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

### Convergence in law to the multiple fractional integral

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