Advertisement

Application of functional integrals to stochastic equations

  • E. A. AyryanEmail author
  • A. D. Egorov
  • D. S. Kulyabov
  • V. B. Malyutin
  • L. A. Sevastyanov
Article
  • 33 Downloads

Abstract

Representing a probability density function (PDF) and other quantities describing a solution of stochastic differential equations by a functional integral is considered in this paper. Methods for the approximate evaluation of the arising functional integrals are presented. Onsager–Machlup functionals are used to represent PDF by a functional integral. Using these functionals the expression for PDF on a small time interval Δt can be written. This expression is true up to terms having an order higher than one relative to Δt. A method for the approximate evaluation of the arising functional integrals is considered. This method is based on expanding the action along the classical path. As an example the application of the proposed method to evaluate some quantities to solve the equation for the Cox–Ingersol–Ross type model is considered.

Keywords

stochastic differential equations Onsager-Machlup functionals functional integrals 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. S. Kulyabov and A. V. Demidova, “Introduction of self-consistent term in stochastic population model equation,” Vestn. RUDN, Ser. Mat. Inform. Fiz., No. 3, 69–78 (2012).Google Scholar
  2. 2.
    A. V. Demidova, “The equations of population dynamics in the form of stochastic differential equations,” Vestn. RUDN, Ser. Mat. Inform. Fiz., No. 1, 67–76 (2013).Google Scholar
  3. 3.
    P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992).CrossRefzbMATHGoogle Scholar
  4. 4.
    D. F. Kuznetsov, Numerical Integration of Stochastic Differential Equations (SPb Gos. Politekh. Univ., St. Petersburg, 2001) [in Russian].Google Scholar
  5. 5.
    H. Risken, The Fokker-Plank Equation: Methods of Solution and Applications (Springer, Berlin, 1984).CrossRefzbMATHGoogle Scholar
  6. 6.
    R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).zbMATHGoogle Scholar
  7. 7.
    H. Kleinert, Path Integrals in Quantum Mechanics, Statistics Polymer Physics, and Financial Markets (World Scientific, Singapore, 2004).CrossRefzbMATHGoogle Scholar
  8. 8.
    N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Nauka, Moscow, 1976; Wiley, New York, 1980).zbMATHGoogle Scholar
  9. 9.
    J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View (Springer, Berlin, Heidelberg, New York, 1981).zbMATHGoogle Scholar
  10. 10.
    A. D. Egorov, E. P. Zhidkov, and Yu. Yu. Lobanov, An Introduction to the Theory and Applications of Functional Integration (Fizmatlit, Moscow, 2006) [in Russian].Google Scholar
  11. 11.
    F. Langouche, D. Roekaerts, and E. Tirapegui, Functional Integration and Semi-Classical Expansions (D. Reidel, Dordrecht, 1982).CrossRefzbMATHGoogle Scholar
  12. 12.
    H. S. Wio, Application of Path Integration to Stochastic Process: An Introduction (World Scientific, Singapore, 2013).CrossRefGoogle Scholar
  13. 13.
    L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953).CrossRefGoogle Scholar
  14. 14.
    J. W. Lamperti, “Semi-stable stochastic processes,” Trans. Am. Math. Soc. 104, 62–78 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).zbMATHGoogle Scholar
  16. 16.
    A. D. Egorov, P. I. Sobolevskii, and L. A. Yanovich, Approximate Methods for Continual Integrals Computation (Nauka Tekhnika, Moscow, 1985) [in Russian].Google Scholar
  17. 17.
    A. D. Egorov, P. I. Sobolevsky, and L. A. Yanovich, Functional Integrals: Approximate Evaluation and Applications (Kluwer Academic, Dordrecht, 1993).CrossRefGoogle Scholar
  18. 18.
    V. I. Krylov, V. V. Bobkov, and P. I. Monastyrnyi, Numerical Methods of Higher Mathematics (Vysheish. Shkola, Minsk, 1975), Vol. 2 [in Russian].Google Scholar
  19. 19.
    C. W. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences, Springer Series in Synergetics (Springer, New York, 1986).Google Scholar
  20. 20.
    A. V. Demidova, M. N. Gevorkian, A. D. Egorov, D. S. Kuliabov, A. V. Korolkova, and L. A. Sevastianov, “Influence of stochastization on one-step models,” Vestn. RUDN, Ser. Mat. Inform. Fiz., No. 1, 71–85 (2014).Google Scholar
  21. 21.
    G. A. Gottwald and J. Harlim, “The role of additive and multiplicative noise in filtering complex dynamical systems,” Proc. R. Soc. A: Math., Phys. Eng. Sci. 469, 20130096 (2013).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • E. A. Ayryan
    • 1
    • 3
    Email author
  • A. D. Egorov
    • 2
  • D. S. Kulyabov
    • 1
    • 3
  • V. B. Malyutin
    • 2
  • L. A. Sevastyanov
    • 3
    • 4
  1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussia
  2. 2.Institute of MathematicsNational Academy of Sciences of BelarusMinskBelarus
  3. 3.Peoples’ Friendship University of RussiaRUDN UniversityMoscowRussia
  4. 4.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubnaRussia

Personalised recommendations