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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- J. JACOD
- A. JAKUBOWSKI
**J. MÉMIN**

**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*