Weak convergence to the multiple Stratonovich integral.

Xavier Bardina and Maria Jolis

Departament de Matemàtiques, Edifici C, Universitat Auṭnoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Abstract

We have considered the problem of the weak convergence, as $\varepsilon$ tends to zero, of the multiple integral processes $$\{\int_0^t\cdots \int_0^t f(t_1,\ldots,t_n) d\eta_{\varepsilon}(t_1)\cdots d\eta_{\varepsilon}(t_n), \quad t\in [0,1]\}$$ in the space $C([0,1])$, where $f\in L^2([0,1]^n)$ is a given function, and $\{\eta_{\varepsilon}(t)\}_{\varepsilon >0}$ is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when $n\ge 2$ and $ f(t_1,\ldots,t_n) = 1_{\{t_1<t_2<\cdots<t_n\} }$, we cannot expect that these multiple integrals converge to the multiple Itô-Wiener integral of $f$, because the quadratic variation of the $\eta_{\varepsilon}$ are nul. We have obtained the existence of the limit for any $\{\eta_{\varepsilon}\}$, when $ f $ is given by a multimeasure, and under some conditions on $\{\eta_{\varepsilon}\}$ when $ f $ is a continuous function and when $f(t_1,\ldots,t_n)= f_1(t_1) \cdots f_n(t_n) 1_{\{t_1<t_2<\cdots<t_n\} }$, with $f_i\in L^2([0,1])$ for any $i=1\ldots n$. In all these cases the limit process is the multiple Stratonovich integral of the function $f$.

Published in: Stochastic Processes and their Applications, 90 (2), 277-300, 2000