Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 ``Probabilités et Modèles Aléatoires''

### Convergence of likelihood ratios for some discontinuous processes

Auteur(s):

Code(s) de Classification MSC:

• 60F99 None of the above but in this section
• 60J75 Jump processes
• 62A10 The likelihood approach

Résumé: We consider here a sequence of parametric models associated with the observation at times \$i/n,~1\le i\le n\$ of the solution to the equation \$~dX_t=\vth~[dW_t+f(X_{t^-})dY_t],~\$ where \$\vth\$ is an unknown parameter. \$W\$ is a standard Brownian motion and \$Y\$ is a compound Poisson with L\'evy measure \$F\$ having no singular diffuse part. Under some regularity assumptions on \$f\$, we prove a convergence theorem for the sequence of local density processes of \$P^{\tnh}\$ with respect to \$P^\vth.~\$ A corollary of this result is the LAMN property in the case \$f=1,\$ providing an asymptotic lower bound for the variance of the estimation. We give also some sequences of estimators achieving this lower bound.

Mots Clés: Local asymptotic mixed normality ; stable convergence in law ; limit theorem for semimartingale

Date: 2000-06-16

Prépublication numéro: PMA-600