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A method for the calculation of characteristics for the solution to stochastic differential equations

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Veröffentlicht/Copyright: 30. Juni 2017

Abstract

In this work, a new numerical method to calculate the characteristics of the solution to stochastic differential equations is presented. This method is based on the Fokker–Planck equation for the transition probability function and the representation of the transition probability function by means of eigenfunctions of the Fokker–Planck operator. The results of the numerical experiments are presented.

MSC 2010: 65C30; 60H35

Award Identifier / Grant number: ü F15-035

Funding statement: This work is supported by Belarusian Republican Foundation for Fundamental Research (grant F15-035).

References

[1] E. Bennati, M. Rosa-Clot and S. Taddei, A path integral approach to derivative security pricing. I: Formalism and analytical results, Int. J. Theor. Appl. Finance 2 (1999), no. 4, 381–407. 10.1142/S0219024999000200Suche in Google Scholar

[2] A. Egorov and K. Sabelfeld, Approximate formulas for expectations of functionals of solutions to stochastic differential equations, Monte Carlo Methods Appl. 16 (2010), no. 2, 95–127. 10.1515/mcma.2010.003Suche in Google Scholar

[3] A. D. Egorov, P. I. Sobolevsky and L. A. Yanovich, Functional Integrals: Approximate Evaluation and Applications, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-011-1761-6Suche in Google Scholar

[4] A. D. Egorov and A. V. Zherelo, Approximations of functional integrals with respect to measure generated by solutions of stochastic differential equations, Monte Carlo Methods Appl. 10 (2004), no. 3–4, 257–264. 10.1515/mcma.2004.10.3-4.257Suche in Google Scholar

[5] I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968. Suche in Google Scholar

[6] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer, Berlin, 1981. 10.1007/978-1-4684-0121-9Suche in Google Scholar

[7] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1999. Suche in Google Scholar

[8] P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of Stochastic Differential Equations Through Computer Experiments, Springer, Berlin, 1994. 10.1007/978-3-642-57913-4Suche in Google Scholar

[9] V. I. Krylov, V. V. Bobkov and P. I. Monastyrnyi, Computational Methods of Higher Mathematics, Izd. Vysshaya Shkola, Minsk, 1972. Suche in Google Scholar

[10] D. F. Kuznetsov, Numerical Integration of Stochastic Differential Equations, Polytechnical University Publishing House, Saint Petersburg, 2001. Suche in Google Scholar

[11] J. W. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62–78. 10.1090/S0002-9947-1962-0138128-7Suche in Google Scholar

[12] A. J. Lotka, Undamped oscillations derived from the law of mass action, J. Amer. Chem. Soc. 42 (1920), no. 8, 1595–1599. 10.1021/ja01453a010Suche in Google Scholar

[13] G. N. Milstein, The Numerical Integration Solution of Stochastic Differential Equations, Kluwer Academic, Dordrecht, 1995. 10.1007/978-94-015-8455-5Suche in Google Scholar

[14] H. Risken, The Fokker–Plank Equation: Methods of Solution and Applications, Springer, Berlin, 1984. 10.1007/978-3-642-96807-5Suche in Google Scholar

[15] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007. 10.1016/B978-044452965-7/50006-4Suche in Google Scholar

[16] V. S. Vladimirov, Equations of Mathematical Physics, Nauka, Moscow, 1981. Suche in Google Scholar

[17] V. Volterra, Mathematical Theory of Struggle for Existence, Nauka, Moscow, 1976. Suche in Google Scholar

[18] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. Suche in Google Scholar

Received: 2017-2-5
Accepted: 2017-5-18
Published Online: 2017-6-30
Published in Print: 2017-9-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 24.4.2026 von https://www.degruyterbrill.com/document/doi/10.1515/mcma-2017-0110/html?lang=de
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