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We consider a model of dynamic inspection/surveillance of a number of facilities in different geographical locations. The inspector in this process travels from one facility to another and performs an inspection at each facility he visits. His aim is to devise an inspection/ travel schedule which minimizes the losses to society (or to his employer) resulting both from undetected violations of the regulations and from the costs of the policing operation. This model is formulated as a noncooperative, single-controller, stochastic game. The existence of stationary Nash equilibria is established as a consequence of aggregating all the inspectees into a single "aggregated inspectee." It is shown that such player aggregation causes no loss of generality under very mild assumptions. A notion of an "optimal Nash equilibrium" for the inspector is introduced and proven to be well-defined in this context. The issue of the inspector's power to "enforce" such an equilibrium is also discussed.