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2013 | OriginalPaper | Chapter

3. Non-asymptotic Error Estimates for Monte Carlo Methods

Authors : Carl Graham, Denis Talay

Published in: Stochastic Simulation and Monte Carlo Methods

Publisher: Springer Berlin Heidelberg

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Abstract

In order to effectively implement Monte Carlo methods, the random approximation errors must be controlled. For this purpose, theoretical results are provided for the estimation of the number of simulations necessary to obtain a desired accuracy with a prescribed confidence interval. Therefore absolute, i.e., non-asymptotic, versions of the Central Limit Theorem (CLT) are developed: Berry–Esseen’s and Bikelis’ theorems, as well as concentration inequalities obtained from logarithmic Sobolev inequalities. The difficult subject of variance reduction techniques for Monte Carlo methods arises naturally in this context, and is discussed at the end of this chapter.

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previous chapter Strong Law of Large Numbers and Monte Carlo Methods
next chapter Poisson Processes as Particular Markov Processes
Footnotes
1
Itô, K., McKean Jr., H.P.: Diffusion Processes and Their Sample Paths. Die Grundlehren der Mathematischen Wissenschaften, vol. 125, 2nd edn. Springer, Berlin (1974).
 
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Arouna, B.: Robbins-Monro algorithms and variance reduction in finance. J. Comput. Finance 7(2), 35–61 (2003)
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Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability, vol. 57. Springer, New York (2007)
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Jacod, J., Protter, P.: Probability Essentials, 2nd edn. Universitext. Springer, Berlin (2004) CrossRef
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Petrov, V.V.: Sums of Independent Random Variables. Springer, New York (1975). Translated from the Russian by A.A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82 CrossRef
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Shiryayev, A.N.: Probability. Graduate Texts in Mathematics, vol. 95. Springer, New York (1984). Translated from the Russian by R.P. Boas MATHCrossRef
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Williams, D.: Probability with Martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge (1991) MATHCrossRef
Metadata
Title
Non-asymptotic Error Estimates for Monte Carlo Methods
Authors
Carl Graham
Denis Talay
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Stochastic Simulation and Monte Carlo Methods

Print ISBN: 978-3-642-39362-4

Electronic ISBN: 978-3-642-39363-1

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-39363-1_3