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.
Similar content being viewed by others
References
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).
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).
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992).
D. F. Kuznetsov, Numerical Integration of Stochastic Differential Equations (SPb Gos. Politekh. Univ., St. Petersburg, 2001) [in Russian].
H. Risken, The Fokker-Plank Equation: Methods of Solution and Applications (Springer, Berlin, 1984).
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics Polymer Physics, and Financial Markets (World Scientific, Singapore, 2004).
N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Nauka, Moscow, 1976; Wiley, New York, 1980).
J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View (Springer, Berlin, Heidelberg, New York, 1981).
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].
F. Langouche, D. Roekaerts, and E. Tirapegui, Functional Integration and Semi-Classical Expansions (D. Reidel, Dordrecht, 1982).
H. S. Wio, Application of Path Integration to Stochastic Process: An Introduction (World Scientific, Singapore, 2013).
L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953).
J. W. Lamperti, “Semi-stable stochastic processes,” Trans. Am. Math. Soc. 104, 62–78 (1962).
J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).
A. D. Egorov, P. I. Sobolevskii, and L. A. Yanovich, Approximate Methods for Continual Integrals Computation (Nauka Tekhnika, Moscow, 1985) [in Russian].
A. D. Egorov, P. I. Sobolevsky, and L. A. Yanovich, Functional Integrals: Approximate Evaluation and Applications (Kluwer Academic, Dordrecht, 1993).
V. I. Krylov, V. V. Bobkov, and P. I. Monastyrnyi, Numerical Methods of Higher Mathematics (Vysheish. Shkola, Minsk, 1975), Vol. 2 [in Russian].
C. W. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences, Springer Series in Synergetics (Springer, New York, 1986).
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).
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.A. Ayryan, A.D. Egorov, D.S. Kulyabov, V.B. Malyutin, L.A. Sevastyanov, 2016, published in Matematicheskoe Modelirovanie, 2016, Vol. 28, No. 11, pp. 113–125.
Rights and permissions
About this article
Cite this article
Ayryan, E.A., Egorov, A.D., Kulyabov, D.S. et al. Application of functional integrals to stochastic equations. Math Models Comput Simul 9, 339–348 (2017). https://doi.org/10.1134/S2070048217030024
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070048217030024