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Teor. Veroyatnost. i Primenen., 2009, Volume 54, Issue 1, Pages 39–62 (Mi tvp2498)  

This article is cited in 3 scientific papers (total in 3 papers)

Moderate Deviations for a Diffusion-Type Process in a Random Environment

R. Sh. Liptsera, P. Chiganskyb

a Tel Aviv University
b Tel Aviv University, Department of Electrical Engineering-Systems

Abstract: Let $\sigma(u)$, $u\in\mathbf{R}$, be an ergodic stationary Markov chain, taking a finite number of values $a_1,\ldots,a_m$, and let $b(u)=g(\sigma(u))$, where $g$ is a bounded and measurable function. We consider the diffusion-type process
$$ dX^\varepsilon_t = b(\frac{X^\varepsilon_t}{\varepsilon}) dt+\varepsilon^\kappa\sigma(\frac{X^\varepsilon_t}{\varepsilon}) dB_t,\qquad t\le T, $$
subject to $X^\varepsilon_0=x_0$, where $\varepsilon$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $\sigma$, and $\kappa>0$ is a fixed constant.
We show that for $\kappa<\frac16$, the family $\{X^\varepsilon_t\}_{\varepsilon\to 0}$ satisfies the large deviation principle (LDP) of Freidlin–Wentzell type with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by
$$ \mathbf{b}=\sum_{i=1}^m\frac{g(a_i)}{a^2_i} \pi_i/ \sum_{i=1}^m\frac{1}{a^2_i} \pi_i, \quad \mathbf{a}=1/\sum_{i=1}^m\frac{1}{a^2_i} \pi_i, $$
where $\{\pi_1,\ldots,\pi_m\}$ is the invariant distribution of the chain $\sigma(u)$.

Keywords: random environment, moderate deviations, diffusion-type processes, Freidlin–Wentzell large deviation principle

DOI: https://doi.org/10.4213/tvp2498

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English version:
Theory of Probability and its Applications, 2010, 54:1, 29–50

Bibliographic databases:

Received: 17.03.2007
Revised: 12.10.2008

Citation: R. Sh. Liptser, P. Chigansky, “Moderate Deviations for a Diffusion-Type Process in a Random Environment”, Teor. Veroyatnost. i Primenen., 54:1 (2009), 39–62; Theory Probab. Appl., 54:1 (2010), 29–50

Citation in format AMSBIB
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\paper Moderate Deviations for a Diffusion-Type Process in a Random Environment
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\pages 39--62
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\jour Theory Probab. Appl.
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\vol 54
\issue 1
\pages 29--50
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    This publication is cited in the following articles:
    1. Jagers P., Klebaner F.C., “Population-Size-Dependent, Age-Structured Branching Processes Linger Around their Carrying Capacity”, J. Appl. Probab., 48A:SI (2011), 249–260  crossref  mathscinet  zmath  isi  scopus
    2. Hamza K., Jagers P., Klebaner F.C., “on the Establishment, Persistence, and Inevitable Extinction of Populations”, J. Math. Biol., 72:4, SI (2016), 797–820  crossref  mathscinet  zmath  isi  scopus
    3. Morse M.R., Spiliopoulos K., “Moderate Deviations For Systems of Slow-Fast Diffusions”, Asymptotic Anal., 105:3-4 (2017), 97–135  crossref  mathscinet  zmath  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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