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A dichotomy for sampling barrier-crossing events of random walks with regularly varying tails

Part of: Algorithms - Computer Science Operations research and management science Stochastic processes

Published online by Cambridge University Press:  30 November 2017

A. B. Dieker  and
Guido R. Lagos
A. B. Dieker*
Affiliation:
Columbia University
Guido R. Lagos*
Affiliation:
Universidad de Chile
*
* Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA.
** Postal address: Center for Mathematical Modeling, Universidad de Chile, Beauchef 851, Torre Norte oficina 705, Santiago, RM 8370456, Chile. Email address: guido.lagos.barrios@gmail.com
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Abstract

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are independent and identically distributed with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index α of the right tail lies in the interval (1, 3/2).

Keywords

Conditional samplingperfect samplingefficiencyrandom walkheavy tailregularly varying tailchange of measurelikelihood ratio

MSC classification

Primary: 68U20: Simulation 60G50: Sums of independent random variables; random walks 68W40: Analysis of algorithms
Secondary: 90B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1213 - 1232
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York. Google Scholar
[2] Asmussen, S. and Binswanger, K. (1997). Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27, 297–318. Google Scholar
[3] Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York. Google Scholar
[4] Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125. CrossRefGoogle Scholar
[5] Asmussen, S., Binswanger, K. and Højgaard, B. (2000). Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303–322. Google Scholar
[6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press. Google Scholar
[7] Blanchet, J. and Chen, X. (2015). Steady-state simulation of reflected Brownian motion and related stochastic networks. Ann. Appl. Prob. 25, 3209–3250. Google Scholar
[8] Blanchet, J. and Glynn, P. (2008). Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378. Google Scholar
[9] Blanchet, J. and Liu, J. (2012). Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks. Stoch. Process. Appl. 122, 2994–3031. Google Scholar
[10] Blanchet, J. H. and Sigman, K. (2011). On exact sampling of stochastic perpetuities. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A), Applied Probability Trust, Sheffield, pp. 165–182. Google Scholar
[11] Blanchet, J. and Wallwater, A. (2015). Exact sampling of stationary and time-reversed queues. ACM Trans. Modeling Comput. Simul. 25, 26. Google Scholar
[12] Blanchet, J., Chen, X. and Dong, J. (2017). ε-strong simulation for multidimensional stochastic differential equations via rough path analysis. Ann. Appl. Prob. 27, 275–336. Google Scholar
[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer, Berlin. CrossRefGoogle Scholar
[14] Ensor, K. B. and Glynn, P. W. (2000). Simulating the maximum of a random walk. J. Statist. Planning Infer. 85, 127–135. CrossRefGoogle Scholar
[15] Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, New York. Google Scholar
[16] Gudmundsson, T. and Hult, H. (2014). Markov chain Monte Carlo for computing rare-event probabilities for a heavy-tailed random walk. J. Appl. Prob. 51, 359–376. CrossRefGoogle Scholar
[17] Juneja, S. and Shahabuddin, P. (2002). Simulating heavy tailed processes using delayed hazard rate twisting. ACM Trans. Modeling Comput. Simul. 12, 94–118. Google Scholar
[18] Liu, Z., Blanchet, J. H., Dieker, A. B. and Mikosch, T. (2016). Optimal exact simulation of max-stable and related random fields. Preprint. Available at https://arxiv.org/abs/1609.06001. Google Scholar
[19] Murthy, K. R. A., Juneja, S. and Blanchet, J. (2014). State-independent importance sampling for random walks with regularly varying increments. Stoch. Systems 4, 321–374. CrossRefGoogle Scholar
[20] Resnick, S. I. (1997). Heavy tail modeling and teletraffic data. Ann. Statist. 25, 1805–1869. Google Scholar