  We study the large deviations for Cox-Ingersoll-Ross (CIR) processes with
small noise and state-dependent fast switching via associated Hamilton-Jacobi
equations. As the separation of time scales, when the noise goes to $0$ and the
rate of switching goes to $\infty$, we get a limit equation characterized by
the averaging principle. Moreover, we prove the large deviation principle (LDP)
with an action-integral form rate function to describe the asymptotic behavior
of such systems. The new ingredient is establishing the comparison principle in
the singular context. The proof is carried out using the nonlinear semigroup
method coming from Feng and Kurtz's book.
