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The comparison principle and the existence of the solution of the integro-differential equation with L{\'e}vy operators, in the framework of the viscosity solution, are shown in this paper. For the one dimensional case, a detailed estimate of the H{\"o}lder continuity of solutions is presented, by localizing the singularity of the L{\'e}vy measure.

A Localization of the Lévy Operators Arising in Mathematical Finances