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Mathematical Modeling of Distributed Catastrophic and Terrorist Risks1

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Cybernetics and Systems Analysis Aims and scope

An Erratum to this article was published on 01 March 2015

Abstract

The paper shows how the mathematical tools of the theory of controlled Markov fields can be applied to model catastrophic risks caused by natural events or terrorist threats. The examples of problem statements of long-term investment in security are given. A survey of solution methods for stochastic optimal control problems is proposed. It is shown that these problems can be reduced to finite-dimensional stochastic programming problems and can be solved by the stochastic quasigradient method.

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References

  1. C. D. Daykin, T. Pentikainen, and M. Pesonen, Practical Risk Theory for Actuaries, Chapman and Hall, London–New York (1993).

    Google Scholar 

  2. R. Kaas, M. Goovaerts, J. Dhaene, and M. Denuit, Modern Actuarial Risk Theory: Using R, Springer-Verlag, Berlin–Heidelberg (2008).

    Book  Google Scholar 

  3. G. Walker, “Current developments in catastrophe modelling,” in: N. R. Britton and J. Oliver (eds.), Financial Risks Management for Natural Catastrophes, Griffith. Univ., Brisbane, Australia (1997), pp. 17–35.

    Google Scholar 

  4. A. Amendola et al. (eds.), “Integrated catastrophe risk modeling: Supporting policy processes,” Advances in Natural and Technological Hazards Research, 32, Springer, Dordrecht (2013).

  5. Yu. M. Ermol’yev, T. Yu. Ermol’yeva, G. McDonald, and V. I. Norkin, “Problems on insurance of catastrophic risks,” Cybern. Syst. Analysis, 37, No. 2, 220–234 (2001).

    Article  MATH  Google Scholar 

  6. Yu. M. Ermoliev, Stochastic Programming Methods [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  7. Yu. Ermoliev and R. J.-B. Wets (eds.), Numerical Techniques for Stochastic Optimization, Springer Series in Computational Mathematics, 10, Springer-Verlag, Berlin (1988).

  8. A. Ruszczyński and A. Shapiro (eds.), Stochastic Programming, Handbooks in OR & MS, Elsvier, Amsterdam (2003).

    Google Scholar 

  9. R. Bellman, Dynamic Programming, Dover Publ. (2003).

  10. R. A. Howard, Dynamic Programming and Markov Processes, Technology Press and Wiley, New York–London (1960).

    MATH  Google Scholar 

  11. C. Derman, Finite State Markovian Decision Processes, Academic Press, New York–London (1970).

    MATH  Google Scholar 

  12. E. B. Dynkin and A. A. Yushkevich, Controlled Markov Processes and their Applications [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  13. H. Mine and S. Osaki, Markovian Decision Processes, Elsevier Sci. Ltd. (1970).

    MATH  Google Scholar 

  14. I. I. Gikhman and A. V. Skorokhod, Controlled Random Processes [in Russian], Naukova Dumka, Kyiv (1977).

    Google Scholar 

  15. D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete-Time Case, Athena Scientific (2000).

  16. A. B. Piunovskii, “Controlled random sequences: Methods of convex analysis and problems with functional constraints,” Uspekhi Mat. Nauk, 53, Issue 6, 129–192 (1998).

    Article  MathSciNet  Google Scholar 

  17. N. B. Vasil’ev and O. K. Kozlov, “Reversible Markov chains with local interaction,” in: Multicomponent Random Systems [in Russian], Nauka, Moscow (1978), pp. 83–100.

  18. O. K. Kozlov and N.B. Vasilyev, “Reversible Markov chains with local interactions,” in: R. L. Dobrushin and Y. G. Sinai (eds.), Multicomponent Random Systems. Advances in Probability and Related Topics, 6, Marcel Dekker, New York, pp. 451–469 (1980).

  19. M. B. Averintsev, “Description of Markov random fields with the help of the Gibbs conditional probabilities,” Probab. Theory and its Application, 17, No. 1, 21–35 (1972).

