Issues in high-dimensional quasi-monte carlo sampling

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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.