Issues in high-dimensional quasi-monte carlo sampling

假设一位金融分析师想要为一种复杂的产品定价。为此，可以使用蒙特卡罗方法。它使用随机抽样来“模拟”可能的金融情景，并对每一种情景，研究中产品的相应价值。通过多次重复这个过程，创建了产品的可能值的样本，然后可以使用该样本进行推断，例如，其平均值给出了产品价格的估计值。准蒙特卡罗方法旨在通过用更结构化的抽样形式取代蒙特卡罗固有的随机抽样来改进这一最终估计阶段。这种改进的采样基于尝试以非常统一的方式在域中放置点的构造的使用。这些方法在过去的10到15年里得到了大量的关注，因为它们被证明在金融中的高维度难题上是有用的，例如，涉及在很长一段时间内模拟几个金融资产。更准确地说，它们可以使用相同的计算量，为估计器提供比应用蒙特卡洛方法获得的误差更小的误差。由于这一成功，这些方法现在被用于日益复杂的应用程序。

Suppose a financial analyst wants to price a complex product. The Monte Carlo method can be used for that purpose. It uses random sampling in order to "simulate" possible financial scenarios, and for each, the corresponding value of the product under study. By repeating this process several times, a sample of possible values for the product is created, which can then be used for inference, e.g., its mean gives an estimator for the product's price. Quasi-Monte Carlo methods aim at improving this final estimation phase by replacing the random sampling inherent in Monte Carlo by a more structured form of sampling. This improved sampling is based on the use of constructions that attempt to place points in a domain in a very uniform way. These methods have gained a lot of attention in the last 10 to 15 years, as they have proven to be useful on difficult high-dimensional problems in finance, e.g., involving the simulation of several financial assets over long periods of time. More precisely, they can provide estimators with a smaller error than those obtained by applying Monte Carlo, using the same amount of computational effort. Because of this success, these methods are now used in applications that are becoming increasingly complex.