Asymptotic equivalence of jumps L\'evy processes and their discrete counterpart

TL;DR

This paper proves that, as the number of observations increases, observing a pure jumps Lévy process is statistically equivalent to observing a finite set of Poisson variables linked to its Lévy measure, simplifying analysis.

Contribution

It establishes the asymptotic equivalence between continuous Lévy process observations and discrete Poisson-based data, justifying simplified statistical procedures.

Findings

Asymptotic equivalence holds as the number of observations grows.

Knowing jump counts in intervals is statistically as informative as full path observation.

Results apply to non-parametric Lévy measures with fixed observation time.

Abstract

We establish the global asymptotic equivalence between a pure jumps L\'evy process $\{X_t\}$ on the time interval $[0,T]$ with unknown L\'evy measure $\nu$ belonging to a non-parametric class and the observation of $2m^2$ Poisson independent random variables with parameters linked with the L\'evy measure $\nu$. The equivalence result is asymptotic as $m$ tends to infinity. The time $T$ is kept fixed and the sample path is continuously observed. This result justifies the idea that, from a statistical point of view, knowing how many jumps fall into a grid of intervals gives asymptotically the same amount of information as observing $\{X_t\}$.