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): **

**Code(s) de Classification MSC:**

- 60G15 Gaussian processes
- 60H05 Stochastic integrals

**Résumé:** We study the convergence in law in $\mathcal C_0([0,1])$, as
$\varepsilon\to0$, of the family of continuous processes
$\{I_{\eta_\varepsilon}(f)\}_{\varepsilon>0}$ defined by the
multiple integrals
$$I_{\eta_{\varepsilon}}(f)_t=\int_0^t\cdots \int_0^t
f(t_1,\ldots,t_n)d\eta_{\varepsilon}(t_1)\cdots
d\eta_{\varepsilon}(t_n); \quad t\in [0,1],$$ where $f$ is a
deterministic function and $\{\eta_{\varepsilon}\}_{\varepsilon
>0}$ is a family of processes, with absolutely continuous paths,
converging in law in $\mathcal C_0([0,1])$ to the fractional
Brownian motion with Hurst parameter $H>\frac12$. When $f$ is
given by a multimeasure and for any family $\{\eta_\varepsilon\}$
with trajectories absolutely continuous whose derivatives are in
$L^2([0,1])$, we prove that $\{I_{\eta_\varepsilon}(f)\}$
converges in law to the multiple fractional integral of $f$. This
last integral is a multiple Stratonovich-type integral defined by
Dasgupta and Kallianpur (1999a) on the space $L^2(\tilde\mu_n)$,
where $\tilde\mu_n$ is a measure on $[0,1]^n$.
Finally, we have shown that, for two natural families
$\{\eta_\varepsilon\}$ converging in law in $\mathcal C_0([0,1])$
to the fractional Brownian motion, the family
$\{I_{\eta_\varepsilon}(f)\}$ converges in law to the multiple
fractional integral for any $f\in L^2(\tilde\mu_n)$.
In order to prove the convergence, we have shown that the integral
introduced by Dasguta and Kallianpur (1999a) can be seen as an
integral in the sense of Sol\'{e} and Utzet (1990).

**Mots Clés:** *fractional Brownian motion ; multiple stochastic ; integrals ; weak convergence*

**Date:** 2002-01-17

**Prépublication numéro:** *PMA-787*

Pdf file: PMA-787.pdf