In this paper, we develop an extended constant elasticity of variance (CEV) model with stochastic volatility to study an optimal investment strategy problem. This extended CEV model remedies the shortcoming of classical CEV model. In CEV model, the volatility term is an power function of stock price, which only covers firm-specific risks. Consequently, we consider the coefficient of volatility with the mean reverting process to make the volatility involve the market risks to improve the classical CEV model. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the extended CEV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we successfully employ a dual method, Legendre transformation, and an asymptotic expansion technique to approach an asymptotic solution. The numerical examples indicate the optimal strategy is an increasing function of the expectation of stock returns and correlations between the two market risks. Besides, it is a decreasing function of interest rate and risk aversion coefficient. In addition, by statistical analysis, we find that the power parameter, the expectation of stock returns and interest rate are all significant factors affecting the investment strategy.

在本文中，我们开发了一个具有随机波动率的扩展常数方差弹性 (CEV) 模型来研究最优投资策略问题。这种扩展的 CEV 模型弥补了经典 CEV 模型的缺点。在 CEV 模型中，波动率项是股价的幂函数，仅涵盖公司特定风险。因此，我们考虑均值回归过程的波动系数，使波动包含市场风险，以改进经典的CEV模型。对于绝对风险规避(CARA)效用函数为常数的最优投资目标，由于偏微分方程的复杂非线性，无法得到扩展CEV模型下的解析解。在本文中，我们成功地采用了对偶方法勒让德变换，和渐近扩展技术来逼近渐近解。数值例子表明最优策略是股票收益预期和两种市场风险之间相关性的递增函数。此外，它是利率和风险厌恶系数的减函数。此外，通过统计分析，我们发现功率参数、股票收益预期和利率都是影响投资策略的重要因素。