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Modern Stochastics and Applications pp 203–219Cite as

Large Deviations for Random Evolutions in the Scheme of Asymptotically Small Diffusion

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 90))

Abstract

The theory of large deviations deals with the asymptotic estimations for probabilities of rare events. The method, used in majority of classical works, is based on the change of measure and application of the variational formula to the cumulant of the process under study. Here the large deviations problem is considered for random evolutions in the scheme of asymptotically small diffusion. The method of asymptotic analysis for the exponential generator of the Markov process is used. The limit exponential generators are calculated for random evolution with the ergodic Markov switching (Sect. 3) and with the split-and-double merging switching Markov process (Sect. 4). The method proposed here may have applications for the finite dimensional models arising in the theory of random evolutions in R d, queuing theory, etc.

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Authors and Affiliations

  1. Institute of Mathematics NASU, Tereschenkivska street, 3, Kyiv, Ukraine

    Volodymyr S. Korolyuk & Igor V. Samoilenko

Authors
  1. Volodymyr S. Korolyuk
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  2. Igor V. Samoilenko
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Corresponding author

Correspondence to Igor V. Samoilenko .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Volodymyr Korolyuk

  2. Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, Compiegne Cedex, France

    Nikolaos Limnios

  3. Dept. of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Yuliya Mishura

  4. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Lyudmyla Sakhno

  5. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Georgiy Shevchenko

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Korolyuk, V.S., Samoilenko, I.V. (2014). Large Deviations for Random Evolutions in the Scheme of Asymptotically Small Diffusion. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_12

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