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Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

Part of: Stochastic processes Algorithms - Computer Science

Published online by Cambridge University Press:  14 July 2016

Jose Blanchet  and
Jingchen Liu
Jose Blanchet*
Affiliation:
Columbia University
Jingchen Liu*
Affiliation:
Columbia University
*
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, S. W. Mudd Building, 500 West 120th Street, New York, NY 10027-6699, USA. Email address: [email protected]
∗∗Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, Room 1030, New York, NY 10027, USA. Email address: [email protected]
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Abstract

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We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

Keywords

State dependentimportance samplingregularly varyingrandom walkheavy tailstrong efficiencyrare-event simulationmixture familymultidimension

MSC classification

Secondary: 60G50: Sums of independent random variables; random walks 68W40: Analysis of algorithms 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 301 - 322
Copyright
Copyright © Applied Probability Trust 2010 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Blanchet, J. and Glynn, P. (2008). Efficient rare event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378.CrossRefGoogle Scholar
Blanchet, J. H. and Liu, J. (2007). Rare-event simulation for a multidimensional random walk with t distributed increments. In Proc. 39th Conf. Winter Simulation, IEEE, pp. 395–402.Google Scholar
Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for regularly varying random walks. Adv. Appl. Prob. 40, 1104–1128.CrossRefGoogle Scholar
Blanchet, J., Glynn, P. and Liu, J. (2007). Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue. Queueing Systems 57, 99–113.CrossRefGoogle Scholar
Collamore, J. (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
Dupuis, P., Leder, K. and Wang, H. (2006). Notes on importance sampling for random variables with regularly varying tails. Preprint.Google Scholar
Embrechts, P., Lindskog, F. and McNeil, A. (2003). Modeling dependence with copulas and applications to risk management. In Handbook of Heavy Tailed Distributions in Finance, ed. Rachev, S., Elsevier, pp. 329–384.CrossRefGoogle Scholar
Glasserman, P. and Juneja, S. (2008). Uniformly efficient importance sampling for the tail distribution of sums of random variables. Math. Operat. Res. 33, 36–50.CrossRefGoogle Scholar
Glynn, P. W. and Iglehart, D. L. (1989). Importance sampling for stochastic simulations. Manag. Sci. 35, 1367–1392.CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2006). Heavy-tailed insurance portfolios: buffer capital and ruin probabilities. Tech. Rep. 1441, School of ORIE, Cornell University.Google Scholar
Hult, H., Lindskog, F., Mikosch, T. and Samorodnitsky, G. (2005). Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Prob. 15, 2651–2680.CrossRefGoogle Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie, Berlin.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle 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.CrossRefGoogle Scholar