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We extend to the multidimensional case a Wong-Zakai-type theorem proved by Huand {\O}ksendal in [7] for scalar quasi-linear It\^o stochastic differentialequations (SDEs). More precisely, with the aim of approximating the solution ofa quasilinear system of It\^o's SDEs, we consider for any finite partition ofthe time interval $[0,T]$ a system of differential equations, where themultidimensional Brownian motion is replaced by its polygonal approximation andthe product between diffusion coefficients and smoothed white noise isinterpreted as a Wick product. We remark that in the one dimensional case thistype of equations can be reduced, by means of a transformation related to themethod of characteristics, to the study of a random ordinary differentialequation. Here, instead, one is naturally lead to the investigation of asemilinear hyperbolic system of partial differential equations that we utilizefor constructing a solution of the Wong-Zakai approximated systems. We showthat the law of each element of the approximating sequence solves in the senseof distribution a Fokker-Planck equation and that the sequence converges to thesolution of the It\^o equation, as the mesh of the partition tends to zero.

Wong-Zakai approximations for quasilinear systems of It\^o's type stochastic differential equations