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。蒙特卡罗方法可用于此目的。它使用随机抽样，以“模拟”可能的金融情景，并为每一个，相应的价值的产品研究。通过多次重复该过程，创建产品的可能值的样本，然后可以将其用于推断，例如，它的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方法中固有的随机抽样来改善这一最终估计阶段。这种改进的采样是基于尝试以非常均匀的方式将点放置在域中的构造的使用。这些方法在过去的10到15年里得到了很多关注，因为它们已经被证明对金融中困难的高维问题很有用，例如，涉及在很长一段时间内模拟几种金融资产。更确切地说，它们可以提供估计量，其误差小于通过应用蒙特卡罗获得的误差，使用相同的计算量。由于这一成功，这些方法现在被用于越来越复杂的应用中。

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.