Application of Advanced Integrodifferential Equations in Insurance Mathematics and Process Engineering
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In this paper we consider a dual risk model with general inter-arrival time distribution and general size distribution. A special Gerber–Shiu discounted penalty function is defined and an integral equation is derived for it in the case of dependent inter-arrival times and sizes. We prove the existence and uniqueness of the solution of the integral equation in the set of bounded functions and we show that the solution tends to zero exponentially. If the density function of the inter-arrival time satisfies a linear differential equation with constant coefficients, the integral equation is transformed into an integrodifferential equation with advances in arguments and an explicit solution is given without any assumption on the size distribution. We also present a link between the Lundberg fundamental equations of the Sparre Andersen risk model and the dual risk model.
KeywordsDual risk model Advanced integrodifferential equation Process engineering LODE-type distribution