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A Two-Step Branching Splitting Model Under Cost Constraint for Rare Event Analysis

Part of: Markov processes Integral transforms, operational calculus

Published online by Cambridge University Press:  14 July 2016

Agnès Lagnoux-Renaudie
Agnès Lagnoux-Renaudie*
Affiliation:
Institut de Mathématique de Toulouse
*
∗Postal address: Institut de Mathématique de Toulouse, Université Paul Sabatier, 31062 Toulouse, France. Email address: lagnoux@cict.fr
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Abstract

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In this paper we consider the splitting method first introduced in rare event analysis. In this technique, the sample paths are split into R multiple copies at various stages to speed up the simulation. Given the cost, the optimization of the algorithm suggests taking all the transition probabilities to be equal; nevertheless, in practice, these quantities are unknown. We address this problem by presenting an algorithm in two phases.

Keywords

Splitting methodsimulation; cost functionLaplace transformGalton–Watson branching processiterated functionrare event

MSC classification

Secondary: 44A10: Laplace transform 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 429 - 452
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic (Appl. Math. Sci. 77). Springer, New York.Google Scholar
[2] Aldous, D. and Vazirani, U. V. (1994). ‘Go with the winners’ algorithms. In Proc. 35th IEEE Symp. Foundations Comput. Sci., IEEE Computer Society Press, Silver Spring, MD, pp. 492–501.Google Scholar
[3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
[4] Cörou, F. and Guyader, A. (2005). Adaptive multilevel splitting for rare event analysis. Tech. Rep. 5710, INRIA.Google Scholar
[5] De Boer, P. T. (2000). Analysis and efficient simulation of queueing models of telecommunication systems. , University of Twente.Google Scholar
[6] Del Moral, P. (2004). Feynman–Kac Formulae. Springer, New York.Google Scholar
[7] Diaconis, P. and Holmes, S. (1995). Three examples of Monte-Carlo Markov chains: at the interface between statistical computing, computer science, and statistical mechanics. In Discrete Probability and Algorithms (Minneapolis, MN, 1993; IMA Vol. Math. Appl. 72), Springer, New York, pp. 43–56.Google Scholar
[8] Doucet, A., De Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo Methods in Practice, Springer, New York, pp. 3–14.Google Scholar
[9] Harris, T. E. (2002). The Theory of Branching Processes. Corrected republication of the 1963 edition. Dover, New York.Google Scholar
[10] Heidelberger, P. (1995). Fast simulation of rare events in queueing and reliability models. ACM Trans. Model. Comput. Simul. 5, 43–85.Google Scholar
[11] Jerrum, M. and Sinclair, A. (1997). The Markov chain Monte Carlo method: an approach to approximate counting and integration. In Approximation Algorithms for NP-hard Problems. PWS Publishing, Boston, MA, pp. 482–520.Google Scholar
[12] Lagnoux, A. (2006). Rare event simulation. Prob. Eng. Inf. Sci. 20, 45–66.Google Scholar
[13] Lagnoux-Renaudie, A. (2008). Effective branching splitting method under cost constraint. Stoch. Process. Appl. 118, 1820–1851.CrossRefGoogle Scholar
[14] LeGland, F. and Oudjane, N. (2006). A sequential particle algorithm that keeps the particle system alive. In Stochastic Hybrid Systems (Lecture Notes Control Inf. Sci. 337), Springer, Berlin, pp. 351–389.Google Scholar
[15] Lyons, R. and Peres, Y. (2009). Probability on Trees and Networks. Cambridge University Press.Google Scholar
[16] Sadowsky, J. S. (1996). On Monte Carlo estimation of large deviations probabilities. Ann. Appl. Prob. 6, 399–422.Google Scholar
[17] Villén-Altamirano, M. and Villén-Altamirano, J. (1991). Restart: a method for accelerating rare event simulations. In Queueing Performance and Control in ATM, Elsevier Science Publishers, North-Holland, pp. 71–76.Google Scholar
[18] Villén-Altamirano, M. and Villén-Altamirano, J. (1997). Restart: an efficient and general method for fast simulation of rare event. Tech. Rep. 7, Universidad Politöcnica de Madrid.Google Scholar