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Asymptotic Behaviour of Ruin Probabilities in a General Discrete Risk Model Using Moment Indices

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  • Jaakko Lehtomaa

    (University of Helsinki)

Abstract

We study the rough asymptotic behaviour of a general economic risk model in a discrete setting. Both financial and insurance risks are taken into account. Loss during the first $$n$$ n years is modelled as a random variable $$B_1+A_1B_2+\cdots +A_1\ldots A_{n-1}B_n$$ B 1 + A 1 B 2 + ⋯ + A 1 … A n - 1 B n , where $$A_i$$ A i corresponds to the financial risk of the year $$i$$ i and $$B_i$$ B i represents the insurance risk, respectively. Risks of the same year $$i$$ i are not assumed to be independent. The main result shows that ruin probabilities exhibit power law decay under general assumptions. Our objective is to give a complete characterisation of the relevant quantities that describe the speed at which the ruin probability vanishes as the amount of initial capital grows. These quantities can be expressed as maximal moments, called moment indices, of suitable random variables. In addition to the study of ultimate ruin, the case of finite time interval ruin is considered. Both of these investigations make extensive use of the new properties of moment indices developed during the first half of the paper.

Suggested Citation

  • Jaakko Lehtomaa, 2015. "Asymptotic Behaviour of Ruin Probabilities in a General Discrete Risk Model Using Moment Indices," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1380-1405, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0547-y
    DOI: 10.1007/s10959-014-0547-y
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    References listed on IDEAS

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    1. Paulsen, Jostein, 1998. "Ruin theory with compounding assets -- a survey," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 3-16, May.
    2. Tang, Qihe & Tsitsiashvili, Gurami, 2003. "Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 299-325, December.
    3. Nyrhinen, Harri, 1999. "On the ruin probabilities in a general economic environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 319-330, October.
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    5. Daley, D.J. & Goldie, Charles M., 2006. "The moment index of minima (II)," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 831-837, April.
    6. Baltrunas, A. & Daley, D. J. & Klüppelberg, C., 2004. "Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 237-258, June.
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