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How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

Author

Listed:
  • Jing Wang

    (Department of Mathematical Sciences, Institute for Financial and Actuarial Mathematics, University of Liverpool, Liverpool L69 7ZL, UK
    These authors contributed equally to this work.)

  • Zbigniew Palmowski

    (Department of Applied Mathematics, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
    These authors contributed equally to this work.)

  • Corina Constantinescu

    (Department of Mathematical Sciences, Institute for Financial and Actuarial Mathematics, University of Liverpool, Liverpool L69 7ZL, UK
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Jing Wang & Zbigniew Palmowski & Corina Constantinescu, 2021. "How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability," Risks, MDPI, vol. 9(9), pages 1-17, August.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:9:p:157-:d:622795
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    References listed on IDEAS

    as
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