Research On Portfolio Optimization Problem With Delay

In this paper,we study several kinds of portfolio optimization models with delay.In our model,the price process of the risky asset is generated by stochastic differential delay equation and this process takes the influence of stochastic factors into consideration,which are different from the classical Merton’s model.Aiming at maximizing the expected discounted utility for wealth and/or consumption,we are looking for the optimal investment and consumption strategies both in complete delay of infinite interval and in bounded delay and stochastic factor of finite interval.For two types of problems,the Constant Absolute Risk Aversion(CARA)utility function and the Constant Relative Risk Aversion(CRRA)utility function are used,respectively.Under some certain conditions,we establish the associated Hamilton-Jacobi-Bellman(HJB)equation with the principle of dynamic programming and work out the explicit solution to the optimal investment and consumption strategies and value function.In addition,Hyperbolic Absolute Risk Aversion(HARA)utility is used for the second problem to obtain more general results including common utility functions such as CRRA utility,CARA utility and Logarithmic utility.On the other hand,because of the complexity of HARA utility structure,we use Legendre transform to derive the linear dual HJB equation to obtain the explicit expression of optimal investment and consumption strategies under HARA utility.On this basis,a special case is proposed to derive explicit solutions of CRRA utility,CARA utility and Logarithmic utility respectively,which are compared with the previous results.Finally,some numerical examples and chart are presented to illustrate the conclusions obtained and explore the impact of model parameters on the optimal strategies.