Finite-order weights imply tractability of multivariate integration
Journal of Complexity
Journal of Complexity
Finite-order weights imply tractability of linear multivariate problems
Journal of Approximation Theory
Randomly shifted lattice rules for unbounded integrands
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Exact cubature for a class of functions of maximum effective dimension
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Low discrepancy sequences in high dimensions: How well are their projections distributed?
Journal of Computational and Applied Mathematics
Comparison of Point Sets and Sequences for Quasi-Monte Carlo and for Random Number Generation
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications
On efficient Monte Carlo-based statistical static timing analysis of digital circuits
Proceedings of the 2008 IEEE/ACM International Conference on Computer-Aided Design
On the approximation error in high dimensional model representation
Proceedings of the 40th Conference on Winter Simulation
Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing
INFORMS Journal on Computing
Generalized Halton sequences in 2008: A comparative study
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
A practical view of randomized quasi-Monte Carlo: invited presentation, extended abstract
Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools
Finite-order weights imply tractability of linear multivariate problems
Journal of Approximation Theory
Dimension-wise integration of high-dimensional functions with applications to finance
Journal of Complexity
Correlation controlled sampling for efficient variability analysis of analog circuits
Proceedings of the Conference on Design, Automation and Test in Europe
A model framework for greedy routing in a sensor network with a stochastic power scheme
ACM Transactions on Sensor Networks (TOSN)
Advanced variance reduction and sampling techniques for efficient statistical timing analysis
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids
Journal of Computational Physics
SIAM Journal on Scientific Computing
Adaptive ANOVA decomposition of stochastic incompressible and compressible flows
Journal of Computational Physics
Error Estimates for the ANOVA Method with Polynomial Chaos Interpolation: Tensor Product Functions
SIAM Journal on Scientific Computing
Enhancing Quasi-Monte Carlo Methods by Exploiting Additive Approximation for Problems in Finance
SIAM Journal on Scientific Computing
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Many problems in mathematical finance can be formulated as high-dimensional integrals, where the large number of dimensions arises from small time steps in time discretization and/or a large number of state variables. Quasi--Monte Carlo (QMC) methods have been successfully used for approximating such integrals. To understand this success, this paper focuses on investigating the special features of some typical high-dimensional finance problems, namely, option pricing and bond valuation. We provide new insight into the connection between the effective dimension and the efficiency of QMC, and we present methods to analyze the dimension structure of a function. We confirm the observation of Caflisch, Morokoff, and Owen that functions from finance are often of low effective dimension, in the sense that they can be well approximated by their low-order ANOVA (analysis of variance) terms, usually just the order-1 and order-2 terms. We explore why the effective dimension is small for many integrals from finance. By deriving explicit forms of the ANOVA terms in simple cases, we find that the importance of each dimension is naturally weighted by certain hidden weights. These weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher-order ANOVA terms. We study the variance ratios captured by low-order ANOVA terms and their asymptotic properties as the dimension tends to infinity, and we show that with the increase of dimension the lower-order terms continue to play a significant role and the higher-order terms tend to be negligible. This provides some insight into high-dimensional problems from finance and explains why QMC algorithms are efficient for problems of this kind.