M-Estimation based on quasi-processes from discrete samples of Levy processes

TL;DR

This paper introduces a new quasi-process based M-estimation method for Levy processes using discrete data, offering a computationally efficient alternative to traditional simulation-based approaches.

Contribution

It develops a novel pseudo-path construction from observed increments, proving its convergence and establishing the consistency and asymptotic normality of the estimator.

Findings

Quasi-processes converge weakly to true Levy processes.

The M-estimator is consistent and asymptotically normal.

Method enables inference from single trajectories efficiently.

Abstract

We propose a novel estimation framework for path-dependent functionals of Levy processes from discretely observed data. Traditional approaches rely on Monte Carlo simulation of full paths, which requires complete model specification and heavy computation. In contrast, our quasi-process method constructs pseudo-paths directly from observed increments by random permutation, preserving the increment distribution while enabling repeated evaluation of functionals. Under a high-frequency, long-term sampling regime, we establish weak convergence of the quasi-process to the true Levy process and prove consistency and asymptotic normality of the resulting $M$-estimator. This bootstrap-like approach provides a practical and computationally efficient tool for inference from a single trajectory and offers promising extensions to multivariate modeling, machine learning integration, and risk management.