Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
Quasi-Monte Carlo methods in numerical finance
Management Science
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Strong tractability of multivariate integration using quasi-Monte Carlo algorithms
Mathematics of Computation
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Sufficient conditions for fast quasi-Monte Carlo convergence
Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
Low discrepancy sequences in high dimensions: How well are their projections distributed?
Journal of Computational and Applied Mathematics
New Brownian bridge construction in quasi-Monte Carlo methods for computational finance
Journal of Complexity
Derivative based global sensitivity measures and their link with global sensitivity indices
Mathematics and Computers in Simulation
SIAM Journal on Scientific Computing
On the necessity of low-effective dimension
Journal of Complexity
Smoothness and dimension reduction in Quasi-Monte Carlo methods
Mathematical and Computer Modelling: An International Journal
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Quasi-Monte Carlo (QMC) methods are important numerical tools in computational finance. Path generation methods (PGMs), such as Brownian bridge and principal component analysis, play a crucial role in QMC methods. Their effectiveness, however, is problem-dependent. This paper attempts to understand how a PGM interacts with the underlying function and affects the accuracy of QMC methods. To achieve this objective, we develop efficient methods to assess the impact of PGMs. The first method is to exploit a quadratic approximation of the underlying function and to analyze the effective dimension and dimension distribution (which can be done analytically). The second method is to carry out a QMC error analysis on the quadratic approximation, establishing an explicit relationship between the QMC error and the PGM. Equalities and bounds on the QMC errors are established, in which the effect of the PGM is separated from the effect of the point set (in a similar way to the Koksma-Hlawka inequality). New measures for quantifying the accuracy of QMC methods combining with PGMs are introduced. The usefulness of the proposed methods is demonstrated on two typical high-dimensional finance problems, namely, the pricing of mortgage-backed securities and Asian options (with zero strike price). It is found that the success or failure of PGMs that do not take into account the underlying functions (such as the standard method, Brownian bridge and principal component analysis) strongly depends on the problem and the model parameters. On the other hand, the PGMs that take into account the underlying function are robust and powerful. The investigation presents new insight on PGMs and provides constructive guidance on the implementation and the design of new PGMs and new QMC rules.