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The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier. (English) Zbl 1273.91456
Summary: We consider the classical surplus process with interest and a constant dividend barrier. Under constant interest, we derive an integro-differential equation for the Gerber-Shiu expected discounted penalty function. Following an idea of X. S. Lin et al. [Insur. Math. Econ. 33, No. 3, 551–566 (2003; Zbl 1103.91369)], we obtain the solution to the integro-differential equation which is in the form of an infinite series. In some special cases with exponential claims, we are able to find closed-form expressions for the Gerber-Shiu expected discounted penalty function. Finally, we extend the integro-differential equation to the case where the surplus is invested in an investment portfolio with stochastic return on investments.

##### MSC:
 91G50 Corporate finance (dividends, real options, etc.) 91B30 Risk theory, insurance (MSC2010) 45J05 Integro-ordinary differential equations
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