Portfolio Optimization Under Stochastic Utility

Learning from the work of Holger Kraft[1] and Merton’s classical model of deter-ministic utility with constant discount rate, we try to construct a stochastic utility func-tion depending on the risky asset.We will study the portfolio optimization problem withconsumption in the environment of the constant interest rate as well as the risky assetdescribed by the Schwartz mean-reversion model.By using the method of LIE symmetry analysis proposed in the field of diferentialequation, we achieve the dimension reduction of the HJB equation derived from dynamicprogramming principle. Combined with the measure transformation and Feynman-Kacformula, we finally obtain the fundamental form of the solution. Furthermore, we give amore visible solution with some special parameters and also try the numerical simulationto get general appearance of the solution in usual circumstances.