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Appell pseudopolynomials and Erlang-type risk models. (English) Zbl 1337.60227

Summary: Appell polynomials are known to play a key role in certain first-crossing problems. The present paper considers a rather general insurance risk model where the claim interarrival times are independent and exponentially distributed with different parameters, the successive claim amounts may be dependent and the premium income is an arbitrary deterministic function. It is shown that the non-ruin (or survival) probability over a finite horizon may be expressed in terms of a remarkable family of functions, named pseudopolynomials, that generalize the classical Appell polynomials. The presence of that underlying algebraic structure is exploited to provide a closed formula, almost explicit, for the non-ruin probability.

MSC:

60K10 Applications of renewal theory (reliability, demand theory, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
12E10 Special polynomials in general fields
91B30 Risk theory, insurance (MSC2010)
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