Numerical Evaluation of Multiple Integrals

Haber, Seymour

doi:10.1137/1012102

Cited by 213 publications

(119 citation statements)

References 58 publications

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“…Positive weights imply that I(Q·) is also a positive functional. In [22] we can find again such two desiderable properties for a numerical integration formula: (P 1 ) the points used lie in the integration domain; (P 2 ) the coefficients be positive. Regarding property (P 1 ), the function to be integrated may not be defined outside the domain of integration.…”

Section: Theorem 4 For Any Function Fmentioning

confidence: 98%

“…This is not the case of rules based on the operators S 2 and W 2 . However, if we define the quantity [22] …”

Section: Theorem 4 For Any Function Fmentioning

confidence: 99%

“…We recall [22] that a region Ω ⊂ R 2 is said to be "fully symmetrical" if, whenever (x, y) is a point of the integration domain, all points of the form (±x, ±y) and (±y, ±x) are also integration points.…”

Section: Remarkmentioning

confidence: 99%

“…Similarly a numerical integration formula is "fully symmetrical" [6,22] if, whenever it contains the point (x, y) with associated coefficient w, it contains all the points (±x, ±y) and (±y, ±x) with the same associated coefficient.…”

Section: Remarkmentioning

confidence: 99%

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Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Lamberti

2009

Bit Numer Math

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In this paper we generate and study new cubature formulas based on spline quasi-interpolants defined as linear combinations of C 1 bivariate quadratic B-splines on a rectangular domain Ω , endowed with a non-uniform criss-cross triangulation, with discrete linear functionals as coefficients. Such B-splines have their supports contained in Ω and there is no data point outside this domain. Numerical results illustrate the methods.

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Section: Theorem 4 For Any Function Fmentioning

confidence: 98%

“…This is not the case of rules based on the operators S 2 and W 2 . However, if we define the quantity [22] …”

Section: Theorem 4 For Any Function Fmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Lamberti

2009

Bit Numer Math

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“…Good reviews of the Monte Carlo method and different variance reduction techniques such as antithetic variables, covariates, stratified sampling, importance sampling can be found in many sources such as Hammersley and Hanscomb (1964), Zaremba (1968), Haber (1970), Kalos and Whitlock (1986), and references thereof.…”

Section: Relation To Other Workmentioning

confidence: 99%

Numerical Valuation of High Dimensional Multivariate European Securities

Barraquand¹

1995

Management Science

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an acknowledgement of the authors and individual contributors to the work; and all applicable portions of the copyright notice. Copying, reproducing, or republishing for any other purpose shall require a license with payment of fee to the Paris Research Laboratory. All rights reserved. ii AbstractWe consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process.Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the number of risk sources.In this paper, we present an efficient numerical technique that combines Monte Carlo simulation with a particular partitioning method of the underlying assets space, which we call Stratified State Aggregation (SSA). Using this technique we can compute accurate approximations of prices of American securities with an arbitrary number of underlying assets. Our numerical experiments show that the method is practical for pricing American claims depending on up to 400 risk sources. On all problems for which we could compare the method with known optimal solutions, the price computed through stratified state aggregation was indistinguishable from the optimal theoretical price. Several numerical examples are presented and discussed.iii Acknowledgements This research benefited from discussions with Bruno Langlois. We wish to thank Thierry Pudet for reviewing an earlier draft.iv

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Approximation of Integrals

Khuri

2002

Advanced Calculus With Applications in Statistics

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Numerical Evaluation of Multiple Integrals

Cited by 213 publications

References 58 publications

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Numerical Valuation of High Dimensional Multivariate European Securities

Approximation of Integrals

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