Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : 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 $X_t$ under
$0.5<\kappa<1$ and appropriate $H$. We mean a decomposition with
``corrector'':
$$
\frac{1}{t^\kappa}\int_0^tH(X_s)ds={\rm
corrector}+\frac{1}{t^\kappa}\underbrace{M_t}_{\rm martingale}.
$$
and show that, as in the CLT analysis, the corrector is negligible but
in the MD scale, and the main contribution in the MD brings the
family ``$ \frac{1}{t^\kappa}M_t, \ t\to\infty. $'' Starting from
Bayer and Freidlin, \cite{BF}, and finishing by Wu's papers
\cite{Wu1}\cite{WuH}, in the MD study Laplace's transform
dominates. In the paper, we replace the Laplace technique by one,
admitting to give the conditions, providing the MD, in terms of
``driftdiffusion'' 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 ...''. 





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