Lie Symmetry Analysis Of An Optimal Portfolio Model With Defined Contribution Pension Plans

In recent years,the acceleration of the process of population aging in China has made the the pension problem more and more prominent.The issue of the optimal investment portfolio has also received widespread attention,and such models often boil down to a terminal value problem for a partial differential equation.This paper considers the optimal investment strategy of defined-contribution pension under a constant elasticity of variance model based on the Lie symmetry analysis method.Firstly,we establish the(2+1)-dimensional HJB equation of pension optimal investment by using stochastic optimal control theory.Secondly,according to the principle of invariance,we obtain the symmetry infinitesimal generator of the equation,and discuss similar transformations that maintain the terminal condition under the power utility function and the exponential utility function to transform the original(2+1)-dimensional equation into a(1+1)-dimensional nonlinear equation,the infiniteparameter symmetry possessed by this(1+1)-dimensional nonlinear equation is used to linearize it,and then based on the variable separation method,we obtain an explicit optimal strategy.At last,the properties and sensitivity analysis of the optimal strategy are discussed by numerical simulations.Compared with the constructive and speculative methods in the literature,this paper uses the systematic method of Lie symmetry analysis,which is convenient for generalization to other problems.