Asymptotic equivalence of jumps Lévy processes and their discrete counterpart

Pierre Etoré (1) , Sana Louhichi (2) , Ester Mariucci (2)

1 MATHFI - Mathématiques financières
2 IPS - Inférence Processus Stochastiques

We establish the global asymptotic equivalence between a pure jumps Lévy process $\{X_t\}$ on the time interval $[0,T]$ with unknown Lévy measure $\nu$ belonging to a non-parametric class and the observation of $2m^2$ Poisson independent random variables with parameters linked with the Lévy 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 $\{X_t\}$.

Mots clés: asymptotic equivalence; Lévy processes.; deficiency distance; Non-parametric experiments; Lévy processes

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Pierre Etoré, Sana Louhichi, Ester Mariucci. Asymptotic equivalence of jumps Lévy processes and their discrete counterpart. 2013. ⟨hal-00827173v2⟩