Design and analysis of efficient quasi-Monte Carlo sampling methods

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In a large number of scientific disciplines, stochastic models are used to represent systems for which certain quantities of interest must be evaluated. For example, a production manager might need to compare different inventory policies in terms of their expected cost, assuming that supply and demand are both subjected to some form of randomness, or a telecommunication network designer might want to compare different designs based on their ability to handle large stochastic flows of information without incurring losses or slow-downs. The Monte Carlo method can be used to answer questions of that nature for complex systems which feature several random interacting components. This method uses random sampling in order to "simulate" possible scenarios for the system under study, and for each, the corresponding value of the quantity of interest is evaluated. By repeating this process several times, a sample of possible values for this quantity is created, which can then be used for inference, e.g., the sample mean can be used as an estimator for the quantity of interest. Quasi-Monte Carlo methods aim at improving upon Monte Carlo by replacing the random sampling inherent to the Monte Carlo method by a more structured form of sampling. This improved sampling is based on the use of low-discrepancy sequences, which are constructions that attempt to place points in a very uniform way in the space over which they are defined. These methods have gained considerable attention over the last 15 to 20 years, as they have proven to be useful for solving difficult high-dimensional problems in finance, e.g., involving the simulation of several financial assets over long periods of time. More precisely, with the same computational effort, they can provide estimators with a smaller error than those obtained by applying the Monte Carlo method. ***In this research program, I plan to contribute to the design and analysis of quasi-Monte Carlo sampling schemes, by working with students at the undergraduate, Master's, and doctoral levels on the three following objectives: 1) to improve our ability to efficiently design quasi-Monte Carlo sampling schemes (i.e., low-discrepancy sequences) that work well for a given problem; 2) to propose novel ways to analyze low-discrepancy sequences so that new insight can be gained into the performance of these sequences, and 3) to expand the class of models that can be tackled by quasi-Monte Carlo methods, so that models featuring complex dependence structures can be handled by these methods.*** **