Large Deviations Principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections

TL;DR

This paper proves a Large Deviations Principle for diffusions with oblique Lipschitz reflections using a novel viscosity solution approach, linking PDEs, control problems, and probabilistic estimates.

Contribution

It introduces a new viscosity solution method to establish large deviations for reflected diffusions, providing a rigorous PDE-based framework.

Findings

Established a Large Deviations Principle for reflected diffusions.

Identified the rate functional as a value function of a control problem.

Proved the rate functional is good and well-defined.

Abstract

We establish a Large Deviations Principle for stochastic processes with Lipschitz continuous oblique reflections on regular domains. The rate functional is given as the value function of a control problem and is proved to be good. The proof is based on an original viscosity solution approach. The idea consists in interpreting the probabilities as the solutions of some PDEs, make the logarithmic transform, pass to the limit, and then identify the action functional as the solution of the limiting equation.