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Convergence of stochastic integrals and SDE’s associated to approximations of the Gaussian white noise. (English) Zbl 1168.60326

Summary: The Gaussian white noise in \(d\) dimensions is approximated in a smooth way by a process \(W_\varepsilon\) which happens to be a semimartingale random measure. Using this approximation, the integral of an associated process to \(W_\varepsilon\) is taken with respect to \(W_\varepsilon\) and also the limit of these integrals is taken as \(\varepsilon\) approaches zero. A Wong and Zakai factor is obtained in the limit. This result is the analogous of a time-smooth approximation of the Brownian motion. Finally, an application to convergence of SDE’s is given, the limit of a sequence of solutions to a stochastic differential equation is taken and the limit is analyzed.

MSC:

60F05 Central limit and other weak theorems
60F25 \(L^p\)-limit theorems
60F15 Strong limit theorems
60G51 Processes with independent increments; Lévy processes
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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