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

TL;DR

This paper introduces a symbolic method to derive asymptotic expressions for ruin probabilities and penalty functions in surplus-dependent premium insurance models, utilizing boundary problems for differential equations.

Contribution

It presents a novel symbolic technique for asymptotic analysis in surplus-dependent premium risk models, including closed-form solutions for specific cases.

Findings

Developed a symbolic approach for asymptotic expressions

Derived exponential-type expansions and Cramér-type asymptotics

Obtained closed-form solutions for specific premium functions

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