Sampling per mode for rare event simulation in switching diffusions. (English) Zbl 1260.65008
A switching diffusion process \(X_t\) is considered which is described by a stochastic differential equation with coefficients depending on a Markov jump process. The problem is to estimate the probabilities connected with reaching a critical region by \(X_t\) in the case when it is a rare event. A Monte Carlo technique with multilevel splitting is considered for this purpose. Using the Feyman-Kac flows and interacting particles systems theory, the authors establish a law of large numbers and a central limit theorem for their estimate in the case when the number of particles tends to infinity. It is demonstrated that the proposed adaptive algorithm of particles resampling improves the asymptotic variance of the estimate.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 65C35 | Stochastic particle methods |
| 60F05 | Central limit and other weak theorems |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60J60 | Diffusion processes |
| 60J05 | Discrete-time Markov processes on general state spaces |
Keywords:
multilevel splitting; central limit theorem; Feyman-Kac distribution flow; interacting particle system approximation; switching diffusion process; stochastic differential equation; Markov jump process; rare event; Monte Carlo technique; law of large numbers; algorithmSoftware:
RESTARTReferences:
| [1] | H. Blom, Stochastic hybrid processes with hybrid jumps, in: IFAC Conference on Analysis and Design of Hybrid Systems (ADHS03), April 2003, HYBRIDGE deliverable R2.3, http://www.nlr.nl/public/hostedsites/hybridge; H. Blom, Stochastic hybrid processes with hybrid jumps, in: IFAC Conference on Analysis and Design of Hybrid Systems (ADHS03), April 2003, HYBRIDGE deliverable R2.3, http://www.nlr.nl/public/hostedsites/hybridge |
| [2] | Blom, H.; Bakker, B.; Krystul, J., Rare event estimation for a large-scale stochastic hybrid system with air traffic application, (Rubino, G.; Tuffin, B., Rare Event Simulation Using Monte Carlo Methods (2009), Wiley), 194-214 · Zbl 1279.90034 |
| [3] | Cérou, F.; Del Moral, P.; Guyader, A., A non asymptotic variance theorem for unnormalized Feynman-Kac particle models, Ann. Inst. H. Poincaré Probab. Statist., 47, 3 (2011) · Zbl 1233.60047 |
| [4] | Cérou, F.; Del Moral, P.; Le Gland, F.; Lezaud, P., Genetic genealogical models in rare event analysis, ALEA, Latin American Journal of Probability and Mathematical Statistics, 1, 181-203 (2006), Paper 01-08 · Zbl 1104.60044 |
| [5] | Del Moral, P., (Feynman-Kac Formulae. Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications, Probability and its Applications (2004), Springer: Springer New York) · Zbl 1130.60003 |
| [6] | Del Moral, P.; Lezaud, P., Branching and interacting particle interpretation of rare event probabilities, (Blom, H.; Lygeros, J., Stochastic Hybrid Systems: Theory and Safety Critical Applications. Stochastic Hybrid Systems: Theory and Safety Critical Applications, Lecture Notes in Control and Information Sciences (2006), Springer: Springer Berlin), 277-323 · Zbl 1114.65004 |
| [7] | M.J.J. Garvels, The splitting method in rare event simulation, Ph.D. Thesis, Faculty of Mathematical Sciences, University of Twente, Enschede, 2000.; M.J.J. Garvels, The splitting method in rare event simulation, Ph.D. Thesis, Faculty of Mathematical Sciences, University of Twente, Enschede, 2000. |
| [8] | Glasserman, P.; Heidelberger, P.; Shahabuddin, P.; Zajic, T., Multilevel splitting for estimating rare event probabilities, Operations Research, 47, 585-600 (1999) · Zbl 0985.65006 |
| [9] | Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes (2003), Springer · Zbl 1018.60002 |
| [10] | J. Krystul, Modelling of stochastic hybrid systems with applications to accident risk assessment, Ph.D. Thesis, Faculty of Mathematical Sciences, University of Twente, Enschede, 2006.; J. Krystul, Modelling of stochastic hybrid systems with applications to accident risk assessment, Ph.D. Thesis, Faculty of Mathematical Sciences, University of Twente, Enschede, 2006. |
| [11] | Lagnoux, A., Rare event simulation, Probability in the Engineering and Information Sciences, 20, 45-66 (2006) · Zbl 1101.65005 |
| [12] | L’Écuyer, P.; Le Gland, F.; Lezaud, P.; Tuffin, B., Splitting techniques, (Rubino, G.; Tuffin, B., Rare Event Simulation Using Monte Carlo Methods (2009), Wiley), 39-61 · Zbl 1168.65309 |
| [13] | L’Écuyer, P.; Mandjes, M.; Tuffin, B., Importance sampling, (Rubino, G.; Tuffin, B., Rare Event Simulation Using Monte Carlo Methods (2009), Wiley), 17-38 · Zbl 1165.62027 |
| [14] | Le Gland, F.; Oudjane, N., A sequential algorithm that keeps the particle system alive, (Blom, H.; Lygeros, J., Stochastic Hybrid Systems: Theory and Safety Critical Applications. Stochastic Hybrid Systems: Theory and Safety Critical Applications, Lecture Notes in Control and Information Sciences (2006), Springer: Springer Berlin), 351-389 · Zbl 1130.93053 |
| [15] | Liu, J. S., (Monte Carlo Strategies in Scientific Computing. Monte Carlo Strategies in Scientific Computing, Springer Series in Statistics (2004), Springer) · Zbl 1132.65003 |
| [16] | M. Villén-Altamirano, J. Villén-Altamirano, RESTART: a straightforward method for fast simulation of rare events, in: Proceedings of the 1994 Winter Simulation Conference, Orlando, 1994, pp. 282-289.; M. Villén-Altamirano, J. Villén-Altamirano, RESTART: a straightforward method for fast simulation of rare events, in: Proceedings of the 1994 Winter Simulation Conference, Orlando, 1994, pp. 282-289. |
| [17] | Villén-Altamirano, M.; Villén-Altamirano, J., Analysis of RESTART simulation: theoretical basis and sensitivity study, European Transactions on Telecommunications, 13, 373-385 (2002) |
| [18] | Villén-Altamirano, M.; Villén-Altamirano, J., On the efficiency of RESTART for multidimensional state systems, ACM Transactions on Modeling and Computer Simulation, 16, 251-279 (2006) · Zbl 1390.65026 |
| [19] | Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46, 1155-1179 (2007) · Zbl 1140.93045 |
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