# Observing a L\'evy process up to a stopping time

## TL;DR

This paper proves that the probability law of a killed Lévy process observed up to a stopping time uniquely determines its entire law, with some exceptions involving compound Poisson components and killing.

## Contribution

It establishes a uniqueness result for Lévy processes based on partial observations up to stopping times, including cases with killing and compound Poisson components.

## Key findings

- Law of Lévy process up to stopping time determines entire law
- Unique identification except for compound Poisson components and killing
- Results apply to both up to and strictly before stopping times

## Abstract

It is proved that the law of a possibly killed L\'evy process $X$, seen up to and including (resp. up to strictly before) a stopping time, determines already the law of $X$ (resp. up to a compound Poisson component and killing).