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On the convergence to the multiple Wiener-Itô integral. (English) Zbl 1163.60022

Summary: We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in \(\mathcal C_0([0,T])\). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function \(f\in L^{2}([0,T]^n)\). We prove also the weak convergence in the space \(\mathcal C_0([0,T])\) to the second-order integral for two important families of processes that converge to a standard Brownian motion.

MSC:

60H05 Stochastic integrals
60B10 Convergence of probability measures
60F05 Central limit and other weak theorems
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References:

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