On the Assessment of Random and Quasi-Random Point Sets

Hellekalek, Peter

doi:10.1007/978-1-4612-1702-2_2

Cited by 46 publications

(36 citation statements)

References 64 publications

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“…Given that the goal is to minimize the variance, this expression tells us that for a given f , the most relevant discrepancy is exactly the expression (3.1). This suggests a general class of discrepancies (or figures of merit) for lattice rules, of the form [28,64,65]…”

Section: Randomization Discrepancies and Parameters Selectionmentioning

confidence: 99%

Quasi-Monte Carlo methods with applications in finance

L’Ecuyer

2009

Finance Stoch

123

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We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.

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Section: Randomization Discrepancies and Parameters Selectionmentioning

confidence: 99%

Quasi-Monte Carlo methods with applications in finance

L’Ecuyer

2009

Finance Stoch

123

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“…The quadrature methods based on low discrepancy sets are called quasi-Monte Carlo methods. They are discussed in several review articles [3,16,41,61] and monographs [28,45,55].…”

Section: Introductionmentioning

confidence: 99%

Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature

Hickernell

Hong

L’Ecuyer³

et al. 2000

SIAM J. Sci. Comput.

105

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Abstract. Integration lattices are one of the main types of low discrepancy sets used in quasiMonte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first b m of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

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“…The uniformity of a set Ψ I is typically assessed by measuring the discrepancy between the empirical distribution of its points and the uniform distribution over (0,1) t (Niederreiter, 1992;Hellekalek and Larcher, 1998;L'Ecuyer and Lemieux, 2002). Discrepancy measures are equivalent to goodness-of-fit test statistics for the multivariate uniform distribution.…”

Section: Random Number Generationmentioning

confidence: 99%