Abstract
In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.
Highlights
Risk models with surplus-dependent premiums capture the dynamics of the surplus of an insurance company better
We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company
We start by recalling the asymptotic result (38) for risk models with linear premiums ψl,3 (u) ∼C1
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The Reference Albrecher et al (2013) extended the method to surplus-dependent premium models with exponential arrivals, for which they derived exact and asymptotic results for a few premium functions, when the claims were exponentially distributed. We extend it to renewal models and Erlang claims.
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