Robust Portfolio Selection Problems With Delay

In the classical Merton problem,we generally believe that the price of the asset is only related to the current moment but not to the past,and assume that the appreciation rate and volatility of the risk asset are constants.However,in the real market,the price of risk asset is affected by historical information,and it is difficult to accurately estimate the appreciation rate and volatility of risk asset.Therefore,the paper studies two robust portfolio selection problems with delay.In the first problem,we discuss the investment problem under power utility.The goal of the investor is to choose the investment strategy that maximizes the minimum expected power utility of terminal wealth.In the second problem,we consider the investment problem under the Heston model and exponential utility.We assume that the risky asset price not only follows the stochastic delay differential equation,but also depends on a stochastic factor,which is characterized by the Cox Ingersoll Ross(CIR)process.The goal of the investor is to choose an investment strategy to maximize the minimum expected exponential utility of terminal wealth.For these two problems,we derive the corresponding HamiltonJacobi-Bellman(HJB)equation by dynamic programming method,and obtain the explicit solution of the value function and the optimal investment strategy.Besides,the verification theorem is provided to illustrate that the solution of HJB equation is the value function.Finally,for the second problem,we remove the random factor and get the robust problem with delay under exponential utility,which is compared with the first problem.Through numerical analysis,the influence of the parameters in the second problem on the optimal investment strategy is provided.