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We consider reduced-form models for portfolio credit risk with interacting default intensities. In this class of models the impact of default of some firm on the default intensities of surviving firms is exogenously specified and the dependence structure of the default times is endogenously determined. We construct and study the model using Markov process techniques. We analyze in detail a model where the interaction between firms is of the mean-field type. Moreover, we discuss the pricing of portfolio related credit products such as basket default swaps and CDOs in our model.