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OPTIMAL MEAN–VARIANCE REINSURANCE WITH COMMON SHOCK DEPENDENCE

Part of: Mathematical economics Stochastic systems and control

Published online by Cambridge University Press:  30 August 2016

ZHIQIN MING ,
ZHIBIN LIANG  and
CAIBIN ZHANG
ZHIQIN MING
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, PR China email xiaoming011204@qq.com, liangzhibin111@hotmail.com, liangzhibin111@hotmail.com
ZHIBIN LIANG*
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, PR China email xiaoming011204@qq.com, liangzhibin111@hotmail.com, liangzhibin111@hotmail.com
CAIBIN ZHANG
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, PR China email xiaoming011204@qq.com, liangzhibin111@hotmail.com, liangzhibin111@hotmail.com
*
✉ liangzhibin111@hotmail.com
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Abstract

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We consider the optimal proportional reinsurance problem for an insurer with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Using the technique of stochastic linear–quadratic control theory and the Hamilton–Jacobi–Bellman equation, we derive the explicit expressions for the optimal reinsurance strategies and value function, and present the verification theorem within the framework of the viscosity solution. Furthermore, we extend the results in the linear–quadratic setting to the mean–variance problem, and obtain an efficient strategy and frontier. Some numerical examples are given to show the impact of model parameters on the efficient frontier.

Keywords

common shock componentcompound Poisson processstochastic linear–quadratic problemHamilton–Jacobi–Bellman equationproportional reinsurance

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 91B30: Risk theory, insurance
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 2 , October 2016 , pp. 162 - 181
Copyright
© 2016 Australian Mathematical Society 

References

Bai, L., Cai, J. and Zhou, M., “Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting”, Insurance Math. Econom. 53 (2013) 664–670; doi:10.1016/j.insmatheco.2013.09.008.CrossRefGoogle Scholar
Bäuerle, N., “Benchmark and mean-variance problems for insurers”, Math. Methods Oper. Res. 62 (2005) 159–165; doi:10.1007/s00186-005-0446-1.CrossRefGoogle Scholar
Bertsekas, D., Nedić, A. and Ozdaglar, A. E., Convex analysis and optimization (Athena Scientific, Belmont, MA, 2003).Google Scholar
Bi, J. and Guo, J., “Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer”, J. Optim. Theory Appl. 157 (2013) 252–275; doi:10.1007/s10957-012-0138-y.CrossRefGoogle Scholar
Centeno, M., “Dependent risks and excess of loss reinsurance”, Insurance Math. Econom. 37 (2005) 229–238; doi:10.1016/j.insmatheco.2004.12.001.CrossRefGoogle Scholar
Fleming, W. and Soner, H., Controlled Markov processes and viscosity solutions, 2nd edn (Springer, New York, 2006).Google Scholar
Irgens, C. and Paulsen, J., “Optimal control of risk exposure, reinsurance and investments for insurance portfolios”, Insurance Math. Econom. 35 (2004) 21–51; doi:10.1016/j.insmatheco.2004.04.004.CrossRefGoogle Scholar
Kass, R., Goovaerts, M., Dhaene, J. and Denuit, M., Modern actuarial risk theory, 2nd edn (Springer, Berlin, 2008).CrossRefGoogle Scholar
Li, D. and Ng, W. L., “Optimal dynamic portfolio selection: Multi-period mean-variance formulation”, Math. Finance 10 (2000) 387–406; doi:10.1111/1467-9965.00100.CrossRefGoogle Scholar
Liang, Z. and Bayraktar, E., “Optimal proportional reinsurance and investment with unobservable claim size and intensity”, Insurance Math. Econom. 55 (2014) 156–166; doi:10.1016/j.insmatheco.2014.01.011.CrossRefGoogle Scholar
Liang, Z. and Yuen, K. C., “Optimal dynamic reinsurance with dependent risks: variance premium principle”, Scand. Actuar. J. 1 (2016) 18–36; doi:10.1080/03461238.2014.892899.CrossRefGoogle Scholar
Liang, Z., Yuen, K. C. and Guo, J., “Optimal proportional reinsurance and investment in a stock market with Ornstein–Uhlenbeck process”, Insurance Math. Econom. 49 (2011) 207–215; doi:10.1016/j.insmatheco.2011.04.005.CrossRefGoogle Scholar
Markowitz, H., “Portfolio selection”, J. Finance 7 (1952) 77–91; doi:10.2307/2975974.Google Scholar
Øksendal, B., Stochastic differential equations: an introduction with applications, 5th edn (Springer, Berlin, 1998).CrossRefGoogle Scholar
Promislow, D. and Young, V., “Minimizing the probability of ruin when claims follow Brownian motion with drift”, N. Am. Actuar. J. 9 (2005) 109–128; doi:10.1080/10920277.2005.10596214.Google Scholar
Schmidli, H., “Optimal proportional reinsurance policies in a dynamic setting”, Scand. Actuar. J. 1 (2001) 55–68; doi:10.1080/034612301750077338.CrossRefGoogle Scholar
Yuen, K. C., Guo, J. and Wu, X., “On a correlated aggregate claim model with Poisson and Erlang risk process”, Insurance Math. Econom. 31 (2002) 205–214; doi:10.1016/S0167-6687(02)00150-6.CrossRefGoogle Scholar
Yuen, K. C., Guo, J. and Wu, X., “On the first time of ruin in the bivariate compound Poisson model”, Insurance Math. Econom. 38 (2006) 298–308; doi:10.1016/j.insmatheco.2005.08.011.CrossRefGoogle Scholar
Zhou, X. and Li, D., “Continuous-time mean-variance portfolio selection: a stochastic LQ framework”, Appl. Math. Optim. 42 (2000) 19–33; doi:10.1007/s002450010003.CrossRefGoogle Scholar