  We study a class of reflected backward stochastic differential equations with
nonpositive jumps and upper barrier. Existence and uniqueness of a minimal
solution is proved by a double penalization approach under regularity
assumptions on the obstacle. In a suitable regime switching diffusion
framework, we show the connection between our class of BSDEs and fully
nonlinear variational inequalities. Our BSDE representation provides in
particular a Feynman-Kac type formula for PDEs associated to general zero-sum
stochastic differential controller-and-stopper games, where control affect both
drift and diffusion term, and the diffusion coefficient can be degenerate.
Moreover, we state a dual game formula of this BSDE minimal solution involving
equivalent change of probability measures, and discount processes. This gives
in particular a new representation for zero-sum stochastic differential
controller-and-stopper games.
