In this paper we study a system of variational inequalities where theoperator is non-local, possibly degenerate and of second order. A special caseof this type of problem occurs in the context of optimal switching problemswhen the dynamics of the underlying state variables is described by anN-dimensional Levy process. We establish a general comparison principle forviscosity sub- and supersolutions to the system under mild regularity, growthand structural assumptions on the data. Using the comparison principle we thenprove the existence of a unique viscosity solution to the system by Perron'smethod. Our main contribution is that we establish existence and uniqueness ofviscosity 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 xas well as t.
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