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On the renewal risk process with stochastic interest

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  • Yuen, Kam C.
  • Wang, Guojing
  • Wu, Rong

Abstract

In this paper, we consider the renewal risk process with stochastic interest. For this risk process, we derive exact expressions and integral equations for the Gerber-Shiu expected discounted penalty function and the ultimate ruin probability. When the interest is received at a constant rate and the inter-occurrence times of claims follow an Erlang distribution, we obtain an integro-differential equation for the expected discounted penalty function. We also give lower and upper bounds for the ultimate ruin probability. Finally, we present exact expressions for the discounted density associated with the expected discounted penalty function in two special cases of stochastic interest processes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:10:p:1496-1510
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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. 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. Sundt, Bjorn & Teugels, Jozef L., 1995. "Ruin estimates under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 7-22, April.
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    5. 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.
    6. Cai, Jun & Dickson, David C. M., 2002. "On the expected discounted penalty function at ruin of a surplus process with interest," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 389-404, June.
    7. Sundt, Bjorn & Teugels, Jozef L., 1997. "The adjustment function in ruin estimates under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 19(2), pages 85-94, April.
    8. Kam-Chuen Yuen & Guojing Wang, 2005. "Some Ruin Problems for a Risk Process with Stochastic Interest," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(3), pages 129-142.
    9. 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.
    10. 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.
    11. 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.
    12. Wu, Rong & Wang, Guojing & Zhang, Chunsheng, 2005. "On a joint distribution for the risk process with constant interest force," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 365-374, June.
    13. 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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    Cited by:

    1. 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.
    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. Franck Adékambi & Kokou Essiomle, 2021. "Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model," Risks, MDPI, vol. 9(7), pages 1-22, June.
    4. Lu, Zhaoyang & Xu, Wei & Zhang, Yan & Sun, Yingling, 2009. "On the ruin probability for the Cox correlated risk model perturbed by diffusion," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 381-389, February.
    5. Li, Jinzhu, 2016. "Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 195-204.
    6. Adekambi Franck, 2013. "The Asymptotic Ruin Problem in Health Care Insurance with Interest," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 7(2), pages 143-162, July.
    7. 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.
    8. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    9. Adekambi Franck & Mamane Salha, 2012. "Health Care Insurance Pricing Using Alternating Renewal Processes," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 7(1), pages 1-14, December.
    10. 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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