  In this paper we study a system of variational inequalities where the
operator is non-local, possibly degenerate and of second order. A special case
of this type of problem occurs in the context of optimal switching problems
when the dynamics of the underlying state variables is described by an
N-dimensional Levy process. We establish a general comparison principle for
viscosity sub- and supersolutions to the system under mild regularity, growth
and structural assumptions on the data. Using the comparison principle we then
prove the existence of a unique viscosity solution to the system by Perron's
method. Our main contribution is that we establish existence and uniqueness of
viscosity solutions, in the setting of Levy processes and non-local operators,
with no sign assumption on the switching costs and allowing them to depend on x
as well as t.
