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