2 research outputs found
Large Deviations for Small Noise Diffusions in a Fast Markovian Environment
A large deviation principle is established for a two-scale stochastic system
in which the slow component is a continuous process given by a small noise
finite dimensional It\^{o} stochastic differential equation, and the fast
component is a finite state pure jump process. Previous works have considered
settings where the coupling between the components is weak in a certain sense.
In the current work we study a fully coupled system in which the drift and
diffusion coefficient of the slow component and the jump intensity function and
jump distribution of the fast process depend on the states of both components.
In addition, the diffusion can be degenerate. Our proofs use certain stochastic
control representations for expectations of exponential functionals of finite
dimensional Brownian motions and Poisson random measures together with weak
convergence arguments. A key challenge is in the proof of the large deviation
lower bound where, due to the interplay between the degeneracy of the diffusion
and the full dependence of the coefficients on the two components, the
associated local rate function has poor regularity properties.Comment: 42 page