The paper proposes a new second-order weak approximation scheme for hypoelliptic diffusions or degenerate systems of stochastic differential equations satisfying a certain Hörmander condition. The scheme is constructed by a Gaussian process and a stochastic polynomial weight through a technique based on Malliavin calculus, and is implemented by a Monte Carlo method and a quasi-Monte Carlo method. A variance analysis for the Monte Carlo method is discussed, and further control variate methods are introduced to reduce the variance. The effectiveness of the proposed scheme is illustrated through numerical experiments for some hypoelliptic diffusions.

中文翻译：

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退化随机微分方程组的二阶离散化

针对满足一定Hörmander条件的次椭圆扩散或退化微分方程组，提出了一种新的二阶弱逼近方案。该方案通过基于Malliavin微积分的技术由高斯过程和随机多项式权重构造而成，并通过蒙特卡洛方法和准蒙特卡洛方法实现。讨论了蒙特卡洛方法的方差分析，并引入了更多的控制变量方法以减少方差。通过一些次椭圆形扩散的数值实验说明了该方案的有效性。