Examples of moderate deviation principle for diffusion processes

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_t^κ = 1/t ^κ ∫₀^t H(X_s)ds, t → ∞ for an ergodic diffusion process X_t, provided that 0.5 < κ < 1, and appropriate H. We use a well known decomposition with "corrector": 1/t^κ ∫₀^t H(X_s)ds = corrector + 1/t^κ M_tmartingale . and show that, as in the CLT analysis, the corrector is negligible, and the main contribution in the MD brings the family "1/t^κM _t, t → ∞." 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 ...".