Skip to main content
SpringerLink
Log in

Dynamical insurance models with investment: Constrained singular problems for integrodifferential equations

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Previous and new results are used to compare two mathematical insurance models with identical insurance company strategies in a financial market, namely, when the entire current surplus or its constant fraction is invested in risky assets (stocks), while the rest of the surplus is invested in a risk-free asset (bank account). Model I is the classical Cramér–Lundberg risk model with an exponential claim size distribution. Model II is a modification of the classical risk model (risk process with stochastic premiums) with exponential distributions of claim and premium sizes. For the survival probability of an insurance company over infinite time (as a function of its initial surplus), there arise singular problems for second-order linear integrodifferential equations (IDEs) defined on a semiinfinite interval and having nonintegrable singularities at zero: model I leads to a singular constrained initial value problem for an IDE with a Volterra integral operator, while II model leads to a more complicated nonlocal constrained problem for an IDE with a non-Volterra integral operator. A brief overview of previous results for these two problems depending on several positive parameters is given, and new results are presented. Additional results are concerned with the formulation, analysis, and numerical study of “degenerate” problems for both models, i.e., problems in which some of the IDE parameters vanish; moreover, passages to the limit with respect to the parameters through which we proceed from the original problems to the degenerate ones are singular for small and/or large argument values. Such problems are of mathematical and practical interest in themselves. Along with insurance models without investment, they describe the case of surplus completely invested in risk-free assets, as well as some noninsurance models of surplus dynamics, for example, charity-type models.

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. J. Grandell, Aspects of Risk Theory (Springer-Verlag, Berlin, 1991).

    Book  MATH  Google Scholar 

  2. V. Yu. Korolev, V. E. Bening, and S. Ya. Shorgin, Mathematical Foundations of Risk Theory (Fizmatlit, Moscow, 2007) [in Russian].

    MATH  Google Scholar 

  3. N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt, Actuarial Mathematics (Soc. of Actuaries, Itasca, IL, 1986; Yanus-K, Moscow, 2001).

    MATH  Google Scholar 

  4. S. Asmussen and H. Albrecher, Ruin Probabilities (World Scientific, Singapore, 2010).

    MATH  Google Scholar 

  5. T. A. Belkina, N. B. Konyukhova, and A. O. Kurkina, “Optimal investment problem in dynamic insurance models: II. Cramér–Lundberg model with exponential claim size distribution,” Obozr. Prikl. Promyshl. Mat. (Sekts. Finans. Strakh. Mat.) 17 (1), 3–24 (2010).

    Google Scholar 

  6. T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Singular initial value problem for linear integrodifferential equation arising in insurance models,” Int. Sci. J. Spectral Evolution Probl. 21 (1), 40–54 (2011).

    Google Scholar 

  7. T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution,” Comput. Math. Math. Phys. 52 (10), 1384–1416 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. T. Belkina, N. Konyukhova, and S. Kurochkin, “Singular problems for integro-differential equations in dynamic insurance models,” in Differential and Difference Equations with Applications (Springer, Berlin, 2013), Vol. 47, pp. 27–44.

    Chapter  Google Scholar 

  9. T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Singular initial and boundary value problems for integro- differential equations in dynamic insurance models with investment,” Sovrem. Mat. Fundam. Napravl. 53, 5–29 (2014).

    Google Scholar 

  10. T. A. Belkina, “Sufficiency theorems for survival probability in dynamic insurance models with investment,” in Analysis and Modeling of Economic Processes, Ed. by V. Z. Belen’kii (Tsentr. Ekonomiko-Mat. Inst. Ross. Akad. Nauk, Moscow, 2011), Vol. 8, pp. 61–74 (http://wwwcemirssiru/publication/books/).

    Google Scholar 

  11. T. A. Belkina, “Risky investment for insurers and sufficiency theorems for the survival probability,” Markov Processes Related Fields 20, 505–525 (2014).

