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Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics by Paolo Brandimarte

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Chapter Nine

Low-Discrepancy Sequences

In the previous chapter we have reviewed the main variance reduction strategies that are used in financial and economic applications. Since those approaches are based on probabilistic concepts, they implicitly assume that random sampling in Monte Carlo methods is really random. However, the pseudorandom numbers produced by an LCG or by more sophisticated algorithms are not random at all. Hence, one could take a philosophical view and wonder about the very validity of variance reduction methods, and even the Monte Carlo approach itself. Taking a more pragmatic view, and considering the fact that Monte Carlo methods have proven their value over the years, we should conclude that this shows that there are some deterministic number sequences that work well in generating samples. It is also useful to remember that the aim of a Monte Carlo simulation is actually to estimate a multidimensional integral on the unit hypercube:

equation

The function h(·) may be so complicated that we cannot express it analytically, but this is of no concern conceptually. We need a stream of i.i.d. random numbers to fill the integration domain in a satisfactory manner. When a regular grid derived from classical product rules for numerical integration (see Chapter 2) is not feasible, we may fill it by random numbers, but we could also resort to alternative deterministic sequences ...

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