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Asymptotic Behavior of Solutions of Stochastic Differential Equations

  • Valeriĭ V. Buldygin
  • Karl-Heinz Indlekofer
  • Oleg I. Klesov
  • Josef G. Steinebach
Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 91)

Abstract

This chapter aims at finding nonrandom approximations (a precise definition is given below) of solutions of a general class of stochastic differential equations. We follow the setting by Gihman and Skorohod [149], however the results below are more general.

References

  1. 15.
    J.A.D. Appleby, G. Berkolaiko, and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic homogeneous difference equations, in Discrete dynamics and difference equations, World Sci. Publ., Hackensack, NJ, 2010, pp. 155–162.zbMATHGoogle Scholar
  2. 16.
    J.A.D. Appleby, J.P. Gleeson, and A. Rodkina, On asymptotic stability and instability with respect to a fading stochastic perturbation, Applicable Analysis 8 (2009), no. 4, 579–603.MathSciNetCrossRefGoogle Scholar
  3. 17.
    J.A.D. Appleby and H. Wu, Solutions of stochastic differential equations obeying the law of the iterated logarithm, with applications to financial markets, Electronic J. Probab. 14 (2009), no. 33, 912–959.MathSciNetCrossRefGoogle Scholar
  4. 25.
    A. Barchielli and A. Paganoni, On the asymptotic behaviour of some stochastic differential equations for quantum states, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 2, 223–243.MathSciNetCrossRefGoogle Scholar
  5. 29.
    G. Berkolaiko and A. Rodkina, Almost sure convergence of solutions to nonhomogeneous stochastic difference equation, J. Difference Equ. Appl. 12 (2006), no. 6, 535–553.MathSciNetCrossRefGoogle Scholar
  6. 52.
    D. Brigo and F. Mercurio, Interest Rate Models. Theory and Practice, second edition, Springer-Verlag, Berlin, 2006.Google Scholar
  7. 63.
    V.V. Buldygin, O.I. Klesov, and J.G. Steinebach, The PRV property of functions and the asymptotic behavior of solutions of stochastic differential equations, Teor. Imov. Mat. Stat. 72 (2005), 10–23 (Ukrainian); English transl. in Theory Probab. Math. Statist. 72 (2006), 11–25.Google Scholar
  8. 64.
    V.V. Buldygin, O.I. Klesov, and J.G. Steinebach, PRV property and the asymptotic behaviour of solutions of stochastic differential equations, Theory Stoch. Process. 11 (27)(2005), no. 3–4, 42–57.Google Scholar
  9. 65.
    V.V. Buldygin, O.I. Klesov, and J.G. Steinebach, On some extensions of Karamata’s theory and their applications, Publ. Inst. Math. (Beograd) (N. S.) 80 (94) (2006), 59–96.MathSciNetCrossRefGoogle Scholar
  10. 67.
    V.V. Buldygin, O.I. Klesov, and J.G. Steinebach, PRV property and the φ-asymptotic behavior of solutions of stochastic differential equations, Liet. Mat. Rink. 77(2007), no. 4, 445–465 (Ukrainian); English transl. in Lithuanian Math. J. 77(2007), no. 4, 361–378.MathSciNetCrossRefGoogle Scholar
  11. 71.
    V.V. Buldygin, O.I. Klesov, J. Steinebach, and O.A. Timoshenko, On the φ-asymptotic behaviour of solutions of stochastic differential equations, Theory Stoch. Process. 14(2008), no. 1, 11–29.Google Scholar
  12. 74.
    V.V. Buldygin and O.A. Tymoshenko, On the asymptotic stability of stochastic differential equations, Naukovi Visti NTUU “KPI” 64 (2007), 126–129. (Ukrainian)Google Scholar
  13. 75.
    V.V. Buldygin and O.A. Tymoshenko, The exact order of growth of solutions of stochastic differential equations, Naukovi Visti NTUU “KPI” 64 (2008), 127–132. (Ukrainian)Google Scholar
  14. 76.
    V.V. Buldygin and O.A. Tymoshenko, On the exact order of growth of solutions of stochastic differential equations with time-dependent coefficients, Theory Stoch. Process. 16(2010), no. 2, 12–22.Google Scholar
  15. 78.
    R.A. Carmona and M.R. Tehranchi, Interest Rate Models: An Infnite Dimensional Stochastic Analysis Perspective, Springer-Verlag, Berlin, 2006.zbMATHGoogle Scholar
  16. 85.
    P.-L. Chow and R.Z. Khas’minskiı̆, Almost sure explosion of solutions to stochastic differential equations, Stochastic Process. Appl. 124 (2014), no. 1, 639–645.MathSciNetCrossRefGoogle Scholar
  17. 95.
    J.C. Cox, J.E. Ingersoll, and S.A. Ross, A theory of the term structure of interest rates, Econometrica53 (1985), 385–407.Google Scholar