    Article  Google Scholar 

  20. T. M. Ligett, Interacting Particle Systems, Springer, Berlin (1985).

    Book  Google Scholar 

  21. T. M. Ligett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes, Springer, Berlin (1999).

    Book  Google Scholar 

  22. G. Daduna, P. S. Knopov, and R. K. Chornei, “Local control of Markovian processes of interaction on a graph with a compact set of states,” Cybern. Syst. Analysis, 37, No. 3, 348–360 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Daduna, P. S. Knopov, and R. K. Chornei, “Controlled semi-Markov fields with graph-structured compact state space,” Theory of Probability and Mathematical Statistics, 69, 39–53 (2004).

    Article  MathSciNet  Google Scholar 

  24. R. K. Chornei, H. Daduna, and P. S. Knopov, Control of Spatially Structured Random Processes and Fields with Applications, Springer, New York (2006).

    MATH  Google Scholar 

  25. R. K. Chornei, H. Daduna, and P. S. Knopov, “Controlled Markov fields with finite space on graphs,” Stochastic Models, 21, 847–874 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  26. Y. Ermoliev, T. Ermolieva, and V. I. Norkin, “Economic growth under shocks: Path dependencies and stabilization,” H. Liljenstroem and U. Svedin (eds.), Micro-Meso-Macro: Addressing Complex Systems Couplings, World Scientific, London (2005), pp. 289–302.

    Chapter  Google Scholar 

  27. V. I. Norkin, “Assessment of the risk of catastrophic decrease in stochastic models of economic growth,” in: Teoriya Optym. Rishen’, V. M. Glushkov Inst. of Cybernetics, NAS Ukr. (2007), pp. 41–50.

  28. A. Manne, “Linear programming and sequential decisions,” Manage. Sci., 6, 259–267 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  29. C. Derman, “Markovian sequential decision processes,” in: Stochastic Processes in Mathematical Physics and Engineering. Proc. of Symposia in Applied Mathematics, 16, AMS, Providence, 281–289 (1964).

  30. Yu. M. Ermoliev and V. I. Norkin, “Solution of nonconvex nonsmooth stochastic optimization problems,” Cybern. Syst. Analysis, 39, No. 5, 701–715 (2003).

    Article  MATH  Google Scholar 

  31. P. Brémaud, Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, New York (1999).

    MATH  Google Scholar 

  32. Y. Ermoliev, A. Gaivoronski, and M. Makowski, “Robust design of networks under risks,” in: K. Marti et al. (eds.), Coping with Uncertainty. Lecture Notes in Economics and Mathematical Systems, 633, Springer-Verlag, Berlin–Heidelberg (2010), pp. 101–137.

  33. D. M. Becker and A. A. Gaivoronski, “Stochastic optimization on social networks with application to service pricing,” Comput. Manag. Sci. (2013), pp. 1–32 (Published online, DOI 10.1007/s10287–013–0201–7).

  34. H. K. Jevne, P. C. Haddow, and A. A. Gaivoronski, “Evolving constrained mean-VaR efficient frontiers,” Evolutionary Computation (CEC), WCCI 2012 IEEE World Congress on Computational Intelligence, Australia, Brisbane, IEEE (2012), pp. 1–8.

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Authors and Affiliations

  1. Norwegian University of Science and Technology, Trondheim, Norway

    O. O. Haivoronskyy

  2. International Institute of Applied Systems Analysis, Laxenburg, Austria

    Yu. M. Ermoliev

  3. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    P. S. Knopov & V. I. Norkin

Authors
  1. O. O. Haivoronskyy
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  2. Yu. M. Ermoliev
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  3. P. S. Knopov
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  4. V. I. Norkin
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Additional information

1The study was financially supported by the grant of the Norwegian Centre for International Cooperation in Education (SIU), Norwegian–Ukrainian Project CPEALA-2012/10052.

Translated from Kibernetika i Sistemnyi Analiz, No. 1, January–February, 2015, pp. 97–110.

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Haivoronskyy, O.O., Ermoliev, Y.M., Knopov, P.S. et al. Mathematical Modeling of Distributed Catastrophic and Terrorist Risks1 . Cybern Syst Anal 51, 85–95 (2015). https://doi.org/10.1007/s10559-015-9700-6

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  • DOI: https://doi.org/10.1007/s10559-015-9700-6

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