[Paper Review] Spectral-free estimation of L\\'evy densities in high-frequency regime

Céline Duval, Ester Mariucci|arXiv (Cornell University)|Feb 28, 2017

Stochastic processes and financial applications3 citations

TL;DR

This paper introduces a spectral-free, pathwise wavelet-based estimator for Lévy densities in high-frequency settings, avoiding reliance on the Lévy–Khintchine formula. It achieves optimal nonparametric rates over Besov balls for both finite and infinite variation Lévy processes, demonstrating robustness to singularity at zero and presence of a Brownian component.

ABSTRACT

We construct an estimator of the L\\'evy density of a pure jump L\\'evy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the L\\'evy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the L\\'evy--Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of $L_p$ loss functions, $p\\geq 1$, over Besov balls. We recover classical nonparametric rates for finite variation L\\'evy processes and for a large nonparametric class of symmetric infinite variation L\\'evy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.

Motivation & Objective

• Develop a non-spectral, pathwise method for estimating Lévy densities from high-frequency discrete sampling of pure jump Lévy processes.
• Overcome limitations of spectral techniques—especially the reliance on the Lévy–Khintchine formula—enabling application to richer classes of jump processes.
• Establish theoretical performance guarantees in terms of $L_p$ loss over Besov balls for both finite and infinite variation Lévy processes.
• Ensure robustness of the estimator when the estimation domain approaches the critical singularity point at zero and under the presence of a Brownian component.

Proposed method

• Use a compound Poisson approximation to the Lévy process as a foundational tool, replacing spectral decomposition.
• Construct a linear wavelet estimator based on jump counts in dyadic intervals, leveraging pathwise properties of the process.
• Decompose the estimation problem into localized regions $A( ho)$ around zero and apply wavelet coefficients to estimate the density $f$.
• Control bias and variance via wavelet thresholding and moment bounds, using the fact that $\mathbb{E}[\widehat{\lambda}_{n,\varepsilon}] \approx \lambda_\varepsilon$.
• Apply concentration inequalities and conditional moment bounds (via Lemma 5) to handle random truncation and sampling variability.
• Use the asymptotic equivalence of infinitely divisible laws to compound Poisson measures (Corollary 8.8 in [33]) to justify the approximation strategy.

Experimental results

Research questions

• RQ1 Can Lévy density estimation be achieved without spectral methods, particularly avoiding the Lévy–Khintchine formula?
• RQ2 What nonparametric rates can be achieved for $L_p$ estimation of Lévy densities using a pathwise wavelet approach in high-frequency sampling?
• RQ3 How does the estimator perform for infinite variation Lévy processes, especially near the origin where the density may blow up?
• RQ4 Is the estimator robust to the presence of a Brownian component or to the estimation domain approaching zero?

Key findings

• The proposed wavelet-based estimator achieves the classical nonparametric rate $2^{-2Js}$ over Besov balls for finite variation Lévy processes.
• For a large class of symmetric infinite variation Lévy processes, the estimator attains the optimal rate $2^{-2Js}$, matching minimax lower bounds.
• The estimator remains robust when the estimation window $A( ho)$ approaches the origin, with the error controlled via $\varepsilon$-dependent norms and moment bounds.
• The method is robust to the presence of a Brownian component, as shown by the stability of the $L_p$ risk under perturbations of the Lévy triplet.
• Theoretical bounds on the $L_p$ risk are derived in terms of wavelet resolution $J$, sample size $n$, and truncation level $\varepsilon$, with explicit dependence on $\mathbf{n}(\varepsilon)$ and $\widetilde{\bm{n}}(\varepsilon)$.
• The error decomposition separates into bias and variance components, with the variance term controlled via moment inequalities and conditional expectation bounds.