Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-15T18:45:14.075Z Has data issue: false hasContentIssue false
  • English
  • Français

On asymptotically efficient simulation of large deviation probabilities

Part of: Special processes Limit theorems Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

A. B. Dieker  and
M. Mandjes
A. B. Dieker*
Affiliation:
CWI and University of Twente
M. Mandjes*
Affiliation:
CWI and University of Amsterdam
*
∗ Postal address: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands.
∗ Postal address: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {νε, ε>0} be a family of probabilities for which the decay is governed by a large deviation principle, and consider the simulation of νε0(A) for some fixed measurable set A and some ε0>0. We investigate the circumstances under which an exponentially twisted importance sampling distribution yields an asymptotically efficient estimator. Varadhan's lemma yields necessary and sufficient conditions, and these are shown to improve on certain conditions of Sadowsky. This is illustrated by an example to which Sadowsky's conditions do not apply, yet for which an efficient twist exists.

Keywords

Asymptotic efficiencyimportance samplinglarge deviation

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60F10: Large deviations 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 539 - 552
Copyright
Copyright © Applied Probability Trust 2005 

Footnotes

Supported by the Netherlands Organization for Scientific Research (NWO) under grant 631.000.002.

References

Asmussen, S. (1989). Risk theory in a Markovian environment. Scand. Actuarial J., 69–100.Google Scholar
Asmussen, S. (2003).Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Rubinstein, R. Y. (1995). “Steady state rare events simulation in queueing models and its complexity properties.” In Advances in Queueing, ed. Dshalalow, J., CRC, Boca Raton, FL, pp. 429–461.Google Scholar
Biggins, J. D. (2004). “Large deviations for mixtures.” Electron. Commun. Prob. 9, 60–71.Google Scholar
Boots, N. K. and Mandjes, M. (2002). “Fast simulation of a queue fed by a superposition of many (heavy-tailed) sources.” Prob. Eng. Inf. Sci. 16, 205–232.Google Scholar
Bucklew, J. A., Ney, P. and Sadowsky, J. S. (1990). “Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains.” J. Appl. Prob. 27, 44–59.CrossRefGoogle Scholar
Chaganty, N. R. (1997). “Large deviations for Joint distributions and statistical applications.” Sankhyā A 59, 147–166.Google Scholar
Collamore, J. F. (2002). “Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors.” Ann. Appl. Prob. 12, 382–421.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). “Large Deviations Techniques and Applications, 2nd edn. ” Springer, New York.CrossRefGoogle Scholar
Deuschel, J.-D. and Stroock, D. W. (1989). “Large Deviations.” Academic Press, Boston, MA.Google Scholar
Dieker, A. B. and Mandjes, M. (2004). “Fast simulation of overflow probabilities in a queue with Gaussian input.” Tech. Rep. PNA-E0404, CWI.Google Scholar
Dinwoodie, I. H. and Zabell, S. L. (1992). “Large deviations for exchangeable random vectors.” Ann. Prob. 20, 1147–1166.Google Scholar
Dinwoodie, I. H. and Zabell, S. L. (1993). “Large deviations for sequences of mixtures.” In Statistics and Probability: A Raghu Raj Bahadur Festschrift, eds Ghosh, J. K., Mitra, S. K., Parthasarathy, K. R. and Prakasa Roa, B. L. S., John Wiley, New Delhi, pp. 171–190.Google Scholar
Dupuis, P. and Wang, H. (2004). “Importance sampling, large deviations, and differential games.” Stoch. Stoch. Reports 76, 481–508.Google Scholar
Feng, J. and Kurtz, T. (2000). “Large deviations for stochastic processes.” Preprint.Google Scholar
Glasserman, P. and Kou, S. (1995). “Analysis of an importance sampling estimator for tandem queues.” ACM Trans. Model. Comput. Simul. 4, 22–42.CrossRefGoogle Scholar
Glasserman, P. and Li, J. (2003). “Importance sampling for portfolio credit risk.” Preprint.Google Scholar
Glasserman, P. and Wang, Y. (1997). “Counterexamples in importance sampling for large deviations probabilities.” Ann. Appl. Prob. 7, 731–746.CrossRefGoogle Scholar
Glasserman, P., Heidelberger, P. and Shahabuddin, P. (1999). “Asymptotically optimal importance sampling and stratification for pricing path-dependent options.” Math. Finance 9, 117–152.CrossRefGoogle Scholar
Glynn, P. W. and Whitt, W. (1992). “The asymptotic efficiency of simulation estimators.” Operat. Res. 40, 505–520.CrossRefGoogle Scholar
Heidelberger, P. (1995). “Fast simulation of rare events in queueing and reliability models.” ACM Trans. Model. Comput. Simul. 5, 43–85.Google Scholar
Lehtonen, T. and Nyrhinen, H. (1992). On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuarial J., 60–75.CrossRefGoogle Scholar
Lehtonen, T. and Nyrhinen, H. (1992). “Simulating level-crossing probabilities by importance sampling.” Adv. Appl. Prob. 24, 858–874.CrossRefGoogle Scholar
Sadowsky, J. S. (1993). “On the optimality and stability of exponential twisting in Monte Carlo estimation.” IEEE Trans. Inf. Theory 39, 119–128.Google Scholar
Sadowsky, J. S. (1996). “On Monte Carlo estimation of large deviations probabilities.” Ann. Appl. Prob. 6, 399–422.Google Scholar
Sadowsky, J. S. and Bucklew, J. A. (1990). “On large deviations theory and asymptotically efficient Monte Carlo estimation.” IEEE Trans. Inf. Theory 36, 579–588.Google Scholar
Siegmund, D. (1976). “Importance sampling in the Monte Carlo study of sequential tests.” Ann. Statist. 4, 673–684.Google Scholar
Szechtman, R. and Glynn, P. W. (2002). “Rare-event simulation for infinite server queues.” In Proc. 34th Conf. Winter Simul. Exploring New Frontiers (San Diego, CA, December 2002), eds Yücesan, E., Chen, C.-H., Snowdon, J. L. and Charmes, J. M., IEEE, Piscataway, pp. 416–423.Google Scholar