Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T16:21:08.220Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Integral transforms, operational calculus Markov processes

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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