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

### About asymptotic errors in discretization of processes

Auteur(s):

Code(s) de Classification MSC:

• 60F17 Functional limit theorems; invariance principles
• 60H99 None of the above but in this section

Résumé: We study the rate at which the time-discretized processes $X^n_t=X_{[nt]/n}$ converge to a given process $X$. When $X$ is a continuous semimartingale it is known that, under appropriate assumptions, the rate is $\sqrt{n}$, so we focus here on the discontinuous case. Then $\alpha_n(X^n-X)$ explodes for any sequence $\alpha_n$ going to infinity, so we consider integrated errors'' of the form $Y^n_t=\int_0^t(X_s-X^n_s)ds$ or $Z^{n,p}_t=\int_0^t|X^n_s-X_s|^pds$ for $p\in(1,\infty)$: we essentially prove that the variables $\sup_{s\leq t}|nY^n_s|$ and $\sup_{s\leq t}|nZ^{n,p}_s|$ are tight for any finite $t$ when $X$ is an arbitrary semimartingale, povided either $p\geq2$ or $p\in(1,2)$ and $X$ has no continuous martingale part and the sum $\sum_{s\leq t}|\D X_s|^p$ converges a.s. for all $t<\infty$. Under suitable additional assumptions, we even prove that the discretized processes $nY^n_{[nt]/n}$ and $nZ^{n,p}_{[nt]/n}$ converge in law to non-trivial processes which are explicitely given.

Mots Clés: Discretization of processes ; Semimartingales

Date: 2001-05-21

Prépublication numéro: PMA-661