Reflected Skorokhod equations and the Neumann boundary value problem for elliptic equations with Levy-type operators

We consider Neumann problem for linear elliptic equations involving integro-differential operators of Levy-type. We show that suitably defined viscosity solutions have probabilistic representations given in terms of the reflected stochastic Skorokhod equation associated with an Ito process and an independent pure-jump Levy process. As an application of the representation we show that viscosity solutions arise a limits of some penalized equations and give some stability results for the viscosity solutions. Our proofs are based on new limit theorems for solutions of penalized stochastic equations with jumps and new estimates on the bounded variation parts of the solutions.