Large deviations for Cox–ingersoll–ross processes with state-dependent fast switching

Abstract We study large deviations for Cox–Ingersoll–Ross processes with small noise and state-dependent fast switching via associated Hamilton–Jacobi–Bellman equations. As time scales separate, 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 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 from Feng and Kurtz’s book [14].