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The expected discounted penalty function: from infinite time to finite time. (English) Zbl 1411.91303

Summary: In this paper we study the finite-time expected discounted penalty function (EDPF) and its decomposition in the classical risk model perturbed by diffusion. We first give the solution to a class of second-order partial integro-differential equations (PIDEs) with certain boundary conditions. We then show that the finite-time EDPFs as well as their decompositions satisfy this specific class of PIDEs so that their explicit expressions are obtained. Furthermore, we demonstrate that the finite-time EDPF may be expressed in terms of its ordinary counterpart (infinite-time) under the same risk model. Especially, the finite-time ruin probability due to oscillations and the finite-time ruin probability caused by a claim may also be expressed in terms of the corresponding quantities under the infinite-time horizon. Numerical examples are given when claims follow an exponential distribution.

MSC:

91B30 Risk theory, insurance (MSC2010)
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
45K05 Integro-partial differential equations
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