Journal of Applied Probability

Optimal dynamic risk control for insurers with state-dependent income

Ming Zhou and Jun Cai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

Article information

J. Appl. Probab., Volume 51, Number 2 (2014), 417-435.

First available in Project Euclid: 12 June 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control
Secondary: 91B30: Risk theory, insurance 91B70: Stochastic models 60H30: Applications of stochastic analysis (to PDE, etc.)

Absolute ruin excess-of-loss reinsurance HJB equation liquid reserve state-dependent income


Zhou, Ming; Cai, Jun. Optimal dynamic risk control for insurers with state-dependent income. J. Appl. Probab. 51 (2014), no. 2, 417--435. doi:10.1239/jap/1402578634.

Export citation


  • \item[] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
  • \item[] Asmussen, S., Højgaard, B. and Taksar, M. (2000). Optimal risk control and dividend distribution policies. Example of excess-of-loss reinsurance for an insurance corporation. Finance Stoch. 4, 299–324.
  • \item[] Borovkov, A. A. (2009). Insurance with borrowing: first- and second-order approximations. Adv. Appl. Prob. 41, 1141–1160.
  • \item[] Cai, J. (2007). On the time value of absolute ruin with debit interest. Adv. Appl. Prob. 39, 343–359.
  • \item[] Cai, J., Feng, R. and Willmot, G. E. (2009). On the expectation of total discounted operating costs up to default and its applications. Adv. Appl. Prob. 41, 495–522.
  • \item[] Embrechts, P. and Schmidli, H. (1994). Ruin estimation for a general insurance risk model. Adv. Appl. Prob. 26, 404–422.
  • \item[] Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.
  • \item[] Gerber, H. U. and Yang, H. (2007). Absolute ruin probabilities in a jump diffusion risk model with investment. N. Amer. Actuarial J. 11, 159–169.
  • \item[] Højgaard, B. and Taksar, M. (1998). Optimal proportional reinsurance policies for diffusion models. Scand. Actuarial J. 2, 166–180.
  • \item[] Luo, S. and Taksar, M. (2011). On absolute ruin minimization under a diffusion approximation model. Insurance Math. Econom. 48, 123–133.
  • \item[] Meng, H. and Zhang, X. (2010). Optimal risk control for the excess of loss reinsurance polices. ASTIN Bull. 40, 179–197.
  • \item[] Pachpatte, B. G. (1973). A note on Gronwall–Bellman inequality. J. Math. Anal. Appl. 44, 758–762.
  • \item[] Schmidli, H. (1994). Diffusion approximations for a risk process with the possibility of borrowing and interest. Commun. Statist. Stoch. Models 10, 365–388.
  • \item[] Schmidli, H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuarial J. 1, 55–68.
  • \item[] Taksar, M. I. (2000). Optimal risk and dividend distribution control models for an insurance company. Math. Methods Operat. Res. 51, 1–42.
  • \item[] Taksar, M. I. and Markussen, C. (2003). Optimal dynamic reinsurance policies for large insurance portfolios. Finance Stoch. 7, 97–121.
  • \item[] Zhang, X., Zhou, M. and Guo, J. Y. (2007). Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting. Appl. Stoch. Models Business Industry 23, 63–71.
  • \item[] Zhou, M. and Yuen, K. C. (2012). Optimal reinsurance and dividend for a diffusion model with capital injection: variance premium principle. Economic Modelling 29, 198–207.
  • \item[] Zhu, J. and Yang, H. (2008). Estimates for the absolute ruin probability in the compound Poisson risk model with credit and debit interest. J. Appl. Prob. 45, 818–830. \endharvreferences