Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Modern Mathematics and Mechanics

Part of the book series: Understanding Complex Systems ((UCS))

  • 1123 Accesses

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.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. 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. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  4. Black, F., Karasinski, P.: Bond and option pricing when short rates are lognormal. Financ. Anal. J. 47, 52–59 (1991)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  6. Buldygin, V.V., Pavlenkov, V.V.: Karamata theorem for regularly log-periodic functions. Ukr. Math. J. 64, 1635–1657 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  13. Buldygin, V.V., Indlekofer, K.-H., Klesov, O.I., Steinebach, J.G.: Pseudo-Regularly Varying Functions and Generalized Renewal Processes. Springer, Berlin (2018)

    Book  Google Scholar 

  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. Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)

    Article  MathSciNet  Google Scholar 

  16. D’Anna, A., Maio, A., Moauro, V.: Global stability properties by means of limiting equations. Nonlinear Anal. 4(2), 407–410 (1980)

    Article  MathSciNet  Google Scholar 

  17. Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)

    Book  Google Scholar 

  18. Hull, J., White, A.: Pricing interest-rate derivative securities. Rev. Financ. Stud. 3, 573–592 (1990)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  20. Kersting, G.: Asymptotic properties of solutions of multidimensional stochastic differential equations. Probab.Theory Relat. Fields 88, 187–211 (1982)

    MathSciNet  MATH  Google Scholar 

  21. Klesov, O.I.: Limit Theorems for Multi-Indexed Sums of Random Variables. Springer, Berlin (2014)

    Book  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Woodhead Publishing, Cambridge (2010)

    Google Scholar 

  25. Mitsui, T.: Stability analysis of numerical solution of stochastic differential equations. Res. Inst. Math. Sci. Kyoto Univ. 850, 124–138 (1995)

    MATH  Google Scholar 

  26. Øksendal, B.K.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)

    Book  Google Scholar 

  27. Rendleman, R., Bartter, B.: The pricing of options on debt securities. J. Financ. Quant. Anal. 15, 11–24 (1980)

    Article  Google Scholar 

  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. Seneta, E.: Regularly Varying Functions. Springer, Berlin (1976)

    Book  Google Scholar 

  30. Strauss, A., Yorke, J.A.: On asymptotically autonomous differential equations. Math. Syst. Theory 1, 175–182 (1967)

    Article  MathSciNet  Google Scholar 

  31. Taniguchi, T.: On sufficient conditions for nonexplosion of solutions to stochastic differential equations, J. Math. Anal. Appl. 153, 549–561 (1990)

    Article  MathSciNet  Google Scholar 

  32. Tymoshenko, O.A.: Generalization of asymptotic behavior of non-autonomous stochastic differential equations, Naukovi Visti NTUU KPI 4, 100–106 (2016)

    Article  Google Scholar 

  33. Vašiček, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

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

    Oleg I. Klesov & Olena A. Tymoshenko

Authors
  1. Oleg I. Klesov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olena A. Tymoshenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleg I. Klesov .

Editor information

Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    Michael Z. Zgurovsky

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Klesov, O.I., Tymoshenko, O.A. (2019). Almost Sure Asymptotic Properties of Solutions of a Class of Non-homogeneous Stochastic Differential Equations. In: Sadovnichiy, V., Zgurovsky, M. (eds) Modern Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-96755-4_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-96755-4_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96754-7

  • Online ISBN: 978-3-319-96755-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation