Research Article Received 13 April 2011, Revised 10 August 2011, Accepted 28 September 2011 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asmb.934 Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump-diffusion risk model Zhibin Liang a * † , Kam Chuen Yuen b and Ka Chun Cheung b In this paper, we consider the jump-diffusion risk model with proportional reinsurance and stock price process following the con- stant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd. Keywords: stochastic control; CEV model; exponential utility; proportional reinsurance; investment 1. Introduction In the traditional risk theory where the focus is on liability, insurers play a passive role, and the claim process is the main source of randomness. In recent years, reinsurance and investment have drawn a great deal of attention in the financial and actuarial literature, and they have been incorporated into the study of risk theory so that the resulting models become more realistic (see, e.g., [1–9]). In these works, they assume that the aggregate claims process is either a compound Poisson pro- cess or a Brownian motion with drift, where variables, such as reinsurance, new business, and investment, are controlled and adjusted dynamically. Under various distributional and regularity assumptions, they are able to obtain closed-form solutions for the optimal strategy and the value function in the sense of maximizing (or minimizing) a certain objec- tive function under different constraints. For example, Browne [1], Schmidli [4], Promislow and Young [10], Liang [8], Luo et al. [11], and Bai and Guo [12] consider one (or both) of the following two controls to minimize the probability of ruin: (i) investing in a risky asset and (ii) purchasing proportional reinsurance. They derive explicit expressions for the optimal values for the Brownian motion risk model. For the compound Poisson risk model, Schmidli [5] discusses the opti- mal investment and proportional reinsurance, which minimize the ruin probability, and is able to obtain some analytical results. For the compound Poisson risk model, an explicit expression for the ruin probability is very difficult to obtain. Hence, estimation of ruin probabilities has been a central topic in risk theory. Liu and Yang [13] and Yang and Zhang [7] discuss the optimal investment strategy and derive the minimum ruin probability by numerical methods. Hipp and Schmidli [14] and Gaier et al. [15] study the asymptotic behavior of the ruin probability and obtain its upper bound. Some other papers focus on finding an optimal strategy to maximize the adjustment coefficient (see, e.g., [16–22] and the references therein). Apart from minimizing ruin probability, maximizing expected utility is another important and commonly adopted objec- tive function in the financial and actuarial literature. Under the criterion of maximizing the expected utility of terminal wealth, Browne [1] (for the Brownian motion case), Yang and Zhang [7] (for the compound Poisson case), Wang [23], and Fern K andez et al. [24] (for a pure jump case, not necessarily compound Poisson) consider the optimal investment a School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210046, China b Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China *Correspondence to: Dr. Zhibin Liang, School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210046, China. † E-mail: liangzhibin111@hotmail.com Copyright © 2011 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2011