We establish a Large Deviations Principle for diffusions 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 a viscosity solution approach. The idea consists in interpreting the probabilities as the solutions to some PDEs, make the logarithmic transform, pass to the limit, and then identify the action functional as the solution of the limiting equation.

We consider a class of ${ℝ}^d$-valued stochastic control systems, with possibly unbounded costs. The systems evolve according to a discrete-time equation $x_{t+1}=Gₙ(x_t,a_t)+{\xi }_t$ (t = 0,1,... ), for each fixed n = 0,1,..., where the ${\xi }_t$ are i.i.d. random vectors, and the Gₙ are given functions converging pointwise to some function $G_{\infty }$ as n → ∞. Under suitable hypotheses, our main results state the existence of stationary control policies that are expected average cost (EAC) optimal and sample path average cost (SPAC) optimal for...