Convergence in law to the multiple fractional integral

Xavier Bardina, Maria Jolis and Ciprian A. Tudor.

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Laboratoire de Probabilités; Université Paris 6; 4, Place Jussieu; 75252 Paris Cedex 05; France.

Abstract

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  ontinuous 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 Solé and Utzet (1990).
 
Published in: Stochastic Processes and their Applications, 105 (2), 315-344, 2003