  Let $\sigma(u)$, $u\in \mathbb{R}$ be an ergodic stationary Markov chain,
taking a finite number of values $a_1,...,a_m$, and $b(u)=g(\sigma(u))$, where
$g$ is a bounded and measurable function.
  We consider the diffusion type process $$ dX^\epsilon_t =
b(X^\epsilon_t/\epsilon)dt +
\epsilon^\kappa\sigma\big(X^\epsilon_t/\epsilon\big)dB_t, t\le T $$ subject to
$X^\epsilon_0=x_0$, where $\epsilon$ 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 $\kappa<1/6$, the family $\{X^\epsilon_t\}_{\epsilon\to 0}$
satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type
with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by
$$ \mathbf{b}=\sum\limits_{i=1}^m\dfrac{g(a_i)}{a^2_i}\pi_i\Big/
\sum\limits_{i=1}^m\dfrac{1}{a^2_i}\pi_i, \quad
\mathbf{a}=1\Big/\sum\limits_{i=1}^m\dfrac{1}{a^2_i}\pi_i, $$ where
$\{\pi_1,...,\pi_m\}$ is the invariant distribution of the chain $\sigma(u)$.
