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OPTIMAL PROPORTIONAL REINSURANCE AND INVESTMENT PROBLEM WITH CONSTRAINTS ON RISK CONTROL IN A GENERAL JUMP-DIFFUSION FINANCIAL MARKET

Part of: Mathematical economics

Published online by Cambridge University Press:  17 February 2016

HUIMING ZHU ,
YA HUANG ,
JIEMING ZHOU ,
XIANGQUN YANG  and
CHAO DENG
HUIMING ZHU*
Affiliation:
College of Business Administration, Hunan University, Changsha 410082, PR China email zhuhuiming@hnu.edu.cn, huangya0219@163.com, dengchaohunan@163.com
YA HUANG
Affiliation:
College of Business Administration, Hunan University, Changsha 410082, PR China email zhuhuiming@hnu.edu.cn, huangya0219@163.com, dengchaohunan@163.com
JIEMING ZHOU
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China, Hunan Normal University, Changsha 410081, PR China email zhjm04101@126.com, xqyang@hunnu.edu.cn
XIANGQUN YANG
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China, Hunan Normal University, Changsha 410081, PR China email zhjm04101@126.com, xqyang@hunnu.edu.cn
CHAO DENG
Affiliation:
College of Business Administration, Hunan University, Changsha 410082, PR China email zhuhuiming@hnu.edu.cn, huangya0219@163.com, dengchaohunan@163.com
*
✉zhuhuiming@hnu.edu.cn
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Abstract

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We study the optimal proportional reinsurance and investment problem in a general jump-diffusion financial market. Assuming that the insurer’s surplus process follows a jump-diffusion process, the insurer can purchase proportional reinsurance from the reinsurer and invest in a risk-free asset and a risky asset, whose price is modelled by a general jump-diffusion process. The insurance company wishes to maximize the expected exponential utility of the terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategy are obtained. A Monte Carlo simulation is conducted to illustrate that the closed-form expressions we derived are indeed the optimal strategies, and some numerical examples are presented to analyse the impact of model parameters on the optimal strategies.

Keywords

jump-diffusion risk modeloptimal investment strategyproportional reinsuranceexponential utilityHamilton–Jacobi–Bellman equation

MSC classification

Secondary: 91B30: Risk theory, insurance
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 3 , January 2016 , pp. 352 - 368
Copyright
© 2016 Australian Mathematical Society 

