This paper studies a mean-variance investment-reinsurance problem under a new stochastic volatility model, namely the 4/2 stochastic volatility model. Solving this problem requires a deep understanding of a class of parabolic partial differential equations (PPDEs). By the parametrix method and the integral transform method, we derive explicit solutions to the PPDEs in several special cases. Through the Lie symmetry analysis, we obtain a four-parameter family of the 4/2 stochastic volatility models such that the corresponding PPDEs have closed-form solutions. The efficient strategy and the efficient frontier of the mean-variance problem are represented by using the closed-form solutions to PPDEs. Numerical examples for the obtained efficient frontier are provided by Monto Carlo method.

本文研究了一种新的随机波动率模型即4/2随机波动率模型下的均值方差投资再保险问题。解决这个问题需要深入了解一类抛物型偏微分方程 (PPDE)。通过参数方法和积分变换方法，我们在几个特殊情况下导出了 PPDEs 的显式解。通过李对称分析，我们得到了 4/2 随机波动率模型的四参数族，使得相应的 PPDE 具有封闭形式的解。均值-方差问题的有效策略和有效边界通过使用 PPDE 的封闭形式解来表示。得到的有效边界的数值例子由蒙托卡罗方法提供。