MDP for integral functionals of fast and slow processes with averaging

Abstract

We establish large deviation principle (LDP) for the family of vector-valued random processes $(X^{\epsilon },Y^{\epsilon }),\epsilon \to 0$ defined as $$X_t^{\epsilon }=\frac{1}{{\epsilon }^{\kappa }}{\int }_0^{t}H({\xi }_s^{\epsilon },Y_s^{\epsilon })ds,d{Y}_t^{\epsilon }=F({\xi }_t^{\epsilon },Y_t^{\epsilon })dt+D{\epsilon }^{1/2-\kappa }G({\xi }_t^{\epsilon },Y_t^{\epsilon })d{W}_t,$$ where $W_t$ is Wiener process and ${\xi }_t^{\epsilon }$ is fast ergodic diffusion. We show that, under $\kappa <1/2$ or less and Veretennikov-Khasminskii type condition for fast diffusion, the LDP holds with rate function of Freidlin-Wentzell's type.

18 pages