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Almost Sure Asymptotic Properties of Solutions of a Class of Non-homogeneous Stochastic Differential Equations

  • Oleg I. KlesovEmail author
  • Olena A. Tymoshenko
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

We study non-homogeneous stochastic differential equation with separation of stochastic and deterministic variables. We express the asymptotic behavior of solutions of such equations in terms of that for the corresponding ordinary differential equation. The general results are discussed for some particular equations, mainly in the field of mathematics of finance.

Notes

Acknowledgements

Supported by the grants from Ministry of Education and Science of Ukraine (projects N 2105 ϕ and M/68-2018).

References

  1. 1.
    Appleby, A.D., Cheng, J.: On the asymptotic stability of a class of perturbed ordinary differential equations with weak asymptotic mean reversion. Electronic J. Qualitative Theory Differ. Equ. Proc. 9th Coll. 1, 1–36 (2011)Google Scholar
  2. 2.
    Appleby, A.D., Cheng, J.: Rodkina, A.: Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation. Discrete Contin. Dyn. Syst. Suppl. 2011, 79–90 (2011)Google Scholar
  3. 3.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  4. 4.
    Black, F., Karasinski, P.: Bond and option pricing when short rates are lognormal. Financ. Anal. J. 47, 52–59 (1991)CrossRefGoogle Scholar
  5. 5.
    Black, F., Derman, E., Toy, W.: A one-factor model of interest rates and its application to treasury bond options. Financ. Anal. J. 46, 24–32 (1990)CrossRefGoogle Scholar
  6. 6.
    Buldygin, V.V., Pavlenkov, V.V.: Karamata theorem for regularly log-periodic functions. Ukr. Math. J. 64, 1635–1657 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Buldygin, V.V., Tymoshenko, O.A.: On the exact order of growth of solutions of stochastic differential equations with time-dependent coefficients. Theory Stoch. Process. 16, 12–22 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Buldygin, V.V., Klesov, O.I., Steinebach, J.G.: On some properties of asymptotically quasi-inverse functions and their applications. I. Theory Probab. Math. Stat. 70, 9–25 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Buldygin, V.V., Klesov, O.I., Steinebach, J.G.: On factorization representations for Avakumović–Karamata functions with nondegenerate groups of regular points. Anal. Math. 30, 161–192 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Buldygin, V.V., Klesov, O.I., Steinebach, J.G.: The PRV property of functions and the asymptotic behavior of solutions of stochastic differential equations. Theory Probab. Math. Stat. 72, 63–78 (2004)Google Scholar
  11. 11.
    Buldygin, V.V., Klesov, O.I., Steinebach, J.G.: On some properties of asymptotically quasi-inverse functions and their applications. II. Theory Probab. Math. Stat. 71, 63–78 (2004)zbMATHGoogle Scholar
  12. 12.
    Buldygin, V.V., Klesov, O.I., Steinebach, J.G., Tymoshenko, O.A.: On the φ-asymptotic behavior of solutions of stochastic differential equations. Theory Stoch. Process. 14, 11–30 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Buldygin, V.V., Indlekofer, K.-H., Klesov, O.I., Steinebach, J.G.: Pseudo-Regularly Varying Functions and Generalized Renewal Processes. Springer, Berlin (2018)CrossRefGoogle Scholar
  14. 14.
    Chen, L.: Stochastic mean and stochastic volatility – a three-factor model of the term structure of interest rates and its application to the pricing of interest rate derivatives. Financ. Mark. Inst. Instrum. 5, 1–88 (1996)Google Scholar
  15. 15.
    Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    D’Anna, A., Maio, A., Moauro, V.: Global stability properties by means of limiting equations. Nonlinear Anal. 4(2), 407–410 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)CrossRefGoogle Scholar
  18. 18.
    Hull, J., White, A.: Pricing interest-rate derivative securities. Rev. Financ. Stud. 3, 573–592 (1990)CrossRefGoogle Scholar
  19. 19.
    Keller, G., Kersting, G., Rösler, U.: On the asymptotic behavior of solutions of stochastic differential equations. Z. Wahrsch. Geb. 68(2), 163–184 (1984)CrossRefGoogle Scholar
  20. 20.
    Kersting, G.: Asymptotic properties of solutions of multidimensional stochastic differential equations. Probab.Theory Relat. Fields 88, 187–211 (1982)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Klesov, O.I.: Limit Theorems for Multi-Indexed Sums of Random Variables. Springer, Berlin (2014)CrossRefGoogle Scholar
  22. 22.
    Klesov, O.I., Tymoshenko, O.A.: Unbounded solutions of stochastic differential equations with time-dependent coefficients. Ann. Univ. Sci. Budapest Sect. Comput. 41, 25–35 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Klesov, O.I., Siren ’ka, I.I., Tymoshenko, O.A.: Strong law of large numbers for solutions of non-autonomous stochastic differential equations. Naukovi Visti NTUU KPI 4, 100–106 (2017)Google Scholar
  24. 24.
    Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Woodhead Publishing, Cambridge (2010)Google Scholar
  25. 25.
    Mitsui, T.: Stability analysis of numerical solution of stochastic differential equations. Res. Inst. Math. Sci. Kyoto Univ. 850, 124–138 (1995)zbMATHGoogle Scholar
  26. 26.
    Øksendal, B.K.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)CrossRefGoogle Scholar
  27. 27.
    Rendleman, R., Bartter, B.: The pricing of options on debt securities. J. Financ. Quant. Anal. 15, 11–24 (1980)CrossRefGoogle Scholar
  28. 28.
    Samoı̆lenko, A.M., Stanzhytskyi, O.M.: Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations. World Scientific Publishing, Hackensack, NJ (2011)Google Scholar
  29. 29.
    Seneta, E.: Regularly Varying Functions. Springer, Berlin (1976)CrossRefGoogle Scholar
  30. 30.
    Strauss, A., Yorke, J.A.: On asymptotically autonomous differential equations. Math. Syst. Theory 1, 175–182 (1967)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Taniguchi, T.: On sufficient conditions for nonexplosion of solutions to stochastic differential equations, J. Math. Anal. Appl. 153, 549–561 (1990)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tymoshenko, O.A.: Generalization of asymptotic behavior of non-autonomous stochastic differential equations, Naukovi Visti NTUU KPI 4, 100–106 (2016)CrossRefGoogle Scholar
  33. 33.
    Vašiček, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”Department of Mathematical Analysis and Probability TheoryKyivUkraine

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