Examples of moderate deviation principle for diffusion processes


Abstract in English

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 $$ S^kappa_t=frac{1}{t^kappa}int_0^tH(X_s)ds, ttoinfty $$ 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, ttoinfty. $ Starting from Bayer and Freidlin, cite{BF}, and finishing by Wus papers cite{Wu1}-cite{WuH}, in the MD study Laplaces transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, 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 ....

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