Constant elasticity of variance model for proportional reinsurance and investment strategies. (English) Zbl 1231.91193

Gu, Mengdi; Yang, Yipeng; Li, Shoude; Zhang, Jingyi

Insur. Math. Econ. 46, No. 3, 580-587 (2010).

Summary: The insurer is allowed to buy reinsurance and invest in a risk-free asset and a risky asset. The claim process is assumed to follow a Brownian motion with drift, while the price process of the risky asset is described by the constant elasticity of variance (CEV) model. The Hamilton-Jacobi-Bellman (HJB) equation associated with the optimal reinsurance and investment strategies is established, and solutions are found for insurers with CRRA or CARRA utility.

MSC:

91B30	Risk theory, insurance (MSC2010)
49L20	Dynamic programming in optimal control and differential games

Keywords:

constant elasticity of variance; reinsurance; Hamilton-Jacobi-Bellman equation; optimal strategies

References:

[1]	Cai, J.; Tan, K. S.; Weng, C.; Zhang, Y., Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43, 185-196 (2008) · Zbl 1140.91417
[2]	Cao, Y.; Wan, N., Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance: Mathematics and Economics, 45, 157-162 (2009) · Zbl 1231.91150
[3]	Cerqueti, R.; Foschi, R.; Spizzichino, F., A spatial mixed Poisson framework for combination of excess-of-loss and proportional reinsurance contracts, Insurance: Mathematics and Economics, 45, 59-64 (2009) · Zbl 1231.91152
[4]	Cox, J. C., The constant elasticity of variance option pricing model, The Journal of Portfolio Management, 22, 16-17 (1996)
[5]	Davydov, D.; Linetsky, V., The valuation and hedging of barrier and lookback option under the CEV process, Management Science, 47, 949-965 (2001) · Zbl 1232.91659
[6]	Detemple, J.; Tian, W. D., The valuation of American options for a class of diffusion processes, Management Science, 48, 917-937 (2002) · Zbl 1232.91660
[7]	Devolder, P.; Bosch, P. M.; Dominguez, F. I., Stochastic optimal control of annuity contracts, Insurance: Mathematics and Economics, 33, 227-238 (2003) · Zbl 1103.91346
[8]	Gao, J., Optimal portfolios for DC pension plans under a CEV model, Insurance: Mathematics and Economics, 44, 479-490 (2009) · Zbl 1162.91411
[9]	Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6, 327-343 (1993) · Zbl 1384.35131
[10]	Hipp, C.; Vogt, M., Optimal dynamic XL reinsurance, ASTIN Bulletin, 33, 193-207 (2003) · Zbl 1059.93135
[11]	Jones, C., The dynamics of the stochastic volatility: evidence from underlying and options markets, Journal of Econometrics, 116, 181-224 (2003) · Zbl 1016.62122
[12]	Lo, C. F.; Yuen, P. H.; Hui, C. H., Constant elasticity of variance option pricing model with time-dependent parameters, International Journal of Theoretical and Applied Finance, 3, 661-674 (2000) · Zbl 1006.91050
[13]	Promislow, D. S.; Young, V. R., Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Juounal, 9, 109-128 (2005) · Zbl 1141.91543
[14]	Taksar, M.; Markussen, C., Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7, 97-121 (2003) · Zbl 1066.91052
[15]	Widdicks, M.; Duck, P.; Andricopoulos, A.; Newton, P., The Black-Scholes equation revisited: symptotic expansions and singular perturbations, Mathematical Finance, 15, 373-391 (2005) · Zbl 1124.91342
[16]	Xiao, J.; Zhai, H.; Qin, C., The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance: Mathematics & Economics, 40, 302-310 (2007) · Zbl 1141.91473