The Annals of Applied Probability

Efficient importance sampling for Monte Carlo evaluation of exceedance probabilities

Hock Peng Chan and Tze Leung Lai

Full-text: Open access

Abstract

Large deviation theory has provided important clues for the choice of importance sampling measures for Monte Carlo evaluation of exceedance probabilities. However, Glasserman and Wang [Ann. Appl. Probab. 7 (1997) 731–746] have given examples in which importance sampling measures that are consistent with large deviations can perform much worse than direct Monte Carlo. We address this problem by using certain mixtures of exponentially twisted measures for importance sampling. Their asymptotic optimality is established by using a new class of likelihood ratio martingales and renewal theory.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 2 (2007), 440-473.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1174323253

Digital Object Identifier
doi:10.1214/105051606000000664

Mathematical Reviews number (MathSciNet)
MR2308332

Zentralblatt MATH identifier
1134.65005

Subjects
Primary: 60F10: Large deviations 65C05: Monte Carlo methods
Secondary: 60J05: Discrete-time Markov processes on general state spaces 65C40: Computational Markov chains

Keywords
Boundary crossing probability importance sampling Markov additive process regeneration

Citation

Chan, Hock Peng; Lai, Tze Leung. Efficient importance sampling for Monte Carlo evaluation of exceedance probabilities. Ann. Appl. Probab. 17 (2007), no. 2, 440--473. doi:10.1214/105051606000000664. https://projecteuclid.org/euclid.aoap/1174323253


Export citation

References

  • Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493–501.
  • Bucklew, J. A., Ney, P. and Sadowsky, J. S. (1990). Monte Carlo simulation and large deviation theory for uniformly recurrent Markov chains. J. Appl. Probab. 27 44–59.
  • Bucklew, J. A., Nitinawarat, S. and Wierer, J. (2004). Universal simulation distributions. IEEE Trans. Inform. Theory 50 2674–2685.
  • Chan, H. P. and Lai, T. L. (2000). Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families. Ann. Statist. 28 1638–1669.
  • Chan, H. P. and Lai, T. L. (2003). Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks. Ann. Appl. Probab. 13 395–429.
  • Chan, H. P. and Lai, T. L. (2005). Importance sampling for generalized likelihood ratio procedures in sequential analysis. Sequential Anal. 24 259–278.
  • Collamore, J. F. (2002). Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors. Ann. Appl. Probab. 12 382–421.
  • Dupuis, P. and Wang, H. (2005). Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 1–38.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Glasserman, P. and Wang, Y. (1997). Counterexamples in importance sampling for large deviation probabilities. Ann. Appl. Probab. 7 731–746.
  • Iscoe, I., Ney, P. and Nummelin, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 373–412.
  • Lehtonen, T. and Nyrhinen, H. (1992). Simulating level-crossing probabilities by importance sampling. Adv. in Appl. Probab. 24 858–874.
  • Lehtonen, T. and Nyrhinen, H. (1992). On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuar. J. 1 60–75.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Ney, P. and Nummelin, E. (1987). Markov additive processes. I. Eigenvalues properties and limit theorems. Ann. Probab. 15 561–592.
  • Ney, P. and Nummelin, E. (1987). Markov additive processes. II. Large deviations. Ann. Probab. 15 593–609.
  • Sadowsky, J. S. and Bucklew, J. A. (1990). On large deviations theory and asymptotically efficient Monte Carlo estimation. IEEE Trans. Inform. Theory 36 579–588.
  • Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673–684.
  • Stone, C.(1965). Local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Statist. 36 546–551.