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Minimal Cost of a Brownian Risk without Ruin

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  • Shangzhen Luo
  • Michael Taksar

Abstract

In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity 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 the optimal reinsurance and cash injection strategy that minimizes the total cost to keep the company's surplus process non-negative, i.e. without ruin, where the cost function is defined as the total discounted value of the injections. The optimal solution is found explicitly by solving the according quasi-variational inequalities (QVIs).

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  • Shangzhen Luo & Michael Taksar, 2011. "Minimal Cost of a Brownian Risk without Ruin," Papers 1112.4005, arXiv.org.
    • Handle: RePEc:arx:papers:1112.4005
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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. 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.
    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. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin, Cambridge University Press, vol. 34(1), pages 49-74, May.
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