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Stochastic Processes and their Applications
Volume 115, Issue 7, July 2005, Pages 1187-1207
 
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doi:10.1016/j.spa.2005.02.008    How to Cite or Link Using DOI (Opens New Window)
Copyright © 2005 Elsevier B.V. All rights reserved.

MDP for integral functionals of fast and slow processes with averaging

A. Guillina, Corresponding Author Contact Information, E-mail The Corresponding Author and R. Liptserb, E-mail The Corresponding Author

aCEREMADE, Université Paris Dauphine, Paris 75016, France bDepartment of Electrical Engineering-Systems, Tel Aviv University, 69978 Tel Aviv, Israel

Received 12 March 2003; 
revised 13 September 2004; 
accepted 18 February 2005. 
Available online 19 March 2005.

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Abstract

We establish the moderate deviation principle (MDP) for the family of

View the MathML source
where 0<κ<0.5 and View the MathML source are slow and fast diffusion processes. We embed the original problem in the MDP study for the pair View the MathML source. The main tool for the MDP analysis is the Poisson equation technique, borrowed from the recent papers of Pardoux and Veretennikov, (Ann. Probab. 29 (3) (2001) 1061; Ann. Probab. 31 (3) (2003) 1166), and a new approach to the large deviation analysis, proposed by Puhalskii, (Large Deviations and Idempotent Probability, 2001), which exploits “fast homogenization” of the drift and diffusion parameters instead of the traditional Laplace transform technique. The obtained MDP for View the MathML source has a typical structure of the Freidlin–Wentzell-type large deviation principle.

Keywords: Moderate deviations; Poisson equation; Puhalskii theorem

MSC: 60J27; 60F10

Article Outline

1. Introduction
2. Notations and assumptions
3. Main result
4. Preliminaries
4.1. Poisson equation
4.2. New family sharing the MDP with View the MathML source
5. MDP analysis for View the MathML source
5.1. The proof of Theorem 5.1 under uniformly nonsingular View the MathML source
5.2. The proof of Theorem 5.1 under nonnegative definite View the MathML source
5.2.1. Verification of (5.14)
5.2.2. Verification of (5.15)
6. The proof of Theorem 3.1
Acknowledgements
Appendix A. Exponential estimates for martingale
Appendix B. Auxiliary results for exponential tightness
References

 
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