Exact and asymptotic results for insurance risk models with surplus-dependent premiums

Abstract

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 Cramér-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.

References in corpus (1)

• Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases

Cited by in corpus (6)

• On Integro-Differential Algebras
• The Gerber-Shiu discounted penalty function: A review from practical perspectives
• At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift
• Constructions of free commutative integro-differential algebras
• On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums
• How much we gain by surplus-dependent premiums -- asymptotic analysis of ruin probability