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Ruin models with investment income

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  • Jostein Paulsen
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    Abstract

    This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].

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    File URL: http://arxiv.org/pdf/0806.4125
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0806.4125.

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    Date of creation: Jun 2008
    Date of revision: Dec 2008
    Publication status: Published in Probability Surveys 2008, Vol. 5, No. 0, 416-434
    Handle: RePEc:arx:papers:0806.4125

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    Web page: http://arxiv.org/

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    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
    2. Paulsen, Jostein, 1998. "Ruin theory with compounding assets -- a survey," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 3-16, May.
    3. Wang, Guojing & Wu, Rong, 2001. "Distributions for the risk process with a stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 329-341, October.
    4. 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.
    5. Cardoso, Rui M. R. & R. Waters, Howard, 2003. "Recursive calculation of finite time ruin probabilities under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 659-676, December.
    6. Sundt, Bjorn & Teugels, Jozef L., 1995. "Ruin estimates under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 7-22, April.
    7. Harrison, J. Michael, 1977. "Ruin problems with compounding assets," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 67-79, February.
    8. Wang, Rongming & Yang, Hailiang & Wang, Hanxing, 2004. "On the distribution of surplus immediately after ruin under interest force and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 35(3), pages 703-714, December.
    9. Brekelmans, R.C.M. & De Waegenaere, A.M.B., 2000. "Approximating the Finite-Time Ruin Probability under Interest Force," Discussion Paper 2000-111, Tilburg University, Center for Economic Research.
    10. 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.
    11. 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.
    12. Paulsen, Jostein & Kasozi, Juma & Steigen, Andreas, 2005. "A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 399-420, June.
    13. 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.
    14. Pergamenshchikov, Serguei & Zeitouny, Omar, 2006. "Ruin probability in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 267-278, February.
    15. Paulsen, Jostein, 1998. "Sharp conditions for certain ruin in a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 135-148, June.
    16. Hailiang Yang & Lihong Zhang, 2006. "Ruin problems for a discrete time risk model with random interest rate," Computational Statistics, Springer, vol. 63(2), pages 287-299, May.
    17. Wang, Guojing & Wu, Rong, 2008. "The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 59-64, February.
    18. Cai, Jun, 2004. "Ruin probabilities and penalty functions with stochastic rates of interest," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 53-78, July.
    19. Delbaen, F. & Haezendonck, J., 1987. "Classical risk theory in an economic environment," Insurance: Mathematics and Economics, Elsevier, vol. 6(2), pages 85-116, April.
    20. Yuen, Kam C. & Wang, Guojing & Wu, Rong, 2006. "On the renewal risk process with stochastic interest," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1496-1510, October.
    21. 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.
    22. de Kok, Ton G., 2003. "Ruin probabilities with compounding assets for discrete time finite horizon problems, independent period claim sizes and general premium structure," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 645-658, December.
    23. Kalashnikov, Vladimir & Konstantinides, Dimitrios, 2000. "Ruin under interest force and subexponential claims: a simple treatment," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 145-149, August.
    24. Chen, Yiqing & Ng, Kai W., 2007. "The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 415-423, May.
    25. Grandits, Peter, 2004. "A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 297-305, April.
    26. Cai, Jun & Dickson, David C. M., 2003. "Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 61-71, February.
    27. Schmidli, Hanspeter, 2005. "On optimal investment and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 25-35, February.
    28. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
    29. Konstantinides, Dimitrios & Tang, Qihe & Tsitsiashvili, Gurami, 2002. "Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 447-460, December.
    30. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, vol. 6(2), pages 227-235.
    31. Albrecher, Hansjorg & Teugels, Jozef L. & Tichy, Robert F., 2001. "On a gamma series expansion for the time-dependent probability of collective ruin," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 345-355, December.
    32. Paulsen, Jostein, 1993. "Risk theory in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 327-361, June.
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    Citations

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    Cited by:
    1. Fu, Ke-Ang & Ng, Cheuk Yin Andrew, 2014. "Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 80-87.
    2. Yin, Chuancun & Wen, Yuzhen, 2013. "An extension of Paulsen–Gjessing’s risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 469-476.
    3. Albrecher, Hansjoerg & Constantinescu, Corina & Thomann, Enrique, 2012. "Asymptotic results for renewal risk models with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3767-3789.
    4. Tang, Qihe & Wang, Guojing & Yuen, Kam C., 2010. "Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 362-370, April.
    5. Henrik Hult & Filip Lindskog, 2011. "Ruin probabilities under general investments and heavy-tailed claims," Finance and Stochastics, Springer, vol. 15(2), pages 243-265, June.
    6. Chuancun Yin & Yuzhen Wen, 2013. "An extension of Paulsen-Gjessing's risk model with stochastic return on investments," Papers 1302.6757, arXiv.org.

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