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Minimal cost of a Brownian risk without ruin

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  • Luo, Shangzhen
  • Taksar, Michael

Abstract

In this paper, we study an optimal stochastic control problem for an insurance company whose surplus process is modeled by a Brownian motion with drift (the diffusion approximation model). The company can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The objective is to find an optimal reinsurance and cash injection strategy that minimizes the total cost to keep the surplus process non-negative (without ruin). Here the cost function is defined as the total discounted value of the injections. The minimal cost function is found explicitly by solving the according quasi-variational inequalities (QVIs). Its associated optimal reinsurance-injection control policy is also found.

Suggested Citation

  • Luo, Shangzhen & Taksar, Michael, 2012. "Minimal cost of a Brownian risk without ruin," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 685-693.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:3:p:685-693
    DOI: 10.1016/j.insmatheco.2012.09.006
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    References listed on IDEAS

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    1. Abel Cadenillas & Tahir Choulli & Michael Taksar & Lei Zhang, 2006. "Classical And Impulse Stochastic Control For The Optimization Of The Dividend And Risk Policies Of An Insurance Firm," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 181-202, January.
    2. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin, Cambridge University Press, vol. 34(1), pages 49-74, May.
    3. Abel Cadenillas & Fernando Zapatero, 2000. "Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rates and Reserves," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 141-156, April.
    4. Taksar, Michael & Zeng, Xudong, 2011. "Optimal non-proportional reinsurance control and stochastic differential games," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 64-71, January.
    5. Harrison, J. Michael & Taylor, Allison J., 1978. "Optimal control of a Brownian storage system," Stochastic Processes and their Applications, Elsevier, vol. 6(2), pages 179-194, January.
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