References

Azcue, P. and Muler, N., “Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem”, Math. Methods Oper. Res. 77 (2013) 177–206; doi:10.1007/s00186-012-0424-3.Google Scholar
Bai, L. and Guo, J., “Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint”, Insurance Math. Econom. 42 (2008) 968–975; doi:10.1016/j.insmatheco.2007.11.002.CrossRefGoogle Scholar
Bai, L. and Guo, J., “Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection”, Sci. China Math. 53 (2010) 1787–1804; doi:10.1007/s11425-010-4033-4.Google Scholar
Bardhan, I. and Chao, X., “Martingale analysis for assets with discontinuous returns”, Math. Oper. Res. 29 (1995) 243–256; doi:10.1287/moor.20.1.243.CrossRefGoogle 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.Google Scholar
Browne, S., “Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin”, Math. Oper. Res. 20 (1995) 937–958; doi:10.1287/moor.20.4.937.Google Scholar
Cao, Y. and Wan, N., “Optimal proportional reinsurance and investment based on Hamilton–Jacobi–Bellman equation”, Insurance Math. Econom. 45 (2009) 157–162; doi:10.1016/j.insmatheco.2009.05.006.Google Scholar
Chen, S., Li, Z. and Li, K., “Optimal investment–reinsurance policy for an insurance company with VaR constraint”, Insurance Math. Econom. 47 (2010) 144–153; doi:10.1016/j.insmatheco.2010.06.002.CrossRefGoogle Scholar
Fleming, W. H. and Soner, H. M., Controlled Markov processes and viscosity solutions (Springer, New York, 1993).Google Scholar
Gerber, H. U., “An introduction to mathematical risk theory”, Volume 8 of S. S. Huebner Foundation Monograph Series (ed. S. S. Huebner), (Foundation for Insurance Education, Philadelphia, PA, 1979).Google Scholar
Gu, M., Yang, Y., Li, S. and Zhang, J., “Constant elasticity of variance model for proportional reinsurance and investment strategies”, Insurance Math. Econom. 46 (2010) 580–587; doi:10.1016/j.insmatheco.2010.03.001.Google Scholar
Guo, W. and Xu, C., “Optimal portfolio selection when stock prices follow a jump-diffusion process”, Math. Methods Oper. Res. 60 (2004) 485–496; doi:10.1007/s00186-006-0139-4.CrossRefGoogle Scholar
Hipp, C. and Plum, M., “Optimal investment for insurers”, Insurance Math. Econom. 27 (2000) 215–228; doi:10.1016/S0167-6687(00)00049-4.Google Scholar
Jeanblanc-Picqué, M. and Pontier, M., “Optimal portfolio for a small investor in a market model with discontinuous prices”, Appl. Math. Optim. 22 (1990) 287–310; doi:10.1007/BF01447332.Google Scholar
Liang, Z., Bai, L. and Guo, J., “Optimal investment and proportional reinsurance with constrained control variables”, Optimal Control Appl. Methods 32 (2011) 587–608; doi:10.1002/oca.965.Google Scholar
Lin, X. and Yang, P., “Optimal investment and reinsurance in a jump diffusion risk model”, ANZIAM J. 52 (2011) 250–262; doi:10.1017/S144618111100068X.Google Scholar
Liu, C. and Yang, H., “Optimal investment for an insurer to minimize its probability of ruin”, N. Am. Actuar. J. 8 (2004) 11–31; doi:10.1080/10920277.2004.10596134.Google Scholar
Luo, S., Taksar, M. and Tsoi, A., “On reinsurance and investment for large insurance portfolios”, Insurance Math. Econom. 42 (2008) 434–444; doi:10.1016/j.insmatheco.2007.04.002.Google Scholar
Øksendal, B. K. and Sulem, A., Applied stochastic control of jump diffusions (Springer, Berlin–Heidelberg, 2005).Google Scholar
Promislow, S. D. and Young, V. R., “Minimizing the probability of ruin when claims follow Brownian motion with drift”, N. Am. Actuar. J. 9 (2005) 110–128; doi:10.1080/10920277.2005.10596214.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J., Stochastic processes for insurance and finance (Wiley, Chichester, 1999).Google Scholar
Schmidli, H., “Optimal proportional reinsurance policies in a dynamic setting”, Scand. Actuar. J. 2001 (2001) 55–68; doi:10.1080/034612301750077338.Google Scholar
Schmidli, H., “On minimizing the ruin probability by investment and reinsurance”, Ann. Appl. Probab. 12 (2002) 890–907; doi:10.1214/aoap/1031863173.Google Scholar
Wang, N., “Optimal investment for an insurer with exponential utility preference”, Insurance Math. Econom. 40 (2007) 77–84; doi:10.1016/j.insmatheco.2006.02.008.Google Scholar
Xu, L., Wang, R. and Yao, D., “On maximizing the expected terminal utility by investment and reinsurance”, J. Ind. Manag. Optim. 4 (2008) 801–815; doi:10.3934/jimo.2008.4.801.Google Scholar
Yang, H. and Zhang, L., “Optimal investment for insurer with jump-diffusion risk process”, Insurance Math. Econom. 37 (2005) 615–634; doi:10.1016/j.insmatheco.2005.06.009.CrossRefGoogle Scholar
Zeng, Y. and Li, Z., “Asset-liability management under benchmark and mean-variance criteria in a jump diffusion market”, J. Syst. Sci. Complex. 24 (2011) 317–327; doi:10.1007/s11424-011-9105-1.Google Scholar
Zhou, J., Deng, Y., Huang, Y. and Yang, X., “Optimal proportional reinsurance and investment for a constant elasticity of variance model under variance principle”, Acta Math. Sci. 35 (2015) 303–312; doi:10.1016/S0252-9602(15)60002-9.Google Scholar