Advertisement

Integral Equations, Quasi-Monte Carlo Methods and Risk Modeling

  • Michael PreischlEmail author
  • Stefan Thonhauser
  • Robert F. Tichy
Chapter

Abstract

We survey a QMC approach to integral equations and develop some new applications to risk modeling. In particular, a rigorous error bound derived from Koksma-Hlawka type inequalities is achieved for certain expectations related to the probability of ruin in Markovian models. The method is based on a new concept of isotropic discrepancy and its applications to numerical integration. The theoretical results are complemented by numerical examples and computations.

Notes

Acknowledgements

The authors are supported by the Austrian Science Fund (FWF) Project F5510 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”).

References

  1. 1.
    Aistleitner, C., Dick, J.: Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality. Acta Arith. 167(2), 143–171 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aistleitner, C., Pausinger, F., Svane, A.M., Tichy, R.F.: On functions of bounded variation. Math. Proc. Camb. Philos. Soc. 162(3), 405–418 (2017)CrossRefGoogle Scholar
  3. 3.
    Albrecher, H., Kainhofer, R.: Risk theory with a nonlinear dividend barrier. Computing 68(4), 289–311 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Albrecher, H., Thonhauser, S.: Optimality results for dividend problems in insurance. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 103(2), 295–320 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Albrecher, H., Constantinescu, C., Pirsic, G., Regensburger, G., Rosenkranz, M.: An algebraic operator approach to the analysis of Gerber–Shiu functions. Insur. Math. Econ. 46(1), 42–51 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Asmussen, S., Albrecher, H.: Ruin Probabilities, 2nd edn. World Scientific, River Edge (2010)Google Scholar
  7. 7.
    Atkinson, K.E.: The numerical solution of Fredholm integral equations of the second kind. SIAM J. Numer. Anal. 4(3), 337–348 (1967)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brandolini, L., Colzani, L., Gigante, G., Travaglini, G.: A Koksma–Hlawka inequality for simplices. In: Trends in Harmonic Analysis, pp. 33–46. Springer, New York (2013)CrossRefGoogle Scholar
  9. 9.
    Brandolini, L., Colzani, L., Gigante, G., Travaglini, G.: On the Koksma–Hlawka inequality. J. Complex. 29(2), 158 – 172 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal. 21(6), 1132–1145 (1984)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brunner, H.: Implicitly linear collocation methods for nonlinear Volterra equations. Appl. Numer. Math. 9(3), 235–247 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Davis, M.H.A.: Markov Models and Optimization. Chapman and Hall, London (1993)CrossRefGoogle Scholar
  13. 13.
    Dick, J., Kritzer, P., Kuo, F.Y., Sloan, I.H.: Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels. J. Complex. 23(4), 752–772 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)Google Scholar
  15. 15.
    Edelsbrunner, H., Pausinger, F.: Approximation and convergence of the intrinsic volume. Adv. Math. 287, 674–703 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gerber, H.U., Shiu, E.S.W.: On the time value of ruin. N. Am. Actuar. J. 2(1), 48–78 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gerber, H.U., Shiu, E.S.W.: The time value of ruin in a Sparre Andersen model. N. Am. Actuar. J. 9(2), 49–84 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Götz, M.: Discrepancy and the error in integration. Monatsh. Math. 136(2), 99–121 (2002)Google Scholar
  19. 19.
    Grigorian, A.: Ordinary Differential Equation. Lecture Notes (2009). Available at https://www.math.uni-bielefeld.de/~grigor/odelec2009.pdf
  20. 20.
    Harman, G.: Variations on the Koksma-Hlawka inequality. Unif. Distrib. Theory 5(1), 65–78 (2010)Google Scholar
  21. 21.
    Hlawka, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl. 54(1), 325–333 (1961)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hua, L.K., Wang, Y.: Applications of Number theory to Numerical Analysis. Springer, Berlin; Kexue Chubanshe (Science Press), Beijing (1981). Translated from the ChineseGoogle Scholar
  23. 23.
    Ikebe, Y.: The Galerkin method for the numerical solution of Fredholm integral equations of the second kind. SIAM Rev. 14(3), 465–491 (1972)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley, New York (1974)Google Scholar
  25. 25.
    Kumar, S., Sloan, I.H.: A new collocation-type method for Hammerstein integral equations. Math. Comput. 48(178), 585–593 (1987)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. J. Complex. 19(3), 301–320 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Leobacher, G., Pillichshammer, F.: Introduction to Quasi-Monte Carlo Integration and Applications. Compact Textbook in Mathematics. Birkhäuser/Springer, Cham (2014)zbMATHGoogle Scholar
  28. 28.
    Lin, X.S., Willmot, G.E., Drekic, S.: The classical risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function. Insur. Math. Econ. 33(3), 551–566 (2003)Google Scholar
  29. 29.
    Lundberg, F.: Approximerad framställning av sannolikhetsfunktionen. Aterförsäkring av kollektivrisker. Akad. Afhandling. Almqvist o. Wiksell, Uppsala (1903)Google Scholar
  30. 30.
    Makroglou, A.: Numerical solution of some second order integro-differential equations arising in ruin theory. In: Proceedings of the third Conference in Actuarial Science and Finance, pp. 2–5 (2004)Google Scholar
  31. 31.
    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume II: Standard Information for Functionals. EMS Tracts in Mathematics, vol. 12. European Mathematical Society (EMS), Zürich (2010)Google Scholar
  32. 32.
    Owen, A.B.: Multidimensional variation for Quasi-Monte Carlo. In: Contemporary Multivariate Analysis and Design of Experiments. Series in Biostatistics, vol. 2, pp. 49–74. World Scientific, Hackensack (2005)CrossRefGoogle Scholar
  33. 33.
    Pausinger, F., Svane, A.M.: A Koksma-Hlawka inequality for general discrepancy systems. J. Complex. 31(6), 773–797 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic processes for insurance and finance. Wiley, Chichester (1999)CrossRefGoogle Scholar
  35. 35.
    Sloan, I.H.: A quadrature-based approach to improving the collocation method. Numer. Math. 54(1), 41–56 (1988)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Sloan, I.H., Lyness, J.N.: The representation of lattice quadrature rules as multiple sums. Math. Comput. 52(185), 81–94 (1989)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sloan, I.H., Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Complex. 17(4), 697–721 (2001)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Stoer, J., Bulirsch, R.: Numerische Mathematik. 2, 4th edn. Springer-Lehrbuch. [Springer Textbook]. Springer, Berlin (2000)Google Scholar
  39. 39.
    Tichy, R.F.: Über eine zahlentheoretische Methode zur numerischen Integration und zur Behandlung von Integralgleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 193(4–7), 329–358 (1984)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Twomey, S.: On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J. Assoc. Comput. Mach. 10(1), 97–101 (1963)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Michael Preischl
    • 1
    Email author
  • Stefan Thonhauser
    • 1
  • Robert F. Tichy
    • 1
  1. 1.Institute of Analysis and Number TheoryGraz University of TechnologyGrazAustria

Personalised recommendations