2024, Vol. 9, Issue 5, Part C

In this study, we analyze an investor's portfolio consisting of two assets: 1) a risk-free asset (bond) with a price process following the Ornstein-Uhlenbeck stochastic interest rate of return, and 2) a risky asset (stock) with a price mechanism drove by the Constant Elasticity of Variance (CEV) model, incorporating consumption. The main objective is to derive a closed-form solution for optimizing the investment in the risky asset and consumption, with a focus on the power utility function. We employ the maximum principle of dynamic optimization to formulate a second-order nonlinear partial differential equation (PDE). Through variable elimination techniques, we obtain analytical solutions for the optimal investment and consumption strategies. Notably, our results show that the values obtained for the logarithmic utility function are dependent on the amount of money available for investment, whereas those for the exponential utility function are not. These findings shed light on the differences in optimal strategies under different utility functions, providing valuable insights for investors seeking to maximize their utility in the presence of market uncertainties.