Lévy过程驱动的随机LQ控制在均值-方差投资组合中的应用

Application of Stochastic LQ Control Driven by Lévy Processes in Mean-Variance Portfolio Selection Problem

文章研究了风险资产价格由Lévy过程和与之独立的多维Brown运动共同驱动的连续时间均值-方差型投资组合选择问题,Lévy过程是由与之相关的Teugles鞅描述。为了求解该问题,首先讨论了由Lévy过程和多维Brown运动共同驱动的非齐次随机系统的线性二次控制问题。借助配方法得到了一个新的随机Riccati方程,若此方程有解,就可以得到系统的最优反馈控制。然后将该理论结果用于求解均值-方差型投资组合问题,在自融资的条件下,得到了最优证券组合的显式表达。最后通过数值算例对比分析有Lévy过程和无Lévy过程情形下投资者的最优投资策略和有效前沿,发现Lévy过程的存在增加了投资者的投资风险,投资者应正确视之。

This paper is concerned w ith a class of mean-variance portfolio selection problem in w hich the price processes of the risky assets are driven by Lévy processes and an independent multidimensional Brow nian motion.Here Lévy processes are presented by Teugels martingale w hich are a family of pairw ise strongly orthonormal martingales associated w ith Lévy processes.In order to obtain the solution of the problem,a stochastic LQ control problem driven by Lévy processes and Brow nian motion is discussed,and by using the Riccati equation approach,the optimal feedback control is obtained.As its application,we transform the mean-variance portfolio selection problem into a stochastic LQ control problem by the w ay of embedded.Moreover,the explicit expression of the optimal portfolio is obtained under the condition of self-financing,and an numerical example is given to compare the optimal portfolio selection strategy and efficient frontier at last.