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Ruin problems with assets and liabilities of diffusion type

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  • Norberg, Ragnar

Abstract

Ruin and related problems are studied for a risk business with compounding assets when the cash flow and the cumulative interest rate are diffusion processes with coefficients depending on the time and on the current cash balance. Differential equations are obtained for the probabilities of ruin at a given date, in finite time, and in infinite time. Some previously known explicit formulas related to Brownian motion come out as special cases. Relationships between crossing probabilities and transition probabilities are investigated and, in particular, existing results on the probability distribution of the running maximum of a Brownian motion and on the relationship between the probability of ruin and on the probability distribution of the discounted total payments are generalized. Proofs rest on a martingale technique.

Suggested Citation

  • Norberg, Ragnar, 1999. "Ruin problems with assets and liabilities of diffusion type," Stochastic Processes and their Applications, Elsevier, vol. 81(2), pages 255-269, June.
    • Handle: RePEc:eee:spapps:v:81:y:1999:i:2:p:255-269
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    References listed on IDEAS

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    1. Garrido, Jose, 1989. "Stochastic differential equations for compounded risk reserves," Insurance: Mathematics and Economics, Elsevier, vol. 8(3), pages 165-173, November.
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    Cited by:

    1. Xing-Fang Huang & Ting Zhang & Yang Yang & Tao Jiang, 2017. "Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks," Risks, MDPI, vol. 5(1), pages 1-14, March.
    2. Yuen, Kam C. & Wang, Guojing & Ng, Kai W., 2004. "Ruin probabilities for a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 259-274, April.
    3. Kalashnikov, Vladimir & Norberg, Ragnar, 2002. "Power tailed ruin probabilities in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 211-228, April.
    4. Tang, Qihe & Vernic, Raluca, 2007. "The impact on ruin probabilities of the association structure among financial risks," Statistics & Probability Letters, Elsevier, vol. 77(14), pages 1522-1525, August.
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    11. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    12. Chiu, Mei Choi & Li, Duan, 2006. "Asset and liability management under a continuous-time mean-variance optimization framework," Insurance: Mathematics and Economics, Elsevier, vol. 39(3), pages 330-355, December.
    13. Xie, Shuxiang, 2009. "Continuous-time mean-variance portfolio selection with liability and regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 148-155, August.
    14. Chen, Yiqing & Liu, Jiajun & Liu, Fei, 2015. "Ruin with insurance and financial risks following the least risky FGM dependence structure," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 98-106.
    15. Zimbidis, Alexandros A., 2014. "Insurance pricing using H∞-control," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 685-697.
    16. Chen, Ping & Yang, Hailiang & Yin, George, 2008. "Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 456-465, December.
    17. Yiqing Chen & Jiajun Liu & Yang Yang, 2023. "Ruin under Light-Tailed or Moderately Heavy-Tailed Insurance Risks Interplayed with Financial Risks," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    18. Haiyang Wang & Zhen Wu, 2014. "Partially Observed Time-Inconsistency Recursive Optimization Problem and Application," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 664-687, May.
    19. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
    20. M. C. Chiu & D. Li, 2009. "Asset-Liability Management Under the Safety-First Principle," Journal of Optimization Theory and Applications, Springer, vol. 143(3), pages 455-478, December.
    21. Xie, Shuxiang & Li, Zhongfei & Wang, Shouyang, 2008. "Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 943-953, June.
    22. Sun, Ying & Wei, Li, 2014. "The finite-time ruin probability with heavy-tailed and dependent insurance and financial risks," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 178-183.
    23. 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.
    24. Chen, Yu & Su, Chun, 2006. "Finite time ruin probability with heavy-tailed insurance and financial risks," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1812-1820, October.

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