Weak convergence to the multiple Stratonovich integral.
The weak convergence, as \(\varepsilon \) tends to zero, of the multiple integral process \[ \int _0^t \dots \int _0^tf(t_1,\dots ,t_n)\,d\eta _\varepsilon (t_1)\dots \,d\eta _\varepsilon (t_n),\quad t\in [0,T], \] in the space \(\mathcal C_0([0,T])\) is studied, where \(f\in L^2([0,T]^n)\) and \((\eta _\varepsilon (t))_{\varepsilon >0}\) is a family of stochastic processes with absolutely continuous paths that converge weakly to the Brownian motion. The limit is shown to exist if \(f\) is a multimeasure. It also exists under some conditions on \((\eta _\varepsilon )\) if \(f\) is a continuous function of the form \(f(t_1,\dots ,t_n)=f_1(t_1)\dots f_n(t_n)1_{[t_1<\dots <t_n]}\). The limit process is the multiple Stratonovich integral of the function \(f\).
scientific article; zbMATH DE number 2102353