Large deviations for solutions of one dimensional Itô equations
Author:
A. V. Logachov
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 90 (2015), 127-137
MSC (2010):
Primary 60F10, 60F17
DOI:
https://doi.org/10.1090/tpms/954
Published electronically:
August 10, 2015
MathSciNet review:
3242025
Full-text PDF Free Access
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Additional Information
Abstract: The large deviations principle for the sequence of stochastic processes \[ \eta _n(t)=x_0+\int _0^t b(n\eta _n(s)) ds+\frac {1}{\varphi (n)}\int _0^t \sigma (n\eta _n(s)) dw(s) \] is proved if the limits of integral means exist for the functions $b(x)\sigma ^{-2}(x)$ and $\sigma ^{-2}(x)$. The rate functional is evaluated.
References
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References
- M. I. Freidlin and R. B. Sowers, A comparison of homogenization and large deviations, with applications to wavefront propagation, Stoch. Process. App. 82 (1999), 23–52. MR 1695068 (2000f:35014)
- S. Ya. Makhno, Large deviations for solutions of stochastic equations, Teor. Veroyatnost. Primenen. 40 (1995), no. 4, 765–785; English transl. in Theory Probab. Appl. 40 (1995), no. 4, 660–678. MR 1405144 (97j:60106)
- A. D. Wentzel and M. I. Freidlin, Fluctuations in Dynamical Systems under the Action of Small Random Perturbations, “Nauka”, Moscow, 1979; English transl., Springer Verlag, Berlin, 1984.
- I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968; English transl., Springer Verlag, Berlin, 1972.
- R. Sh. Liptser and P. Chigansky, Moderate deviations for a diffusion-type process in a random environment, Teor. Veroyatnost. Primenen. 54 (2009), no. 1, 39–62; English transl. in Theory Probab. Appl. 54 (2010), no. 1, 29–50. MR 2766646 (2012d:60080)
- A. A. Pukhalskiĭ, Large Deviations of Stochastic Dynamical Systems, “Fizmatlit”, Moscow, 2005. (Russian)
- A. D. Wentzel, Limit Theorems on Large Deviations for Markov Stochastic Processes, “Nauka”, Moscow, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1990. MR 1135113 (92i:60054)
- I. V. Samoĭlenko, Large deviations for random evolutions with independent increments in the scheme of the Poisson approximation, Teor. Imovirnost. Matem. Statyst. 85 (2011), 95–102; English transl. in Theor. Probability and Math. Statist. 85 (2012), 107–114. MR 2933707
- D. S. Budkov and S. Ya. Makhno, Law of the iterated logarithm for solutions of stochastic equations, Teor. Imovirnost. Matem. Statyst. 83 (2010), 39–48; English transl. in Theor. Probability and Math. Statist. 83 (2011), 47–57. MR 2768847 (2012d:60083)
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Additional Information
A. V. Logachov
Affiliation:
Department of Probability Theory and Mathematical Statistics, Institute for Applied Mathematics and Mechanics, National Academy of Science of Ukraine, R. Luxemburg Street, 74, Donetsk, 83114, Ukraine
Email:
omboldovskaya@mail.ru
Keywords:
Large deviations,
rate functional,
stochastic differential equation
Received by editor(s):
November 15, 2012
Published electronically:
August 10, 2015
Article copyright:
© Copyright 2015
American Mathematical Society
