Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Convergence of likelihood ratios for some discontinuous processes


Code(s) de Classification MSC:

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