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Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model

Author

Listed:
  • Krzysztof Dȩbicki

    (Mathematical Institute, University of Wrocław, 50-137 Wrocław, Poland
    These authors contributed equally to this work.)

  • Lanpeng Ji

    (School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK)

  • Tomasz Rolski

    (Mathematical Institute, University of Wrocław, 50-137 Wrocław, Poland
    These authors contributed equally to this work.)

Abstract

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function P ( u ) for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where u is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of − ln P ( u ) / u as u tends to infinity, which depends essentially on the correlation ρ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.

Suggested Citation

  • Krzysztof Dȩbicki & Lanpeng Ji & Tomasz Rolski, 2019. "Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model," Risks, MDPI, vol. 7(3), pages 1-21, August.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:3:p:83-:d:253948
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    References listed on IDEAS

    as
    1. Li, Junhai & Liu, Zaiming & Tang, Qihe, 2007. "On the ruin probabilities of a bidimensional perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 185-195, July.
    2. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Rolski, Tomasz, 2018. "Extremal behavior of hitting a cone by correlated Brownian motion with drift," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4171-4206.
    Full references (including those not matched with items on IDEAS)

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