    MathSciNet  MATH  Google Scholar 

  12. J. Paulsen and H. K. Gjessing, “Ruin theory with stochastic return on investments,” Adv. Appl. Probab. 29 (4), 965–985 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Frolova, Yu. Kabanov, and S. Pergamenshchikov, “In the insurance business risky investments are dangerous,” Finance Stoch. 6 (2), 227–235 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Pergamenshchikov and O. Zeitouny, “Ruin probability in the presence of risky investments,” Stochastic Process. Appl. 116 (2), 267–278 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. V. Boikov, Candidate’s Dissertation in Mathematics and Physics (Steklov Mathematical Inst., Russ. Acad. Sci., Moscow, 2003).

    Google Scholar 

  16. A. Ramos, PhD Thesis (Univ. Carlos III de Madrid, Madrid, 2009) (http://e-archivouc3mes/haudle/ 10016/5631).

    Google Scholar 

  17. L. Bachelier, “Theorie de la speculation,” Ann. Sci. Ecole Norm. Super. 17, 21–86 (1900).

    MathSciNet  MATH  Google Scholar 

  18. T. A. Belkina, N. B. Konyukhova, and A. O. Kurkina, “Optimal investment problem in dynamic insurance models: I. Investment strategies and ruin probability,” Obozr. Prikl. Promyshl. Mat. (Sekts. Finans. Strakh. Mat.) 16 (6), 961–981 (2009).

    Google Scholar 

  19. R. Bellman, Stability Theory of Differential Equations (McGraw-Hill, New York, 1953; Inostrannaya Literatura, Moscow, 1954).

    Google Scholar 

  20. M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Nauka, Moscow, 1983; Springer, Berlin, 1993).

    Book  MATH  Google Scholar 

  21. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955; Inostrannaya Literatura, Moscow, 1958).

    MATH  Google Scholar 

  22. W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York, 1965; Mir, Moscow, 1968).

    MATH  Google Scholar 

  23. E. Kamke, Differentialgleichungen: Lösungmethoden und Lösungen: I. Gewöhnlishe Differentialgleishungen (Akademie-Verlag, Leipzig, 1959; Nauka, Moscow, 1971).

    Google Scholar 

  24. E. S. Birger and N. B. Lyalikova (Konyukhova), “Discovery of the solutions of certain systems of differential equations with a given condition at infinity I,” USSR Comput. Math. Math. Phys. 5 (6), 1–17 (1965); “On finding the solutions for a given condition at infinity of certain systems of ordinary differential equations II,” 6 (3), 47–57 (1966).

    Article  MATH  Google Scholar 

  25. N. B. Konyukhova, “Singular Cauchy problems for systems of ordinary differential equations,” USSR Comput. Math. Math. Phys. 23 (3), 72–82 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  26. T. A. Belkina, C. Hipp, S. Luo, and M. Taksar, “Optimal constrained investment in the Cramér–Lundberg model,” Scand. Actuarial J., No. 5, 383–404 (2014).

    Article  MathSciNet  Google Scholar 

  27. N. B. Konyukhova, “Singular Cauchy problems for singularly perturbed systems of nonlinear ordinary differential equations,” I: Differ. Equations 32 (1), 54–63 (1996), II: Differ. Equations 32 (4), 491–500 (1996).

    MathSciNet  MATH  Google Scholar 

  28. Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1965).

  29. H. Gingold and S. Rosenblat, “Differential equations with moving singularities,” SIAM J. Math. Anal. 7 (6), 942–957 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  30. A. V. Boikov, “The Cramér–Lundberg model with stochastic premium process,” Theory Probab. Appl. 47, 489–493 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  31. N. Zinchenko and A. Andrusiv, “Risk processes with stochastic premiums,” Theory Stoch. Processes 14 (3–4), 189–208 (2008).

    MathSciNet  MATH  Google Scholar 

  32. G. Temnov, “Risk models with stochastic premium and ruin probability estimation,” J. Math. Sci. 196 (1), 84–96 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatulina, Introduction to the Theory of Functional Differential Equations (Nauka, Moscow, 1991) [in Russian].

