A discrete-time risk model is proposed that describes the temporal evolution of the surplus of an insurance company at fixed dates. The novelty of the model comes from its total nonhomogeneity in the sense that as in our previous paper [1], the premium income process is deterministic and nonuniform, but in addition, in the present work, the successive claim amounts are independent and nonidentically distributed. Our purpose is to evaluate for this extended model the probability of ruin over any finite time horizon. Rather surprisingly, the methodology which was developed in [1] can be generalized to the ease of nonstationary claim amounts. The key mathematical tool for that is a theory of pseudopolynomials of Appell type. Furthermore, it is shown that a similar approach can be applied to a multirisks version of the model, as well as to a continuous-time version.

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