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Optimal Dividend of Compound Poisson Process under a Stochastic Interest Rate

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  • Linlin Tian
  • Xiaoyi Zhang

Abstract

In this paper we assume the insurance wealth process is driven by the compound Poisson process. The discounting factor is modelled as a geometric Brownian motion at first and then as an exponential function of an integrated Ornstein-Uhlenbeck process. The objective is to maximize the cumulated value of expected discounted dividends up to the time of ruin. We give an explicit expression of the value function and the optimal strategy in the case of interest rate following a geometric Brownian motion. For the case of the Vasicek model, we explore some properties of the value function. Since we can not find an explicit expression for the value function in the second case, we prove that the value function is the viscosity solution of the corresponding HJB equation.

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  • Linlin Tian & Xiaoyi Zhang, 2018. "Optimal Dividend of Compound Poisson Process under a Stochastic Interest Rate," Papers 1807.08081, arXiv.org.
    • Handle: RePEc:arx:papers:1807.08081
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    References listed on IDEAS

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    1. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    2. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    3. Eisenberg, Julia, 2015. "Optimal dividends under a stochastic interest rate," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 259-266.
    4. Azcue, Pablo & Muler, Nora, 2012. "Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 26-42.
    5. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    6. Hans Gerber & Elias Shiu, 2006. "On Optimal Dividend Strategies In The Compound Poisson Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 76-93.
    7. Asmussen, Soren & Taksar, Michael, 1997. "Controlled diffusion models for optimal dividend pay-out," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 1-15, June.
    8. Albrecher, Hansjörg & Thonhauser, Stefan, 2008. "Optimal dividend strategies for a risk process under force of interest," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 134-149, August.
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