Parallel processors for planning under uncertainty
Annals of Operations Research
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Numerical valuation of high dimensional multivariate European securities
Management Science
Quasi-Monte Carlo methods in numerical finance
Management Science
Latin supercube sampling for very high-dimensional simulations
ACM Transactions on Modeling and Computer Simulation (TOMACS) - Special issue on uniform random number generation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Improving the rejection sampling method in quasi-Monte Carlo methods
Journal of Computational and Applied Mathematics
Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates
Mathematics and Computers in Simulation - IMACS sponsored Special issue on the second IMACS seminar on Monte Carlo methods
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Variance Reduction via Lattice Rules
Management Science
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
Good Lattice Rules in Weighted Korobov Spaces with General Weights
Numerische Mathematik
Constructing Robust Good Lattice Rules for Computational Finance
SIAM Journal on Scientific Computing
On the approximation error in high dimensional model representation
Proceedings of the 40th Conference on Winter Simulation
Dimension-wise integration of high-dimensional functions with applications to finance
Journal of Complexity
Smoothness and dimension reduction in Quasi-Monte Carlo methods
Mathematical and Computer Modelling: An International Journal
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The valuation of many financial securities can be formulated as high-dimensional integrals for which quasi-Monte Carlo (QMC) methods are becoming indispensable numerical tools. It is known that some typical high-dimensional problems in finance and in many other fields have a strong additive nature. This paper investigates the enhancement of QMC methods by exploiting the additive approximation in combination with suitable transformation methods. To this end, we study (a) the approximation of a multivariate function by a sum of one-dimensional functions; and (b) the effective use of the possible strong additive property inherent in finance problems to enhance QMC. For problem (a), we establish a relationship between the approximation quality resulting from anchored decomposition and the global sensitivity indices. For problem (b), we propose new methods for efficiency improvement—the additive weighted uniform sampling and the additive control variate, which exactly exploit the strong additive structure. The required importance distribution and control variate are found constructively based on the anchored additive approximation with a suitably chosen anchor. Numerical examples on bond valuation and option pricing illustrate that the proposed methods reduce the variance by huge factors in QMC. Transformation methods play a crucial role, since they may enhance the degree of additivity of a function. Traditional methods, such as Brownian bridge and principal component analysis, do not necessarily work well, whereas methods that take into account the underlying functions are robust and powerful.