Skip to main content
SpringerLink
Log in

Optimization methods for compound poisson risk processes

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

In this article we introduce the notion of the family of compound Poisson risk processes that depend on a finite-dimensional parameter; we describe examples of such families that arise during formalization of surplus, quota, and quota-surplus reinsurance contracts; the optimization problem is stated for the growth rate of insurance company capital subject to a given constraint on the probability of ruin; numerical methods for solving this problem by Lagrange function saddle-point techniques are reviewed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Émbrechts and K. Kluppelberg, “Some aspects of insurance mathematics, “ Teor. Veroyatn. Primen.,38, No. 2, 374–417(1993).

    MATH  Google Scholar 

  2. R.E. Beard, T. Pentikainen, and E. Pesonen, Risk Theory, Methuen, London (1969).

    MATH  Google Scholar 

  3. W.-R. Heilmann, Fundamentals of Risk Theory, Verlag Versicherungswirtschaft, Karlsruhe (1988).

    MATH  Google Scholar 

  4. M. Feilmeier and J. Bertram, Anvendung numerischer Methoden in der Risikotheorie, Verlag Versicherungswirtschaft, Karlsruhe (1987).

    Google Scholar 

  5. C. Hipp and R. Michel, Risikotheorie: stochastische Modelle und statistische Methoden Verlag Versicherungswirtschaft, Karlsruhe (1990).

    MATH  Google Scholar 

  6. A. N. Nakonechnyi, “Optimization of risk processes, “ Kibern. Sist. Anal., No. 5, 42–48 (1996).

  7. M. Bazara and C. Shetty, Nonlinear Programming: Theory and Algorithms [Russian translation], Nauka, Moscow (1982).

    Google Scholar 

  8. D. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods [in Russian], Radio i Svyaz', Moscow (1987).

    MATH  Google Scholar 

  9. E. G. Gol'shtein and N. V. Tret'yakov, Modified Lagrange Functions [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  10. Yu. M. Ermol'ev, Stochastic Programming Methods [in Russian], Nauka, Moscow (1976).

    MATH  Google Scholar 

  11. Yu. M. Ermoliev and R. J.-B. Werts, Numerical Techniques for Stochastic Optimization, Springer-Verlag, Berlin (1988).

    MATH  Google Scholar 

  12. A. N. Nakonechnyi, “Stochastic gradient processes: a survey of convergence theory using Lyapunov second method,” Kibern. Sist. Anal., No. 1, 46–61 (1995).

  13. A. N. Nakonechnyi, “Monte Carlo estimation of the probability of ruin in a compound Poisson model of risk theory,” Kibern. Sist. Anal., No. 6, 160–163 (1995).

  14. I. N. Kovalenko aind A. N. Nakonechnyi, Approximate Calculation and Optimization of Reliability [in Russian], Naukova Dumka, Kiev (1989).

    MATH  Google Scholar 

  15. A. N. Nakonechnyi, “Extremal problems with rare events. I,” Kibernetika, No. 5, 55–58 (1990).

  16. I. N. Kovalenko and N. Yu. Kuznetsov, Calculation Methods for Highly Reliable Systems [in Russian], Radio i Svyaz', Moscow (1988).

    Google Scholar 

  17. Ph. Gill, W. Murray, and M. Wright, Practical Optimization [Russian translation], Mir, Moscow (1985).

    Google Scholar 

  18. J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Minimization and Nonlinear Equations [Russian translation], Mir, Moscow (1988).

    Google Scholar 

  19. A. N. Nakonechnyi, “Iterative processes: a survey of convergence theory using Lyapunov second method, “ Kibern. Sist. Anal., No. 4, 66–85 (1994).

Download references

Authors
  1. G. I. Lyubchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. N. Nakonechnyi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research described in this publication was made possible in part by Grant No. UAL000 from the International Science Foundation.

Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 87–96, March–April, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lyubchenko, G.I., Nakonechnyi, A.N. Optimization methods for compound poisson risk processes. Cybern Syst Anal 34, 230–237 (1998). https://doi.org/10.1007/BF02742072

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02742072

Keywords

Search

Navigation