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Automation and Remote Control

, Volume 78, Issue 8, pp 1438–1448 | Cite as

Nonlinear trend exclusion procedure for models defined by stochastic differential and difference equations

  • V. D. KonakovEmail author
  • A. R. Markova
Stochastic Systems
  • 21 Downloads

Abstract

We consider a diffusion process and its approximation with a Markov chain whose trends contain a nonlinear unbounded component. The usual parametrix method is inapplicable here since the trend is unbounded. We present a procedure that lets us exclude a nonlinear growing trend and pass to a stochastic differential equation with bounded drift and diffusion coefficients. A similar procedure is also considered for a Markov chain.

Keywords

stochastic differential equation diffusion process Markov chains parametrix method 

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References

  1. 1.
    Konakov, V.D. and Markova, A.R., Linear Trend Exclusion for Models Defined with Stochastic Differential and Difference Equations, Autom. Remote Control, 2015, vol. 76, no. 10, pp. 1771–1783.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Il’in, A.M., Kalashnikov, A.S., and Oleinik, O.A., Second Order Linear Equations of Parabolic Type, Usp. Mat. Nauk, 1962, vol. 17, no. 3, pp. 3–146.MathSciNetGoogle Scholar
  3. 3.
    Friedman, A., Partial Differential Equations of Parabolic Type, Englewood Cliffs: Prentice Hall, 1964.zbMATHGoogle Scholar
  4. 4.
    Konakov, V.D., The Parametrix Method for Diffusions and Markov Chains, Preprint, Moscow: Izd. Popech. Soveta Mekhaniko-Matematicheskogo Fakul’teta MGU, Ser. WP BRP “STI,”2012.Google Scholar
  5. 5.
    Arnol’d, V.I., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations), Moscow: Nauka, 1971.Google Scholar
  6. 6.
    Bibikov, Yu.N., Kurs obyknovennykh differentsial’nykh uravnenii (A Course in Ordinary Differential Equations), Moscow: Vysshaya Shkola, 1991.Google Scholar
  7. 7.
    Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve (Stability of Solutions for Differential Equations in a Banach Space), Moscow: Nauka, 1970.Google Scholar
  8. 8.
    Kelley, W. and Peterson, A., The Theory of Differential Equations, Classical and Qualitative, Upper Saddle River: Prentice Hall, 2004.Google Scholar
  9. 9.
    Bally, V. and Rey C., Approximation of Markov Ssemigroups in Total Variation Distance, Electron. J. Probab., 2016, no. 12, pp. 1–44.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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