  18. 100.
    A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993.zbMATHGoogle Scholar
  19. 103.
    S. Ditlevsen and A. Samson, Introduction to stochastic models in biology, in Stochastic Biomathematical Models with Applications to Neuronal Modeling, (eds. M. Bachar, J. Batzel, and S. Ditlevsen), Springer-Verlag, Berlin, 2013, pp. 3–35.zbMATHGoogle Scholar
  20. 134.
    A. Friedman, Limit behavior of solutions of stochastic differential equations, Trans. Amer. Math. Soc. 170 (1972), 359–384.MathSciNetCrossRefGoogle Scholar
  21. 135.
    A. Friedman, Stochastic Differential Equations And Applications, Academic Press, London–New York, 1975.zbMATHGoogle Scholar
  22. 136.
    A. Friedman and M. Pinsky, Behavior of solutions of linear stochastic differential systems, Trans. Amer. Math. Soc. 181 (1973), 1–22.MathSciNetCrossRefGoogle Scholar
  23. 149.
    I.I. Gihman and A.V. Skorohod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968 (Russian); English transl. Springer-Verlag, Berlin–Heidelberg–New York, 1972.Google Scholar
  24. 157.
    Q. Guo, X. Mao, and R. Yue, Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim. 54 (2016), no. 4, 1919–1933.MathSciNetCrossRefGoogle Scholar
  25. 177.
    D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006.CrossRefGoogle Scholar
  26. 189.
    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co.; Kodansha, Ltd., Amsterdam; Tokyo, 1989.Google Scholar
  27. 190.
    P. Imkeller, I. Pavlyuchenko, and T. Wetzel, First exit times for Lévy-driven diffusions with exponentially light jumps, Ann. Probab. 37 (2009), no. 2, 530–564.MathSciNetCrossRefGoogle Scholar
  28. 202.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd edition, Elsevier, Amsterdam, 2003.zbMATHGoogle Scholar
  29. 211.
    G. Keller, G. Kersting, and U. Rösler, On the asymptotic behaviour of solutions of stochastic differential equations, Z. Wahrscheinlichkeitstheorie verw. Gebiete 68 (1984), no. 2, 163–189.MathSciNetCrossRefGoogle Scholar
  30. 212.
    G. Keller, G. Kersting, and U. Rösler, The asymptotic behaviour of discrete time stochastic growth processes, Ann. Probab. 15 (1987), no. 1, 305–343.MathSciNetCrossRefGoogle Scholar
  31. 213.
    G. Keller, G. Kersting, and U. Rösler, On the asymptotic behaviour of first passage times for discussions, Probab. Theory Relat. Fields 77 (1988), 379–395.MathSciNetCrossRefGoogle Scholar
  32. 214.
    G. Kersting, Asymptotic properties of solutions of multi-dimensional stochastic differential Equations, Probab. Theory Relat. Fields 82 (1989), no. 2, 187–211.CrossRefGoogle Scholar
  33. 216.
    H. Kesten, Random difference equations and renewal theory for the product of random Matrices, Acta Mathematica 131 (1973), 207–248.MathSciNetCrossRefGoogle Scholar
  34. 217.
    R.Z. Khas’minskiı̆, Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Teor. Verojatnost. i Primenen. 12 (1967), no. 1, 167–172; English transl. in Theor. Probability Appl. 12 (1967), no. 1, 144–147.Google Scholar
  35. 218.
    R.Z. Khas’minskiı̆, Stochastic stability of differential equations, “Nauka”, Moscow, 1969 (Russian); English transl. Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, 1980.Google Scholar
  36. 221.
    F.C. Klebaner, Stochastic differential equations and generalized Gamma distributions, Ann. Probab. 7 (1989), no. 1, 178–188.MathSciNetCrossRefGoogle Scholar
  37. 231.
    O.I. Klesov and O.A. Timoshenko, Unbounded solutions of stochastic differential equations with time-dependent coefficients, Annales Univ. Sci. Budapest. Sect. Comp. 41 (2013), 25–35.MathSciNetzbMATHGoogle Scholar
  38. 239.
    M.A. Kouritzin and A.J. Heunes, A law of the iterated logarithm for stochastic processes defined by differential equations with small parameter, Ann. Probab. 22 (1994), no. 2, 659–579.MathSciNetCrossRefGoogle Scholar
  39. 240.
    F. Kozin, On almost sure asymptotic sample properties of diffusion processes defined by stochastic differential equation, J. Math. Kyoto Univ. 4 (1964/1965), 515–528.MathSciNetCrossRefGoogle Scholar
  40. 244.
