains constant as the number of observations N grows. In this regime the distribution of the increments is complex, and identifiability of the Lévy triplet is not immediate.

The authors propose a general minimum‑distance estimation framework. For the observed increments ΔXₖ = X_{kΔ} – X_{(k‑1)Δ}, k = 1,…,N, they define the empirical characteristic function (ECF)
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