Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T01:34:11.528Z Has data issue: false hasContentIssue false
  • English
  • Français

Monte Carlo methods for sensitivity analysis of Poisson-driven stochastic systems, and applications

Part of: Geometric probability and stochastic geometry Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Charles Bordenave  and
Giovanni Luca Torrisi
Charles Bordenave*
Affiliation:
University of California, Berkeley
Giovanni Luca Torrisi*
Affiliation:
Istituto per le Applicazioni del Calcolo ‘Mauro Picone’
*
∗ Postal address: Department of Electrical Engineering and Computer Science and Department of Statistics, University of California, 257 Cory Hall, Berkeley, CA 94720-1770, USA.
∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘Mauro Picone’, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, 00161 Roma, Italy. Email address: [email protected]
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.

We extend a result due to Zazanis (1992) on the analyticity of the expectation of suitable functionals of homogeneous Poisson processes with respect to the intensity of the process. As our main result, we provide Monte Carlo estimators for the derivatives. We apply our results to stochastic models which are of interest in stochastic geometry and insurance.

Keywords

Importance samplingmarked Poisson processMonte Carlo estimatorsensitivity analysisstabilizing functionalstopping set

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 65C05: Monte Carlo methods
Secondary: 60G48: Generalizations of martingales 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 293 - 320
Copyright
Copyright © Applied Probability Trust 2008 

References

Albeverio, S., Kondratiev, Y. G. and Röckner, M. (1996). Differential geometry of Poisson spaces. C. R. Acad. Sci. Paris Sér. I 323, 1129–1134.Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
Asmussen, S. and Rubinstein, R. Y. (1999). Sensitivity analysis of insurance risk models. Manag. Sci. 45, 1125–1141.Google Scholar
Baccelli, F. and Brémaud, P. (1993). Virtual customers in sensitivity and light traffic analysis via Campbell's formula for point processes. Adv. Appl. Prob. 25, 221–234.Google Scholar
Baccelli, F., Hasenfuss, S. and Schmidt, V. (1999). Differentiability of functionals of Poisson processes via coupling with applications to queueing theory. Stoch. Process. Appl. 81, 299–321.Google Scholar
Blaszczyszyn, B. (1995). Factorial moment expansion for stochastic systems. Stoch. Process. Appl. 56, 321–335.CrossRefGoogle Scholar
Brémaud, P. (2000). An insensitivity property of Lundberg's estimate for delayed claims. J. Appl. Prob. 37, 914–917.Google Scholar
Brémaud, P. and Vazquez-Abad, F. J. (1992). On the pathwise computation of derivatives with respect to the rate of a point process: the phantom RPA method. Queueing Systems Theory Appl. 10, 249–270.Google Scholar
Bucklew, J. A. (2004). Introduction to Rare Event Simulation. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Decreusefond, L. (1998). Perturbation analysis and Malliavin calculus. Ann. Appl. Prob. 8, 496–523.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, New York.Google Scholar
Foss, S. and Zuyev, S. (1996). On a Voronoi aggregative process related to a bivariate Poisson process. Adv. Appl. Prob. 28, 965–981.Google Scholar
Glassermann, P. (1990). Gradient Estimation via Perturbation Analysis. Kluwer, Dordrecht.Google Scholar
Gut, A. (1974). On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.Google Scholar
Ho, Y. C. and Cao, X. R. (1983). Perturbation analysis and optimization of queueing networks. J. Optimization Theory Appl. 40, 559–582.Google Scholar
Klüppelberg, C. and Mikosch, T. (1995). Delay in claim settlement and ruin probability approximations. Scand. Actuarial J. 2, 154–168.Google Scholar
Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.CrossRefGoogle Scholar
L'Ecuyer, P. (1990). A unified version of the IPA, SF, and LR gradient estimation techniques. Manag. Sci. 36, 1364–1383.Google Scholar
Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees. Ann. Appl. Prob. 7, 996–1020.Google Scholar
Macci, C., Stabile, G. and Torrisi, G. L. (2005). Lundberg parameters for non standard risk processes. Scand. Actuarial J. 6, 417–432.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Molchanov, I. and Zuyev, S. (2000). Variational analysis of functionals of Poisson processes. Math. Operat. Res. 25, 485–508.Google Scholar
Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Penrose, M. and Yukich, J. (2001). Limit theory for random sequential packing and deposition. Ann. Appl. Prob. 12, 272–301.Google Scholar
Penrose, M. and Yukich, J. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar
Reiman, M. I. and Simon, B. (1989). Open queueing systems in light traffic. Math. Operat. Res. 14, 26–59.Google Scholar
Reiman, M. I. and Weiss, A. (1989). Sensitivity analysis for simulations via likelihood ratios. Operat. Res. 37, 830–844.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
Suri, R. and Zazanis, F. M. (1988). Perturbation analysis gives strongly consistent sensitivity estimates for the M/G/1 queues. Manag. Sci. 34, 39–64.Google Scholar
Torrisi, G. L. (2004). Simulating the ruin probability of risk processes with delay in claim settlement. Stoch. Process. Appl. 112, 225–244.Google Scholar
Zazanis, F. M. (1992). Analyticity of Poisson-driven stochastic systems. Adv. Appl. Prob. 24, 532–541.Google Scholar
Zuyev, S. (1993). Russo's formula for Poisson point fields and its applications. Discrete Math. Appl. 3, 63–73.Google Scholar
Zuyev, S. (1999). Stopping-sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.Google Scholar