# Efficient rare-event simulation for the maximum of heavy-tailed random walks

- Published: 01 Aug 2008 Journal: The Annals of Applied Probability, volume 18, pages 1,351-1,378 (issn: 1050-5164, Copyright policy)
- Publisher: Institute of Mathematical Statistics

P (X ≤ s)P (X > t − s) ds Adler, J., Feldman, R. and Taqqu, M., eds. (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkh¨auser, Boston. MR1652283 Asmussen, S. (2003). Applied Probability and Queues. Springer, New York. MR1978607 Asmussen, S. and Glynn, P. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York. MR2331321

Asmussen, S. and Binswanger, K. (1997). Simulation of ruin probabilities for subexponential claims. Ast. Bulletin 27 297-318. [OpenAIRE]

Asmussen, S., Binswanger, K. and Hojgaard, B. (2000). Rare event simulation for heavy-tailed distributions. Bernoulli 6 303-322. MR1748723 Asmussen, S. and Kluppelberg, C. (1996). Large deviation results for subexponential tails, with applications to insurance risk. Stoch. Proc. Appl. 64 103-125. MR1419495 Asmussen, S. and Kroese, D. (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Probab. 38 545-558. MR2264957 Bassamboo, A., Juneja, S. and Zeevi, A. (2006). On the efficiency loss of stateindependent importance sampling in the presence of heavy-tails. Oper. Res. Lett. 34 521-531. [OpenAIRE]

Blanchet, J., Glynn, P. and Liu, J. C. (2007). Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue. QUESTA 57 99-113.

Blanchet, J. and Li, C. (2006). Efficient rare-event simulation for geometric sums. Proc. RESIM, Bamberg. Germany.

Borovkov, A. A. and Borovkov, K. A. (2001). On probabilities of large deviations for random walks. I: Regularly varying distribution tails. Theory Probab. Appl. 49 189-205. [OpenAIRE]

Bucklew, J. (2004). Introduction to Rare-Event Simulation. Springer, New York. MR2045385

Dupuis, P. and Wang, H. (2004). Importance sampling, large deviations, and differential games. Stochastics Stochastics Rep. 76 481-508. MR2100018 Dupuis, P., Leder, K. and Wang, H. (2006). Importance sampling for sums of random variables with regularly varying tails. TOMACS 17.

Embrechts, P., Klu¨ppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, New York. MR1458613 Hammersley, J. and Morton, K. (1954). Poor man's Monte Carlo. J. Roy. Statist. Soc. Ser. B 16 23-38. MR0064475

Juneja, S. and Shahabuddin, P. (2002). Simulating heavy-tailed processes using delayed hazard rate twisting. ACM TOMACS 12 94-118. [OpenAIRE]

Juneja, S. and Shahabuddin, P. (2006). Rare event simulation techniques: An introduction and recent advances. In Handbook on Simulation (S. Henderson and B. Nelson, eds.) 291-350. North-Holland, Amsterdam. [OpenAIRE]

Liu, J. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York. MR1842342

Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Available at http://decision.csl.uiuc.edu/˜meyn/pages/book.html. MR1287609 Rosenbluth, M. and Rosenbluth, A. (1955). Monte Carlo calculation of the average extension of molecular chains. J. Chem. Phys. 23 356-359. [OpenAIRE]

Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673-684. MR0418369 [OpenAIRE]

Department of Management Science and Engineering Stanford University Stanford, California 94305 USA