We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

我们将 Hu 和 Øksendal (1996) 证明的用于标量拟线性 Itô 随机微分方程 (SDE) 的 Wong-Zakai 型定理扩展到多维情况。更准确地说，为了逼近 Itô 的 SDE 拟线性系统的解，我们考虑时间间隔的任何有限分区$[0,吨]$一个微分方程系统，其中多维布朗运动被其多边形近似代替，并且扩散系数和平滑白噪声之间的乘积被解释为 Wick 乘积。我们注意到，在一维情况下，这种类型的方程可以通过与特征方法相关的变换简化为随机常微分方程的研究。在这里，人们自然而然地研究了偏微分方程的半线性双曲系统，我们利用它来构建 Wong-Zakai 近似系统的解。我们证明了近似序列的每个元素的定律在分布意义上求解 Fokker-Planck 方程，并且该序列收敛于 Itô 方程的解，