Large deviations for Cox-Ingersoll-Ross processes with state-dependent fast switching

TL;DR

This paper establishes a large deviation principle for Cox-Ingersoll-Ross processes with small noise and fast, state-dependent switching, using Hamilton-Jacobi equations and the nonlinear semigroup method.

Contribution

It introduces a novel approach to large deviations for CIR processes with state-dependent switching, including a comparison principle in a singular context.

Findings

Proves the LDP with an action-integral rate function.

Characterizes the limit behavior via an averaging principle.

Develops a new comparison principle for singular Hamilton-Jacobi equations.

Abstract

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.