Moderate Deviations for a Diffusion-Type Process in a Random Environment

Let $\sigma(u)$, $u\in\mathbf{R}$, be an ergodic stationary Markov chain, taking a finite number of values $a_1,\ldots,a_m$, and let $b(u)=g(\sigma(u))$, where $g$ is a bounded and measurable function. We consider the diffusion-type process
$$ dX^\varepsilon_t = b\biggl(\frac{X^\varepsilon_t}{\varepsilon}\biggr)\,dt+\varepsilon^\kappa\sigma\biggl(\frac{X^\varepsilon_t}{\varepsilon}\biggr)\,dB_t,\qquad t\le T, $$
subject to $X^\varepsilon_0=x_0$, where $\varepsilon$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $\sigma$, and $\kappa>0$ is a fixed constant.
We show that for $\kappa<\frac16$, the family $\{X^\varepsilon_t\}_{\varepsilon\to 0}$ satisfies the large deviation principle (LDP) of Freidlin–Wentzell type with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by
$$ \mathbf{b}=\sum_{i=1}^m\frac{g(a_i)}{a^2_i}\,\pi_i\Big/ \sum_{i=1}^m\frac{1}{a^2_i}\,\pi_i, \quad \mathbf{a}=1\Big/\sum_{i=1}^m\frac{1}{a^2_i}\,\pi_i, $$
where $\{\pi_1,\ldots,\pi_m\}$ is the invariant distribution of the chain $\sigma(u)$.