July  2006, 6(4): 803-828. doi: 10.3934/dcdsb.2006.6.803

Examples of moderate deviation principle for diffusion processes

1. 

CEREMADE, Université Paris Dauphine and TSI, Ecole nationale des Telecommunications, France

2. 

Electrical Engineering Systems, Tel Aviv University, 69978 - Ramat Aviv, Tel Aviv, Israel

Received  January 2005 Revised  November 2005 Published  April 2006

Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family

$S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, t\to\infty$

for an ergodic diffusion process $X_t$, provided that $0.5<\kappa<1$, and appropriate $H$. We use a well known decomposition with "corrector'':

$\frac{1}{t^\kappa}\int_0^tH(X_s)ds=$corrector$+\frac{1}{t^\kappa}$ ${M_t}$martingale.

and show that, as in the CLT analysis, the corrector is negligible, and the main contribution in the MD brings the family "$ \frac{1}{t^\kappa}M_t, \ t\to\infty. $'' Starting from Freidlin, [7], and finishing by Wu's papers [33]-[37], in the MD study Laplace's transform dominates. In the paper, we replace this technique by "Stochastic exponential'' one, enabling to formulate the MDP conditions in terms of "drift-diffusion'' parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named "Examples ...''.

Citation: A. Guillin, R. Liptser. Examples of moderate deviation principle for diffusion processes. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 803-828. doi: 10.3934/dcdsb.2006.6.803
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