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On the expected discounted penalty functions for two classes of risk processes under a threshold dividend strategy. (English) Zbl 1232.91354

Summary: The discounted penalty (Gerber-Shiu) functions for a risk model involving two independent classes of insurance risks under a threshold dividend strategy are developed. We also assume that the two claim number processes are independent Poisson and generalized Erlang (2) processes, respectively. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two systems of integro-differential equations for discounted penalty functions are derived, based on whether the surplus is above this threshold level. Laplace transformations of the discounted penalty functions when the surplus is below the threshold level are obtained. And we also derive a system of renewal equations satisfied by the discounted penalty function with initial surplus above the threshold strategy via the Dickson-Hipp operator. Finally, analytical solutions of the two systems of integro-differential equations are presented.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
60K05 Renewal theory
44A10 Laplace transform
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