Smoothness and dimension reduction in Quasi-Monte Carlo methods

Moskowitz, B.S.; Caflisch, Russel E.

doi:10.1016/0895-7177(96)00038-6

Cited by 149 publications

(130 citation statements)

References 14 publications

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“…The convergence rate of QMC depends on the smoothness of the summand and it may deteriorate because of the lack of regularity in v k ω j (x k , φ, t). By introducing a linear approximation of v k close to the switch from 1 to 0, better convergence is achieved in [2,25]. The convergence of the MC and QMC methods are compared using Faure sequences [6] for QMC in a numerical example in Sect.…”

Section: Approximation Of the Summationmentioning

confidence: 99%

“…The acceptance-rejection method (A-R) [2,12] and a smoothing alternative [2,25] are tested. The discontinuity in the A-R method is replaced by a linear function between 0 and 1 in [25]. One important consideration is the fact that when for example the A-R method is used to generate numbers from the distribution, fewer evaluation points will be accepted and used than in e.g.…”

Section: Metabolites Controled By Enzymesmentioning

confidence: 99%

“…Raw MC summation with pseudorandom numbers is compared to raw QMC summation using numbers from the Faure sequence generated with the algorithm in [16]. The acceptance-rejection method (A-R) [2,12] and a smoothing alternative [2,25] are tested. The discontinuity in the A-R method is replaced by a linear function between 0 and 1 in [25].…”

Section: Metabolites Controled By Enzymesmentioning

confidence: 99%

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Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with a mesoscopic, stochastic model. The molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules. The macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation. The probability distribution is obtained by the Stochastic Simulation Algorithm due to Gillespie. The equations are coupled via a summation over the mesoscale variables. This summation is approximated by Monte Carlo and Quasi Monte Carlo methods. The error in the approximations is analyzed. The hybrid method is applied to three chemical systems from molecular cell biology.

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Section: Approximation Of the Summationmentioning

confidence: 99%

Section: Metabolites Controled By Enzymesmentioning

confidence: 99%

Section: Metabolites Controled By Enzymesmentioning

confidence: 99%

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Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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“…The effectiveness of quasi-Monte Carlo for path integrals will be regained using an alternative representation of the random walk, the Brownian Bridge Discretization, which was first introduced as a quasi-Monte Carlo technique by [25]. This representation relies on the following Brownian bridge formula [15] for b(t + ∆t 1 ) knowing b(t) and b(T = t + ∆t 1 + ∆t 2 ):…”

Section: 2mentioning

confidence: 99%

Monte Carlo Simulation for American Options

Caflisch¹,

Chaudhary²

2004

A Celebration of Mathematical Modeling

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This paper reviews the basic properties of American options and the difficulties of applying Monte Carlo valuation to American options. Asymptotic results by Keller and co-workers are described for the singularity in the early exercise boundary for time t near the final time T . Recent progress on application of Monte Carlo to American options is described including the following: Branching processes have been constructed to obtain upper and lower bounds on the American option price. A Martingale optimization formulation for the American option price can be used to obtain an upper bound on the price, which is complementary to the trivial lower bound. The Least Squares Monte Carlo (LSM) provides a direct method for pricing American options. Quasirandom sequences have been used to improve performance of LSM; a brief introduction to quasi-random sequences is presented. Conclusions and prospects for future research are discussed. In particular, we expect that the asymptotic results of Keller and co-workers could be useful for improving Monte Carlo methods.

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“…The authors have combined this approach with the so-called bridge sampling (see Moskowitz and Caflisch 1996) and have developed a method for sampling from the gamma process. It has been applied by Avramidis et al (2003) to the gamma process used to randomize the time in the Brownian motion representation of the VG process, a method, named Brownian Gamma Bridge Sampling (BGBS).…”

Section: Introductionmentioning

confidence: 99%