    A.P. Krenevich, Asymptotic equivalence of solutions of linear Itô stochastic systems, Ukrain. Matem. Zh. 58 (2006), no. 10, 1368–1384 (Ukrainian); English transl. in Ukrain. Math. J. 58 (2006), no. 10, 1552–1569.Google Scholar
  41. 245.
    A.P. Krenevich, Asymptotic equivalence of solutions of non-linear Itô stochastic systems, Nonlinear oscillations 9 (2006), no. 2, 213–220. (Ukrainian)Google Scholar
  42. 249.
    H.R. Lerche, Boundary Crossing of Brownian Motion: Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis, Lecture Notes in Statistics 40, Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
  43. 255.
    S.Ya. Makhno, The law of iterated logarithm for solutions of stochastic differential equations, Ukrain. Matem. Zh. 48 (1996), no. 5, 650–655; English transl. in Ukrain. Math. J. 48 (1996), no. 5, 725–732.MathSciNetCrossRefGoogle Scholar
  44. 257.
    X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Woodhead Publishing, Oxford–Cambridge–New Delhi, 2010.Google Scholar
  45. 258.
    X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, Inc., New York, 1994.zbMATHGoogle Scholar
  46. 269.
    R. Metzler, G. Oshanin, and S. Redner (eds.), First-Passage Phenomena and Their Applications, World Scientific, Singapore, 2014.zbMATHGoogle Scholar
  47. 287.
    B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, 6th ed., Springer-Verlag, Berlin–Heidelberg–New York, 2003.Google Scholar
  48. 289.
    E.S. Palamarchuk, Asymptotic behavior of the solution to a linear stochastic differential equation and almost sure optimality for a controlled stochastic process, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 54 (2014), no. 1, 89–103; English transl. in Comput. Math. and Math. Phys. 54 (2014), no. 1, 83–96.MathSciNetCrossRefGoogle Scholar
  49. 295.
    N. Privault, An Elementary Introduction to Stochastic Interest Rate Modeling, second edition, World Scientific, New Jersey, 2012.CrossRefGoogle Scholar
  50. 307.
    G. Rosenkranz, Growth models with stochastic differential equations. An example from tumor immunology, Math. Biosciences 15 (1985), 175–186.MathSciNetCrossRefGoogle Scholar
  51. 314.
    A.M. Samoı̆lenko and O.M. Stanzhytskyi, Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations, Naukova Dumka, Kyiv, 2009 (Ukrainian); English transl. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.Google Scholar
  52. 315.
    A.M. Samoı̆lenko, O.M. Stanzhyts’kyi and I.H. Novak, On asymptotic equivalence of solutions of stochastic and ordinary equations, Ukrain. Matem. Zh. 63 (2011), no. 8, 1103–1127 (Ukrainian); English transl. in Ukrain. Math. J. 63 (2011), no. 8, 1268–1297.Google Scholar
  53. 322.
    C.H. Sciadas, Exact solutions of stochastic differential equations: Gompertz, generalized logistic and revised exponential, Meth. Comput. Appl. Probab. 12 (2010), no. 2, 261–270.MathSciNetCrossRefGoogle Scholar
  54. 351.
    A. Strauss and J.A. Yorke, On asymptotically autonomous differential equations, Math. Systems Theory 1 (1967), 175–182.MathSciNetCrossRefGoogle Scholar
  55. 352.
    P. Sundar, Law of the iterated logarithm for solutions of stochastic differential equations, Stoch. Anal. Appl. 5 (1987), no. 3, 311–321.MathSciNetCrossRefGoogle Scholar
  56. 358.
    O. Vašíček, An equilibrium characterisation of the term structure, J. Financ. Enomics 5 (1977), 177–188.CrossRefGoogle Scholar
  57. 359.
    A.D. Ventsel’, Rough limit theorems on large deviations for Markov stochastic processes. I, Teor. Veroyatnost. i Primenen. 21 (1976), no. 2, 235–252; English transl. in Theory Probab. Appl. 21 (1977), no. 2, 227–242.CrossRefGoogle Scholar
  58. 364.
    J.G. Wang, Law of the iterated logarithm for stochastic integrals, Stoch. Proc. Appl. 47 (1993), 215–228.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Valeriĭ V. Buldygin
    • 1
  • Karl-Heinz Indlekofer
    • 2
  • Oleg I. Klesov
    • 3
  • Josef G. Steinebach
    • 4
  1. 1.Department of Mathematical AnalysisNational Technical University of UkraineKyivUkraine
  2. 2.Department of MathematicsUniversity of PaderbornPaderbornGermany
  3. 3.Department of Mathematical Analysis and Probability TheoryNational Technical University of UkraineKyivUkraine
  4. 4.Mathematical InstituteUniversity of CologneCologneGermany

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