Large deviations for two scaled diffusions

Abstract

We formulate large deviations principle (LDP) for diffusion pair $(X^\epsilon,\xi^\epsilon)=(X_t^\epsilon,\xi_t^\epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for $(X^\epsilon,\nu^\epsilon)$ with $\nu^\epsilon(dt,dz)$ being an occupation type measure corresponding to $\xi_t^\epsilon$. In some sense we obtain a combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem.