Exact and asymptotic results for insurance risk models with surplus-dependent premiums
In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Green's operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cram\'er-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.
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- Landriault, David & Willmot, Gordon, 2008. "On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 600-608, April.
- Albrecher, Hansjörg & Constantinescu, Corina & Pirsic, Gottlieb & Regensburger, Georg & Rosenkranz, Markus, 2010. "An algebraic operator approach to the analysis of Gerber-Shiu functions," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 42-51, February.
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