    Google Scholar 

  34. N. B. Konyukhova, “Singular Cauchy problems for some systems of nonlinear functional-differential equations,” Differ. Equations 31 (8), 1286–1293 (1995).

    MathSciNet  MATH  Google Scholar 

  35. N. B. Konyukhova, “Singular problems for systems of nonlinear functional-differential equations,” Int. Sci. J. Spectral Evolution Probl. 20, 199–214 (2010).

    Google Scholar 

  36. A. A. Abramov, “On the transfer of the condition of boundedness for some systems of ordinary linear differential equations,” USSR Comput. Math. Math. Phys. 1 (4), 875–881 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  37. A. A. Abramov, K. Balla, and N. B. Konyukhova, “Transfer of boundary conditions from singular points for systems of ordinary differential equations,” in Reports on Applied Mathematics (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1981) [in Russian].

    Google Scholar 

  38. A. A. Abramov, N. B. Konyukhova, and K. Balla, “Stable initial manifolds and singular boundary value problems for systems of ordinary differential equations,” Comput. Math. Banach Center Publ. 13, 319–351 (1984).

    MathSciNet  MATH  Google Scholar 

  39. A. A. Abramov and N. B. Konyukhova, “Transfer of admissible boundary conditions from a singular point for systems of linear ordinary differential equations,” in Reports on Applied Mathematics (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1985) [in Russian].

    Google Scholar 

  40. A. A. Abramov and N. B. Konyukhova, “Transfer of admissible boundary conditions from a singular point for systems of linear ordinary differential equations,” Sov. J. Numer. Anal. Math. Model. 1 (4), 245–265 (1986).

    MathSciNet  MATH  Google Scholar 

  41. A. A. Abramov, V. V. Ditkin, N. B. Konyukhova, B. S. Pariiskii, and V. I. Ul’yanova, “Evaluation of the eigenvalues and eigenfunctions of ordinary differential equations with singularities,” USSR Comput. Math. Math. Phys. 20 (5), 63–81 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. A. Abramov, “On the transfer of boundary conditions for systems of ordinary linear differential equations (a variant of the dispersive method),” USSR Comput. Math. Math. Phys. 1 (1), 617–622 (1961).

    MATH  Google Scholar 

  43. N. S. Bakhvalov, Numerical Methods: Analysis, Algebra, Ordinary Differential Equations (Nauka, Moscow, 1973; Mir, Moscow, 1977).

    Google Scholar 

  44. V. Kalashnikov and R. Norberg, “Power tailed ruin probabilities in the presence of risky investments,” Stoch. Proc. Appl. 98, 211–228 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Laubis and J.-E. Lin, “Optimal investment allocation in a jump diffusion risk model with investment: A numerical analysis of several examples,” Proceedings of the 43rd Actuarial Research Conference, 2008 (http://wwwsoaorg/news-and-publications/publications/proceedings/arch/arch-2009-isslaspx).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Central Economics and Mathematics Institute, Russian Academy of Sciences, Nakhimovskii pr. 47, Moscow, 117418, Russia

    T. A. Belkina

  2. International Laboratory of Quantitative Finance, National Research University Higher School of Economics, ul. Shabolovka 31B, Moscow, 115162, Russia

    T. A. Belkina

  3. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    N. B. Konyukhova & S. V. Kurochkin

Authors
  1. T. A. Belkina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. B. Konyukhova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. V. Kurochkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. A. Belkina.

Additional information

Original Russian Text © T.A. Belkina, N.B. Konyukhova, S.V. Kurochkin, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 1, pp. 47–98.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belkina, T.A., Konyukhova, N.B. & Kurochkin, S.V. Dynamical insurance models with investment: Constrained singular problems for integrodifferential equations. Comput. Math. and Math. Phys. 56, 43–92 (2016). https://doi.org/10.1134/S0965542516010073

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542516010073

Keywords

Search

Navigation